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Lower left coordinates % are always set to 0. % /roundpagebbox { 7 dict begin /papersizes [8.5 inch 11 inch 14 inch 17 inch] def /mappapersize { /val exch def /slop .5 inch def /diff slop def /j 0 def 0 1 papersizes length 1 sub { /i exch def papersizes i get val sub abs dup diff le {/diff exch def /j i def} {pop} ifelse } for diff slop lt {papersizes j get} {val} ifelse } def pagebbox 0 0 put pagebbox 1 0 put pagebbox dup 2 get mappapersize 2 exch put pagebbox dup 3 get mappapersize 3 exch put end } bind def %%EndProlog %%BeginSetup mark /rotation 1 def /gotpagebbox true def /linewidth 0.5 def /xoffset 0 def /yoffset 0 def /#copies 1 store /magnification 1 def %%FormsPerPage: 1 /formsperpage 1 def %%Patch from lp %%EndPatch from lp /landscape false def /resolution 720 def setup 2 setdecoding %%EndSetup %%Page: 1 1 %%PageBoundingBox: (atend) /saveobj save def mark 1 pagesetup 12 B f (There Is No Royal Road to Programs)6 1905 1 1927 1230 t (A Trilogy on)2 654 1 2553 1530 t (Raster Ellipses)1 765 1 2497 1680 t (and)2783 1830 w (Programming Methodology)1 1424 1 2168 1980 t 10 I f (M. Douglas McIlroy)2 818 1 2471 2220 t 10 R f (AT&T Bell Laboratories)2 993 1 2383 2400 t (Murray Hill, New Jersey 07974)4 1267 1 2246 2520 t 10 I f (ABSTRACT)2643 2900 w 10 R f (Wherein, with some insight, some formality and some scorn, ellipse-drawing algo-)10 3350 1 1330 3160 t ( been wont unpredictably to stray by a pixel here and there, are brought)13 2875(rithms, which had)2 725 2 1080 3280 t ( attributed to premature ``opti-)4 1284( flawed designs of previous algorithms are)6 1787( The)1 220(to heel.)1 309 4 1080 3400 t ( for a special)3 548(mization'': uncritical reuse of an algorithmic scheme that had been tuned)10 3052 2 1080 3520 t (case \(circles\) beyond the point of no return.)7 1740 1 1080 3640 t 10 I f (There is no royal road to geometry.)6 1424 1 2168 3916 t 10 R f ( to King Ptolomey's expressed desire for a less arduous introduction)10 2780(The legendary answer of Euclid)4 1290 2 970 4108 t (to the)1 225 1 720 4228 t 10 I f (Elements)970 4228 w 10 R f ( is no substitute for precise analysis.)6 1446( There)1 282(carries equal force in programming.)4 1432 3 1361 4228 t ( lies)1 167( Therein)1 363( outline of a solution is clear.)6 1181(The problem of drawing ellipses is simple, and the general)9 2359 4 970 4384 t ( method of solution without fully specifying an objective;)8 2369( get written by specifying a)5 1135( Programs)1 442(a danger.)1 374 4 720 4504 t ( good enough, the pro-)4 927( a few test cases look)5 865( When)1 292(mathematics figures only in deriving arithmetic details.)6 2236 4 720 4624 t ( output)1 290( Its)1 159( is fragile because it lacks well defined mathematical properties.)9 2642( it)1 91( But)1 205(gram is declared done.)3 933 6 720 4744 t ( results follow from)3 797( Better)1 296( subroutine.)1 474(can be looked at but not built upon\320a bad state of affairs for a basic)14 2753 4 720 4864 t (mathematical study of the program as a whole, not just as a collection of isolated statements.)15 3704 1 720 4984 t ( to mimic the most highly optimized algorithms for drawing circles,)10 2823(Published algorithms, which attempt)3 1497 2 720 5140 t ( this small example)3 820( Thus)1 265( lack.)1 231(have failed because the optimization depends on symmetry that ellipses)9 3004 4 720 5260 t ( often ignored truth of software engineering: to extend the functionality of a program, it is)15 3799(illustrates an)1 521 2 720 5380 t ( ``bet-)1 247( Since)1 273( and rebuild, not just remodel.)5 1207(sometimes necessary to back off to a more general starting point)10 2593 4 720 5500 t ( it may be wise to preserve earlier and)8 1542(ter,'' i.e. more highly tuned, programs are likely to be less adaptable,)11 2778 2 720 5620 t (less perfected versions for their evolutionary potential.)6 2182 1 720 5740 t (Jon Bentley, Brian Kernighan, and Chris Van Wyk gave helpful criticism about presentation.)12 3720 1 720 5896 t 10 B f (Contents)720 6136 w 10 I f (Getting Raster Ellipses Right.)3 1222 1 720 6292 t 10 R f (A development of the general algorithm, illustrated with many pictures of)10 3039 1 2001 6292 t (pitfalls, plus an implementation in C.)5 1487 1 720 6412 t 10 I f ( Algorithm.)1 468(Math before Code: A Soundly Derived Ellipse-drawing)6 2275 2 720 6568 t 10 R f ( same)1 242( The)1 217(A more formal treatment.)3 1056 3 3525 6568 t (algorithm is derived by a direct argument undistracted by motivating examples.)10 3177 1 720 6688 t 10 I f (Ellipses Not Yet Made Easy.)4 1200 1 720 6844 t 10 R f ( in)1 120(One of the papers that inspired this work is reproduced and criticized)11 2934 2 1986 6844 t ( written, on an understandable and)5 1375( Accessibly)1 488( the methods by which it was obtained.)7 1560(regard to its result and)4 897 4 720 6964 t (graphic topic, it affords a revealing case study of pitfalls in practical computer science.)13 3467 1 720 7084 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 68 514 691 %%EndPage: 1 1 %%Page: 2 2 %%PageBoundingBox: (atend) /saveobj save def mark 2 pagesetup 10 R f (- 2 -)2 166 1 2797 480 t 12 B f (Getting Raster Ellipses Right)3 1500 1 2130 840 t 10 I f (M. Douglas McIlroy)2 818 1 2471 1080 t 10 R f (AT&T Bell Laboratories)2 993 1 2383 1260 t (Murray Hill, New Jersey 07974)4 1267 1 2246 1380 t (ABSTRACT)2618 1620 w ( raster approximations to ellipses in standard position)7 2149(A concise incremental algorithm for)4 1451 2 1080 1980 t ( or the)2 255(produces approximations that are good to the last pixel even near octant boundaries)12 3345 2 1080 2100 t ( with reflec-)2 489( resulting approximations commute)3 1419( The)1 205(thin ends of highly eccentric ellipses.)5 1487 4 1080 2220 t ( details of)2 400(tion about the diagonal and are mathematically specifiable without reference to)10 3200 2 1080 2340 t (the algorithm.)1 561 1 1080 2460 t 10 B f (1. Introduction)1 670 1 720 2700 t 10 R f ( ellipse is cen-)3 595( The)1 212( approximating an ellipse by lighting pixels on a bitmap.)9 2324(We are concerned with)3 939 4 970 2856 t ( principal axes are parallel to)5 1160( The)1 206( take to be \(0,0\).)4 658(tered on a point of a square grid, which for simplicity we)11 2296 4 720 2976 t ( are)1 148( lengths of the semiaxes)4 964( The)1 206(the grid lines.)2 549 4 720 3096 t 10 I f (a)2614 3096 w 10 R f (and)2691 3096 w 10 I f (b)2862 3096 w 10 R f ( both quantities are positive, the ellipse satis-)7 1813(. When)1 315 2 2912 3096 t (fies the familiar equation,)3 1026 1 720 3216 t 10 I f (a)2573 3526 w 7 R f (2)2634 3486 w 10 I f (x)2576 3386 w 7 R f (2)2631 3346 w 10 S1 f (_ __)1 134 1 2558 3416 t 10 S f (+)2751 3446 w 10 I f (b)2880 3526 w 7 R f (2)2941 3486 w 10 I f (y)2883 3386 w 7 R f (2)2938 3346 w 10 S1 f (_ __)1 134 1 2865 3416 t 10 S f (=)3058 3446 w 10 R f (1 \(1\))1 1878 1 3162 3446 t (When the length of a semiaxis is zero, the ellipse degenerates into a line segment.)14 3261 1 720 3686 t ( incremental approximation algorithms that involve only)6 2350(More particularly, we are concerned with)5 1720 2 970 3842 t ( Accordingly)1 551(integer arithmetic.)1 734 2 720 3962 t 10 I f (a)2032 3962 w 10 R f (and)2108 3962 w 10 I f (b)2278 3962 w 10 R f (are taken to be integers and the grid is taken to be the plane integer)14 2686 1 2354 3962 t (lattice.)720 4082 w (Ideally an approximation to a simple curve drawn by lighting points of the integer lattice should be)16 3957 1 720 4238 t 10 I f (Metrically accurate.)1 819 1 970 4394 t 10 R f (Every point of the approximation should be as close to the curve as possible in)14 3197 1 1843 4394 t (some sense.)1 477 1 970 4514 t 10 I f (Connected.)970 4670 w 10 R f (The approximation should be connected by chess-king moves.)7 2491 1 1472 4670 t 10 I f (Topologically accurate.)1 961 1 970 4826 t 10 R f ( be the same)3 517(The topology of king-move paths in the approximation should)8 2536 2 1987 4826 t (as the topology of the original curve.)6 1473 1 970 4946 t 10 I f (Thin.)970 5102 w 10 R f ( is a corol-)3 436( Thinness)1 414(Each lighted point should have exactly two lighted king-move neighbors.)9 2958 3 1232 5102 t (lary of topological accuracy.)3 1141 1 970 5222 t 10 I f (Symmetric.)970 5378 w 10 R f ( translations,)1 519(Approximation should commute with the symmetry operations of the grid:)9 3048 2 1473 5378 t (rotations through multiples of)3 1192 1 970 5498 t 10 S f (p)2187 5498 w 10 I f (/)2250 5498 w 10 R f (2, and reflections about horizontal, vertical and diagonal axes.)8 2481 1 2286 5498 t 10 I f (Describable.)970 5654 w 10 R f (The approximation should be specifiable mathematically without reference to the)9 3481 1 1559 5654 t (approximating algorithm.)1 1022 1 970 5774 t (These desiderata cannot always be met in full.)7 1846 1 720 5930 t ( original)1 345(Thinness and topological accuracy may not be achievable when the scale of features in the)14 3725 2 970 6086 t ( spacing of the bitmap; then pixels approximating different)8 2401(curve is comparable to or smaller than the grid)8 1919 2 720 6206 t ( particular, figures with)3 949( In)1 136(stretches of the curve may come into adjacency or coincidence.)9 2553 3 720 6326 t 10 I f (tails)4387 6326 w 10 R f (may result;)1 451 1 4589 6326 t ( can save the appearances, however, by understanding coinci-)8 2491( We)1 192( Appendix 2, Lemma 2.)4 965(see Figure 1 and)3 672 4 720 6446 t (dent or irrelevantly adjacent stretches of the approximation to be traced in separate sheets.)13 3597 1 720 6566 t ( metric accuracy at certain pixels called)6 1632(Thinness conflicts with)2 949 2 970 6722 t 10 I f (square corners.)1 636 1 3585 6722 t 10 R f (At a square corner)3 760 1 4280 6722 t ( corners sometimes occur in the approxima-)6 1776( Square)1 331( vertices of a grid square are lighted.)7 1485(the points at three)3 728 4 720 6842 t ( there can be at most one square corner per quadrant,)10 2167( However,)1 446(tions adopted in this paper; see Figure 2a.)7 1707 3 720 6962 t ( shall argue that such square corners are)7 1620( We)1 193(near the point where the magnitude of the ellipse's slope is 1.)11 2507 3 720 7082 t (inevitable: to exorcise them, one would have to sacrifice other critical properties.)11 3235 1 720 7202 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 56 514 764 %%EndPage: 2 2 %%Page: 3 3 %%PageBoundingBox: (atend) /saveobj save def mark 3 pagesetup 10 R f (- 3 -)2 166 1 2797 480 t cleartomark saveobj restore %%BeginGlobal % % Version 3.3.1 drawing procedures for dpost. 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(\347)3557 1220 w (\347)3557 1320 w (\347)3557 1420 w (\347)3472 1012 w (\347)3472 1020 w (\347)3472 1120 w (\347)3472 1220 w (\347)3472 1320 w (\347)3472 1420 w (\347)3388 1012 w (\347)3388 1020 w (\347)3388 1120 w (\347)3388 1220 w (\347)3388 1320 w (\347)3388 1420 w (\347)3303 1012 w (\347)3303 1020 w (\347)3303 1120 w (\347)3303 1220 w (\347)3303 1320 w (\347)3303 1420 w (\347)3218 1012 w (\347)3218 1020 w (\347)3218 1120 w (\347)3218 1220 w (\347)3218 1320 w (\347)3218 1420 w (\347)3134 1012 w (\347)3134 1020 w (\347)3134 1120 w (\347)3134 1220 w (\347)3134 1320 w (\347)3134 1420 w (\347)3049 1012 w (\347)3049 1020 w (\347)3049 1120 w (\347)3049 1220 w (\347)3049 1320 w (\347)3049 1420 w (\347)2964 1012 w (\347)2964 1020 w (\347)2964 1120 w (\347)2964 1220 w (\347)2964 1320 w (\347)2964 1420 w (\347)2880 1012 w (\347)2880 1020 w (\347)2880 1120 w (\347)2880 1220 w (\347)2880 1320 w (\347)2880 1420 w (\347)2795 1012 w (\347)2795 1020 w (\347)2795 1120 w (\347)2795 1220 w (\347)2795 1320 w (\347)2795 1420 w (\347)2710 1012 w (\347)2710 1020 w (\347)2710 1120 w (\347)2710 1220 w (\347)2710 1320 w (\347)2710 1420 w (\347)2625 1012 w (\347)2625 1020 w (\347)2625 1120 w (\347)2625 1220 w (\347)2625 1320 w (\347)2625 1420 w (\347)2541 1012 w (\347)2541 1020 w (\347)2541 1120 w (\347)2541 1220 w (\347)2541 1320 w (\347)2541 1420 w (\347)2456 1012 w (\347)2456 1020 w (\347)2456 1120 w (\347)2456 1220 w (\347)2456 1320 w (\347)2456 1420 w (\347)2371 1012 w (\347)2371 1020 w (\347)2371 1120 w (\347)2371 1220 w (\347)2371 1320 w (\347)2371 1420 w (\347)2287 1012 w (\347)2287 1020 w (\347)2287 1120 w (\347)2287 1220 w (\347)2287 1320 w (\347)2287 1420 w (\347)2202 1012 w (\347)2202 1020 w (\347)2202 1120 w (\347)2202 1220 w (\347)2202 1320 w (\347)2202 1420 w (\347)2117 1012 w (\347)2117 1020 w (\347)2117 1120 w (\347)2117 1220 w (\347)2117 1320 w (\347)2117 1420 w (\347)2032 1012 w (\347)2032 1020 w (\347)2032 1120 w (\347)2032 1220 w (\347)2032 1320 w (\347)2032 1420 w (\347)1948 1012 w (\347)1948 1020 w (\347)1948 1120 w (\347)1948 1220 w (\347)1948 1320 w (\347)1948 1420 w (\347)1863 1012 w (\347)1863 1020 w (\347)1863 1120 w (\347)1863 1220 w (\347)1863 1320 w (\347)1863 1420 w (\347)1778 1012 w (\347)1778 1020 w (\347)1778 1120 w (\347)1778 1220 w (\347)1778 1320 w (\347)1778 1420 w (\347)1694 1012 w (\347)1694 1020 w (\347)1694 1120 w (\347)1694 1220 w (\347)1694 1320 w (\347)1694 1420 w (\347)1609 1012 w (\347)1609 1020 w (\347)1609 1120 w (\347)1609 1220 w (\347)1609 1320 w (\347)1609 1420 w (\347)1524 1012 w (\347)1524 1020 w (\347)1524 1120 w (\347)1524 1220 w (\347)1524 1320 w (\347)1524 1420 w (\347)1440 1012 w (\347)1440 1020 w (\347)1440 1120 w (\347)1440 1220 w (\347)1440 1320 w (\347)1440 1420 w 1638 1148 1609 1166 Dl 4121 1184 4150 1166 Dl 1638 1184 1609 1166 Dl 4121 1148 4150 1166 Dl 1668 1140 1639 1147 Dl 4091 1191 4120 1184 Dl 1668 1191 1639 1184 Dl 4091 1140 4120 1147 Dl 1697 1135 1668 1140 Dl 4062 1196 4091 1191 Dl 1697 1196 1668 1191 Dl 4062 1135 4091 1140 Dl 1727 1130 1698 1134 Dl 4032 1201 4061 1197 Dl 1727 1201 1698 1197 Dl 4032 1130 4061 1134 Dl 1757 1127 1728 1130 Dl 4003 1204 4032 1201 Dl 1757 1204 1728 1201 Dl 4003 1127 4032 1130 Dl 1786 1123 1757 1126 Dl 3973 1208 4002 1205 Dl 1786 1208 1757 1205 Dl 3973 1123 4002 1126 Dl 1816 1119 1787 1122 Dl 3943 1212 3972 1209 Dl 1816 1212 1787 1209 Dl 3943 1119 3972 1122 Dl 1845 1117 1816 1119 Dl 3914 1214 3943 1212 Dl 1845 1214 1816 1212 Dl 3914 1117 3943 1119 Dl 1875 1114 1846 1116 Dl 3884 1217 3913 1215 Dl 1875 1217 1846 1215 Dl 3884 1114 3913 1116 Dl 1905 1112 1876 1114 Dl 3854 1220 3883 1218 Dl 1905 1220 1876 1218 Dl 3854 1112 3883 1114 Dl 1934 1109 1905 1111 Dl 3825 1222 3854 1220 Dl 1934 1222 1905 1220 Dl 3825 1109 3854 1111 Dl 1964 1107 1935 1109 Dl 3795 1224 3824 1222 Dl 1964 1224 1935 1222 Dl 3795 1107 3824 1109 Dl 1994 1106 1965 1107 Dl 3765 1225 3794 1224 Dl 1994 1225 1965 1224 Dl 3765 1106 3794 1107 Dl 2023 1104 1994 1105 Dl 3736 1227 3765 1226 Dl 2023 1227 1994 1226 Dl 3736 1104 3765 1105 Dl 2053 1102 2024 1103 Dl 3706 1229 3735 1228 Dl 2053 1229 2024 1228 Dl 3706 1102 3735 1103 Dl 2083 1100 2054 1101 Dl 3676 1231 3705 1230 Dl 2083 1231 2054 1230 Dl 3676 1100 3705 1101 Dl 2112 1099 2083 1100 Dl 3647 1233 3676 1232 Dl 2112 1233 2083 1232 Dl 3647 1099 3676 1100 Dl 2142 1097 2113 1098 Dl 3617 1234 3646 1233 Dl 2142 1234 2113 1233 Dl 3617 1097 3646 1098 Dl 2172 1096 2143 1097 Dl 3587 1236 3616 1235 Dl 2172 1236 2143 1235 Dl 3587 1096 3616 1097 Dl 2201 1094 2172 1095 Dl 3558 1237 3587 1236 Dl 2201 1237 2172 1236 Dl 3558 1094 3587 1095 Dl 2231 1093 2202 1094 Dl 3528 1238 3557 1237 Dl 2231 1238 2202 1237 Dl 3528 1093 3557 1094 Dl 2261 1092 2232 1093 Dl 3499 1239 3528 1238 Dl 2261 1239 2232 1238 Dl 3499 1092 3528 1093 Dl 2290 1091 2261 1092 Dl 3469 1241 3498 1240 Dl 2290 1241 2261 1240 Dl 3469 1091 3498 1092 Dl 2320 1090 2291 1091 Dl 3439 1242 3468 1241 Dl 2320 1242 2291 1241 Dl 3439 1090 3468 1091 Dl 2349 1090 2320 1090 Dl 3410 1242 3439 1242 Dl 2349 1242 2320 1242 Dl 3410 1090 3439 1090 Dl 2379 1089 2350 1089 Dl 3380 1243 3409 1243 Dl 2379 1243 2350 1243 Dl 3380 1089 3409 1089 Dl 2409 1088 2380 1088 Dl 3350 1243 3379 1243 Dl 2409 1243 2380 1243 Dl 3350 1088 3379 1088 Dl 2438 1087 2409 1087 Dl 3321 1244 3350 1244 Dl 2438 1244 2409 1244 Dl 3321 1087 3350 1087 Dl 2468 1086 2439 1086 Dl 3291 1245 3320 1245 Dl 2468 1245 2439 1245 Dl 3291 1086 3320 1086 Dl 2498 1085 2469 1085 Dl 3261 1246 3290 1246 Dl 2498 1246 2469 1246 Dl 3261 1085 3290 1085 Dl 2527 1085 2498 1085 Dl 3232 1246 3261 1246 Dl 2527 1246 2498 1246 Dl 3232 1085 3261 1085 Dl 2557 1084 2528 1084 Dl 3202 1247 3231 1247 Dl 2557 1247 2528 1247 Dl 3202 1084 3231 1084 Dl 2587 1084 2558 1084 Dl 3172 1248 3201 1248 Dl 2587 1248 2558 1248 Dl 3172 1084 3201 1084 Dl 2616 1083 2587 1083 Dl 3143 1248 3172 1248 Dl 2616 1248 2587 1248 Dl 3143 1083 3172 1083 Dl 2646 1083 2617 1083 Dl 3113 1248 3142 1248 Dl 2646 1248 2617 1248 Dl 3113 1083 3142 1083 Dl 2676 1082 2647 1082 Dl 3083 1249 3112 1249 Dl 2676 1249 2647 1249 Dl 3083 1082 3112 1082 Dl 2705 1082 2676 1082 Dl 3054 1249 3083 1249 Dl 2705 1249 2676 1249 Dl 3054 1082 3083 1082 Dl 2735 1082 2706 1082 Dl 3024 1250 3053 1250 Dl 2735 1250 2706 1250 Dl 3024 1082 3053 1082 Dl 2765 1081 2736 1081 Dl 2995 1250 3024 1250 Dl 2765 1250 2736 1250 Dl 2995 1081 3024 1081 Dl 2794 1081 2765 1081 Dl 2965 1250 2994 1250 Dl 2794 1250 2765 1250 Dl 2965 1081 2994 1081 Dl 10 R f ( elongated ellipse with tails,)4 1119( An)1 172(Figure 1.)1 361 3 1732 1696 t 10 I f (a)3409 1696 w 10 S f (=)3499 1696 w 10 R f (15 ,)1 133 1 3594 1696 t 10 I f (b)3768 1696 w 10 S f (=)3858 1696 w 10 R f (1.)3953 1696 w ( a connected approximation via chess-king moves, guided by a function)10 2915(Incremental algorithms trace)2 1155 2 970 1972 t ( grid lines that intersect the curve a grid point is cho-)11 2133( each of some set of)5 810( On)1 175(that measures goodness of fit.)4 1202 4 720 2092 t (sen to minimize one of these criteria:)6 1481 1 720 2212 t 10 I f (Displacement)970 2368 w 10 R f (of the lighted point from the intersection, measured along the grid line.)11 2833 1 1544 2368 t 10 I f (Distance)970 2488 w 10 R f (of the lighted point from the curve, measured normal to the curve.)11 2637 1 1350 2488 t 10 I f (Residual)970 2608 w 10 R f (of the curve's defining equation evaluated at the lighted point.)9 2485 1 1345 2608 t ( circles with integer radius,)4 1100(The three criteria agree for)4 1079 2 720 2764 t 8 R f (1, 2)1 120 1 2907 2732 t 10 R f (but do not necessarily agree for ellipses; see Fig-)8 1984 1 3056 2764 t (ures 2b and 2c.)3 604 1 720 2884 t ( be classed as a)4 661( approximating point will)3 1056( An)1 184(We shall adopt the minimum-displacement criterion.)5 2169 4 970 3040 t 10 I f (minimum-horizontal-displacement)720 3160 w 10 R f (point or a)2 403 1 2132 3160 t 10 I f (minimum-vertical-displacement)2569 3160 w 10 R f (point according to the direc-)4 1167 1 3873 3160 t ( two classes are not mutually exclusive.)6 1584( The)1 205(tion in which the minimized displacement is measured.)7 2204 3 720 3280 t 2636 4120 69 69 De 3413 4120 69 69 De 2506 4120 69 69 De 2518 4120 46 46 De 2529 4120 23 23 De 3543 4120 69 69 De 3555 4120 46 46 De 3566 4120 23 23 De 3413 4249 69 69 De 3425 4249 46 46 De 3437 4249 23 23 De 2636 3990 69 69 De 2647 3990 46 46 De 2659 3990 23 23 De 2636 4249 69 69 De 2647 4249 46 46 De 2659 4249 23 23 De 3413 3990 69 69 De 3425 3990 46 46 De 3437 3990 23 23 De 3284 4249 69 69 De 3295 4249 46 46 De 3307 4249 23 23 De 2765 3990 69 69 De 2777 3990 46 46 De 2789 3990 23 23 De 2765 4249 69 69 De 2777 4249 46 46 De 2789 4249 23 23 De 3284 3990 69 69 De 3295 3990 46 46 De 3307 3990 23 23 De 3154 4249 69 69 De 3166 4249 46 46 De 3177 4249 23 23 De 2895 3990 69 69 De 2907 3990 46 46 De 2918 3990 23 23 De 2895 4249 69 69 De 2907 4249 46 46 De 2918 4249 23 23 De 3154 3990 69 69 De 3166 3990 46 46 De 3177 3990 23 23 De 3025 4249 69 69 De 3036 4249 46 46 De 3048 4249 23 23 De 3025 3990 69 69 De 3036 3990 46 46 De 3048 3990 23 23 De 10 S1 f (_ _________________________)1 1296 1 2412 3472 t (_ _________________________)1 1296 1 2412 3601 t (_ _________________________)1 1296 1 2412 3731 t (_ _________________________)1 1296 1 2412 3860 t (_ _________________________)1 1296 1 2412 3990 t (_ _________________________)1 1296 1 2412 4120 t (_ _________________________)1 1296 1 2412 4249 t (_ _________________________)1 1296 1 2412 4379 t (_ _________________________)1 1296 1 2412 4508 t (_ _________________________)1 1296 1 2412 4638 t (_ _________________________)1 1296 1 2412 4768 t 10 S f (\347)3708 3572 w (\347)3708 3668 w (\347)3708 3768 w (\347)3708 3868 w (\347)3708 3968 w (\347)3708 4068 w (\347)3708 4168 w (\347)3708 4268 w (\347)3708 4368 w (\347)3708 4468 w (\347)3708 4568 w (\347)3708 4668 w (\347)3708 4768 w (\347)3578 3572 w (\347)3578 3668 w (\347)3578 3768 w (\347)3578 3868 w (\347)3578 3968 w (\347)3578 4068 w (\347)3578 4168 w (\347)3578 4268 w (\347)3578 4368 w (\347)3578 4468 w (\347)3578 4568 w (\347)3578 4668 w (\347)3578 4768 w (\347)3448 3572 w (\347)3448 3668 w (\347)3448 3768 w (\347)3448 3868 w (\347)3448 3968 w (\347)3448 4068 w (\347)3448 4168 w (\347)3448 4268 w (\347)3448 4368 w (\347)3448 4468 w (\347)3448 4568 w (\347)3448 4668 w (\347)3448 4768 w (\347)3319 3572 w (\347)3319 3668 w (\347)3319 3768 w (\347)3319 3868 w (\347)3319 3968 w (\347)3319 4068 w (\347)3319 4168 w (\347)3319 4268 w (\347)3319 4368 w (\347)3319 4468 w (\347)3319 4568 w (\347)3319 4668 w (\347)3319 4768 w (\347)3189 3572 w (\347)3189 3668 w (\347)3189 3768 w (\347)3189 3868 w (\347)3189 3968 w (\347)3189 4068 w (\347)3189 4168 w (\347)3189 4268 w (\347)3189 4368 w (\347)3189 4468 w (\347)3189 4568 w (\347)3189 4668 w (\347)3189 4768 w (\347)3060 3572 w (\347)3060 3668 w (\347)3060 3768 w (\347)3060 3868 w (\347)3060 3968 w (\347)3060 4068 w (\347)3060 4168 w (\347)3060 4268 w (\347)3060 4368 w (\347)3060 4468 w (\347)3060 4568 w (\347)3060 4668 w (\347)3060 4768 w (\347)2930 3572 w (\347)2930 3668 w (\347)2930 3768 w (\347)2930 3868 w (\347)2930 3968 w (\347)2930 4068 w (\347)2930 4168 w (\347)2930 4268 w (\347)2930 4368 w (\347)2930 4468 w (\347)2930 4568 w (\347)2930 4668 w (\347)2930 4768 w (\347)2800 3572 w (\347)2800 3668 w (\347)2800 3768 w (\347)2800 3868 w (\347)2800 3968 w (\347)2800 4068 w (\347)2800 4168 w (\347)2800 4268 w (\347)2800 4368 w (\347)2800 4468 w (\347)2800 4568 w (\347)2800 4668 w (\347)2800 4768 w (\347)2671 3572 w (\347)2671 3668 w (\347)2671 3768 w (\347)2671 3868 w (\347)2671 3968 w (\347)2671 4068 w (\347)2671 4168 w (\347)2671 4268 w (\347)2671 4368 w (\347)2671 4468 w (\347)2671 4568 w (\347)2671 4668 w (\347)2671 4768 w (\347)2541 3572 w (\347)2541 3668 w (\347)2541 3768 w (\347)2541 3868 w (\347)2541 3968 w (\347)2541 4068 w (\347)2541 4168 w (\347)2541 4268 w (\347)2541 4368 w (\347)2541 4468 w (\347)2541 4568 w (\347)2541 4668 w (\347)2541 4768 w (\347)2412 3572 w (\347)2412 3668 w (\347)2412 3768 w (\347)2412 3868 w (\347)2412 3968 w (\347)2412 4068 w (\347)2412 4168 w (\347)2412 4268 w (\347)2412 4368 w (\347)2412 4468 w (\347)2412 4568 w (\347)2412 4668 w (\347)2412 4768 w 2553 4092 2541 4120 Dl 3566 4148 3578 4120 Dl 2553 4148 2541 4120 Dl 3566 4092 3578 4120 Dl 2565 4080 2553 4091 Dl 3554 4159 3566 4148 Dl 2565 4159 2553 4148 Dl 3554 4080 3566 4091 Dl 2578 4072 2566 4080 Dl 3541 4167 3553 4159 Dl 2578 4167 2566 4159 Dl 3541 4072 3553 4080 Dl 2590 4064 2578 4071 Dl 3529 4175 3541 4168 Dl 2590 4175 2578 4168 Dl 3529 4064 3541 4071 Dl 2602 4058 2590 4064 Dl 3517 4181 3529 4175 Dl 2602 4181 2590 4175 Dl 3517 4058 3529 4064 Dl 2615 4053 2603 4058 Dl 3504 4186 3516 4181 Dl 2615 4186 2603 4181 Dl 3504 4053 3516 4058 Dl 2627 4049 2615 4053 Dl 3492 4190 3504 4186 Dl 2627 4190 2615 4186 Dl 3492 4049 3504 4053 Dl 2639 4044 2627 4048 Dl 3480 4195 3492 4191 Dl 2639 4195 2627 4191 Dl 3480 4044 3492 4048 Dl 2652 4039 2640 4043 Dl 3467 4200 3479 4196 Dl 2652 4200 2640 4196 Dl 3467 4039 3479 4043 Dl 2664 4036 2652 4039 Dl 3455 4203 3467 4200 Dl 2664 4203 2652 4200 Dl 3455 4036 3467 4039 Dl 2676 4033 2664 4036 Dl 3443 4206 3455 4203 Dl 2676 4206 2664 4203 Dl 3443 4033 3455 4036 Dl 2689 4029 2677 4032 Dl 3430 4210 3442 4207 Dl 2689 4210 2677 4207 Dl 3430 4029 3442 4032 Dl 2701 4026 2689 4029 Dl 3418 4213 3430 4210 Dl 2701 4213 2689 4210 Dl 3418 4026 3430 4029 Dl 2713 4024 2701 4026 Dl 3406 4215 3418 4213 Dl 2713 4215 2701 4213 Dl 3406 4024 3418 4026 Dl 2725 4021 2713 4023 Dl 3394 4218 3406 4216 Dl 2725 4218 2713 4216 Dl 3394 4021 3406 4023 Dl 2738 4018 2726 4020 Dl 3381 4221 3393 4219 Dl 2738 4221 2726 4219 Dl 3381 4018 3393 4020 Dl 2750 4016 2738 4018 Dl 3369 4223 3381 4221 Dl 2750 4223 2738 4221 Dl 3369 4016 3381 4018 Dl 2762 4013 2750 4015 Dl 3357 4226 3369 4224 Dl 2762 4226 2750 4224 Dl 3357 4013 3369 4015 Dl 2775 4011 2763 4013 Dl 3344 4228 3356 4226 Dl 2775 4228 2763 4226 Dl 3344 4011 3356 4013 Dl 2787 4010 2775 4011 Dl 3332 4229 3344 4228 Dl 2787 4229 2775 4228 Dl 3332 4010 3344 4011 Dl 2799 4008 2787 4009 Dl 3320 4231 3332 4230 Dl 2799 4231 2787 4230 Dl 3320 4008 3332 4009 Dl 2812 4006 2800 4007 Dl 3307 4233 3319 4232 Dl 2812 4233 2800 4232 Dl 3307 4006 3319 4007 Dl 2824 4005 2812 4006 Dl 3295 4234 3307 4233 Dl 2824 4234 2812 4233 Dl 3295 4005 3307 4006 Dl 2836 4003 2824 4004 Dl 3283 4236 3295 4235 Dl 2836 4236 2824 4235 Dl 3283 4003 3295 4004 Dl 2849 4001 2837 4002 Dl 3270 4238 3282 4237 Dl 2849 4238 2837 4237 Dl 3270 4001 3282 4002 Dl 2861 4000 2849 4001 Dl 3258 4239 3270 4238 Dl 2861 4239 2849 4238 Dl 3258 4000 3270 4001 Dl 2873 3999 2861 4000 Dl 3246 4240 3258 4239 Dl 2873 4240 2861 4239 Dl 3246 3999 3258 4000 Dl 2886 3998 2874 3999 Dl 3233 4241 3245 4240 Dl 2886 4241 2874 4240 Dl 3233 3998 3245 3999 Dl 2898 3996 2886 3997 Dl 3221 4243 3233 4242 Dl 2898 4243 2886 4242 Dl 3221 3996 3233 3997 Dl 2910 3996 2898 3996 Dl 3209 4243 3221 4243 Dl 2910 4243 2898 4243 Dl 3209 3996 3221 3996 Dl 2922 3995 2910 3995 Dl 3197 4244 3209 4244 Dl 2922 4244 2910 4244 Dl 3197 3995 3209 3995 Dl 2935 3994 2923 3994 Dl 3184 4245 3196 4245 Dl 2935 4245 2923 4245 Dl 3184 3994 3196 3994 Dl 2947 3994 2935 3994 Dl 3172 4245 3184 4245 Dl 2947 4245 2935 4245 Dl 3172 3994 3184 3994 Dl 2959 3993 2947 3993 Dl 3160 4246 3172 4246 Dl 2959 4246 2947 4246 Dl 3160 3993 3172 3993 Dl 2972 3992 2960 3992 Dl 3147 4247 3159 4247 Dl 2972 4247 2960 4247 Dl 3147 3992 3159 3992 Dl 2984 3992 2972 3992 Dl 3135 4247 3147 4247 Dl 2984 4247 2972 4247 Dl 3135 3992 3147 3992 Dl 2996 3991 2984 3991 Dl 3123 4248 3135 4248 Dl 2996 4248 2984 4248 Dl 3123 3991 3135 3991 Dl 3009 3991 2997 3991 Dl 3110 4248 3122 4248 Dl 3009 4248 2997 4248 Dl 3110 3991 3122 3991 Dl 3021 3991 3009 3991 Dl 3098 4248 3110 4248 Dl 3021 4248 3009 4248 Dl 3098 3991 3110 3991 Dl 3033 3990 3021 3990 Dl 3086 4249 3098 4249 Dl 3033 4249 3021 4249 Dl 3086 3990 3098 3990 Dl 10 R f (\(b\))3024 4944 w 1901 4379 69 69 De 1913 4379 46 46 De 1925 4379 23 23 De 1124 3860 69 69 De 1135 3860 46 46 De 1147 3860 23 23 De 1124 4379 69 69 De 1135 4379 46 46 De 1147 4379 23 23 De 1901 3860 69 69 De 1913 3860 46 46 De 1925 3860 23 23 De 2031 4249 69 69 De 2043 4249 46 46 De 2054 4249 23 23 De 994 3990 69 69 De 1006 3990 46 46 De 1017 3990 23 23 De 994 4249 69 69 De 1006 4249 46 46 De 1017 4249 23 23 De 2031 3990 69 69 De 2043 3990 46 46 De 2054 3990 23 23 De 994 4120 69 69 De 1006 4120 46 46 De 1017 4120 23 23 De 2031 4120 69 69 De 2043 4120 46 46 De 2054 4120 23 23 De 1901 4508 69 69 De 1124 3731 69 69 De 1124 4508 69 69 De 1901 3731 69 69 De 1772 4508 69 69 De 1783 4508 46 46 De 1795 4508 23 23 De 1253 3731 69 69 De 1265 3731 46 46 De 1277 3731 23 23 De 1253 4508 69 69 De 1265 4508 46 46 De 1277 4508 23 23 De 1772 3731 69 69 De 1783 3731 46 46 De 1795 3731 23 23 De 1642 4638 69 69 De 1654 4638 46 46 De 1665 4638 23 23 De 1383 3601 69 69 De 1395 3601 46 46 De 1406 3601 23 23 De 1383 4638 69 69 De 1395 4638 46 46 De 1406 4638 23 23 De 1642 3601 69 69 De 1654 3601 46 46 De 1665 3601 23 23 De 1513 4638 69 69 De 1524 4638 46 46 De 1536 4638 23 23 De 1513 3601 69 69 De 1524 3601 46 46 De 1536 3601 23 23 De 10 S1 f (_ _________________________)1 1296 1 900 3472 t (_ _________________________)1 1296 1 900 3601 t (_ _________________________)1 1296 1 900 3731 t (_ _________________________)1 1296 1 900 3860 t (_ _________________________)1 1296 1 900 3990 t (_ _________________________)1 1296 1 900 4120 t (_ _________________________)1 1296 1 900 4249 t (_ _________________________)1 1296 1 900 4379 t (_ _________________________)1 1296 1 900 4508 t (_ _________________________)1 1296 1 900 4638 t (_ _________________________)1 1296 1 900 4768 t 10 S f (\347)2196 3572 w (\347)2196 3668 w (\347)2196 3768 w (\347)2196 3868 w (\347)2196 3968 w (\347)2196 4068 w (\347)2196 4168 w (\347)2196 4268 w (\347)2196 4368 w (\347)2196 4468 w (\347)2196 4568 w (\347)2196 4668 w (\347)2196 4768 w (\347)2066 3572 w (\347)2066 3668 w (\347)2066 3768 w (\347)2066 3868 w (\347)2066 3968 w (\347)2066 4068 w (\347)2066 4168 w (\347)2066 4268 w (\347)2066 4368 w (\347)2066 4468 w (\347)2066 4568 w (\347)2066 4668 w (\347)2066 4768 w (\347)1936 3572 w (\347)1936 3668 w (\347)1936 3768 w (\347)1936 3868 w (\347)1936 3968 w (\347)1936 4068 w (\347)1936 4168 w (\347)1936 4268 w (\347)1936 4368 w (\347)1936 4468 w (\347)1936 4568 w (\347)1936 4668 w (\347)1936 4768 w (\347)1807 3572 w (\347)1807 3668 w (\347)1807 3768 w (\347)1807 3868 w (\347)1807 3968 w (\347)1807 4068 w (\347)1807 4168 w (\347)1807 4268 w (\347)1807 4368 w (\347)1807 4468 w (\347)1807 4568 w (\347)1807 4668 w (\347)1807 4768 w (\347)1677 3572 w (\347)1677 3668 w (\347)1677 3768 w (\347)1677 3868 w (\347)1677 3968 w (\347)1677 4068 w (\347)1677 4168 w (\347)1677 4268 w (\347)1677 4368 w (\347)1677 4468 w (\347)1677 4568 w (\347)1677 4668 w (\347)1677 4768 w (\347)1548 3572 w (\347)1548 3668 w (\347)1548 3768 w (\347)1548 3868 w (\347)1548 3968 w (\347)1548 4068 w (\347)1548 4168 w (\347)1548 4268 w (\347)1548 4368 w (\347)1548 4468 w (\347)1548 4568 w (\347)1548 4668 w (\347)1548 4768 w (\347)1418 3572 w (\347)1418 3668 w (\347)1418 3768 w (\347)1418 3868 w (\347)1418 3968 w (\347)1418 4068 w (\347)1418 4168 w (\347)1418 4268 w (\347)1418 4368 w (\347)1418 4468 w (\347)1418 4568 w (\347)1418 4668 w (\347)1418 4768 w (\347)1288 3572 w (\347)1288 3668 w (\347)1288 3768 w (\347)1288 3868 w (\347)1288 3968 w (\347)1288 4068 w (\347)1288 4168 w (\347)1288 4268 w (\347)1288 4368 w (\347)1288 4468 w (\347)1288 4568 w (\347)1288 4668 w (\347)1288 4768 w (\347)1159 3572 w (\347)1159 3668 w (\347)1159 3768 w (\347)1159 3868 w (\347)1159 3968 w (\347)1159 4068 w (\347)1159 4168 w (\347)1159 4268 w (\347)1159 4368 w (\347)1159 4468 w (\347)1159 4568 w (\347)1159 4668 w (\347)1159 4768 w (\347)1029 3572 w (\347)1029 3668 w (\347)1029 3768 w (\347)1029 3868 w (\347)1029 3968 w (\347)1029 4068 w (\347)1029 4168 w (\347)1029 4268 w (\347)1029 4368 w (\347)1029 4468 w (\347)1029 4568 w (\347)1029 4668 w (\347)1029 4768 w (\347)900 3572 w (\347)900 3668 w (\347)900 3768 w (\347)900 3868 w (\347)900 3968 w (\347)900 4068 w (\347)900 4168 w (\347)900 4268 w (\347)900 4368 w (\347)900 4468 w (\347)900 4568 w (\347)900 4668 w (\347)900 4768 w 1041 4008 1029 4120 Dl 2054 4232 2066 4120 Dl 1041 4232 1029 4120 Dl 2054 4008 2066 4120 Dl 1053 3962 1041 4007 Dl 2042 4277 2054 4232 Dl 1053 4277 1041 4232 Dl 2042 3962 2054 4007 Dl 1066 3928 1054 3962 Dl 2029 4311 2041 4277 Dl 1066 4311 1054 4277 Dl 2029 3928 2041 3962 Dl 1078 3899 1066 3927 Dl 2017 4340 2029 4312 Dl 1078 4340 1066 4312 Dl 2017 3899 2029 3927 Dl 1090 3875 1078 3899 Dl 2005 4364 2017 4340 Dl 1090 4364 1078 4340 Dl 2005 3875 2017 3899 Dl 1103 3853 1091 3874 Dl 1992 4386 2004 4365 Dl 1103 4386 1091 4365 Dl 1992 3853 2004 3874 Dl 1115 3834 1103 3853 Dl 1980 4405 1992 4386 Dl 1115 4405 1103 4386 Dl 1980 3834 1992 3853 Dl 1127 3816 1115 3833 Dl 1968 4423 1980 4406 Dl 1127 4423 1115 4406 Dl 1968 3816 1980 3833 Dl 1140 3799 1128 3815 Dl 1955 4440 1967 4424 Dl 1140 4440 1128 4424 Dl 1955 3799 1967 3815 Dl 1152 3784 1140 3799 Dl 1943 4455 1955 4440 Dl 1152 4455 1140 4440 Dl 1943 3784 1955 3799 Dl 1164 3771 1152 3784 Dl 1931 4468 1943 4455 Dl 1164 4468 1152 4455 Dl 1931 3771 1943 3784 Dl 1177 3757 1165 3770 Dl 1918 4482 1930 4469 Dl 1177 4482 1165 4469 Dl 1918 3757 1930 3770 Dl 1189 3745 1177 3757 Dl 1906 4494 1918 4482 Dl 1189 4494 1177 4482 Dl 1906 3745 1918 3757 Dl 1201 3734 1189 3745 Dl 1894 4505 1906 4494 Dl 1201 4505 1189 4494 Dl 1894 3734 1906 3745 Dl 1213 3723 1201 3733 Dl 1882 4516 1894 4506 Dl 1213 4516 1201 4506 Dl 1882 3723 1894 3733 Dl 1226 3713 1214 3723 Dl 1869 4526 1881 4516 Dl 1226 4526 1214 4516 Dl 1869 3713 1881 3723 Dl 1238 3704 1226 3713 Dl 1857 4535 1869 4526 Dl 1238 4535 1226 4526 Dl 1857 3704 1869 3713 Dl 1250 3695 1238 3703 Dl 1845 4544 1857 4536 Dl 1250 4544 1238 4536 Dl 1845 3695 1857 3703 Dl 1263 3686 1251 3694 Dl 1832 4553 1844 4545 Dl 1263 4553 1251 4545 Dl 1832 3686 1844 3694 Dl 1275 3679 1263 3686 Dl 1820 4560 1832 4553 Dl 1275 4560 1263 4553 Dl 1820 3679 1832 3686 Dl 1287 3671 1275 3678 Dl 1808 4568 1820 4561 Dl 1287 4568 1275 4561 Dl 1808 3671 1820 3678 Dl 1300 3665 1288 3671 Dl 1795 4574 1807 4568 Dl 1300 4574 1288 4568 Dl 1795 3665 1807 3671 Dl 1312 3658 1300 3664 Dl 1783 4581 1795 4575 Dl 1312 4581 1300 4575 Dl 1783 3658 1795 3664 Dl 1324 3652 1312 3658 Dl 1771 4587 1783 4581 Dl 1324 4587 1312 4581 Dl 1771 3652 1783 3658 Dl 1337 3646 1325 3651 Dl 1758 4593 1770 4588 Dl 1337 4593 1325 4588 Dl 1758 3646 1770 3651 Dl 1349 3641 1337 3646 Dl 1746 4598 1758 4593 Dl 1349 4598 1337 4593 Dl 1746 3641 1758 3646 Dl 1361 3637 1349 3641 Dl 1734 4602 1746 4598 Dl 1361 4602 1349 4598 Dl 1734 3637 1746 3641 Dl 1374 3632 1362 3636 Dl 1721 4607 1733 4603 Dl 1374 4607 1362 4603 Dl 1721 3632 1733 3636 Dl 1386 3627 1374 3631 Dl 1709 4612 1721 4608 Dl 1386 4612 1374 4608 Dl 1709 3627 1721 3631 Dl 1398 3624 1386 3627 Dl 1697 4615 1709 4612 Dl 1398 4615 1386 4612 Dl 1697 3624 1709 3627 Dl 1410 3620 1398 3623 Dl 1685 4619 1697 4616 Dl 1410 4619 1398 4616 Dl 1685 3620 1697 3623 Dl 1423 3616 1411 3619 Dl 1672 4623 1684 4620 Dl 1423 4623 1411 4620 Dl 1672 3616 1684 3619 Dl 1435 3614 1423 3616 Dl 1660 4625 1672 4623 Dl 1435 4625 1423 4623 Dl 1660 3614 1672 3616 Dl 1447 3611 1435 3613 Dl 1648 4628 1660 4626 Dl 1447 4628 1435 4626 Dl 1648 3611 1660 3613 Dl 1460 3609 1448 3611 Dl 1635 4630 1647 4628 Dl 1460 4630 1448 4628 Dl 1635 3609 1647 3611 Dl 1472 3608 1460 3609 Dl 1623 4631 1635 4630 Dl 1472 4631 1460 4630 Dl 1623 3608 1635 3609 Dl 1484 3606 1472 3607 Dl 1611 4633 1623 4632 Dl 1484 4633 1472 4632 Dl 1611 3606 1623 3607 Dl 1497 3604 1485 3605 Dl 1598 4635 1610 4634 Dl 1497 4635 1485 4634 Dl 1598 3604 1610 3605 Dl 1509 3603 1497 3604 Dl 1586 4636 1598 4635 Dl 1509 4636 1497 4635 Dl 1586 3603 1598 3604 Dl 1521 3603 1509 3603 Dl 1574 4636 1586 4636 Dl 1521 4636 1509 4636 Dl 1574 3603 1586 3603 Dl 10 R f (\(a\))1512 4944 w 4738 4217 23 23 De 4742 4217 15 15 De 4746 4217 7 7 De 4537 4120 69 69 De 5185 4768 69 69 De 5196 4768 46 46 De 5208 4768 23 23 De 5055 4638 69 69 De 5067 4638 46 46 De 5078 4638 23 23 De 4925 4508 69 69 De 4937 4508 46 46 De 4949 4508 23 23 De 4925 4379 69 69 De 4937 4379 46 46 De 4949 4379 23 23 De 4796 4249 69 69 De 4807 4249 46 46 De 4819 4249 23 23 De 4666 4120 69 69 De 4678 4120 46 46 De 4689 4120 23 23 De 4537 3990 69 69 De 4548 3990 46 46 De 4560 3990 23 23 De 4407 3990 69 69 De 4419 3990 46 46 De 4430 3990 23 23 De 4277 3860 69 69 De 4289 3860 46 46 De 4301 3860 23 23 De 4148 3731 69 69 De 4159 3731 46 46 De 4171 3731 23 23 De 4018 3731 69 69 De 4030 3731 46 46 De 4041 3731 23 23 De 3889 3601 69 69 De 3900 3601 46 46 De 3912 3601 23 23 De 10 S1 f (_ _________________________)1 1296 1 3924 3472 t (_ _________________________)1 1296 1 3924 3601 t (_ _________________________)1 1296 1 3924 3731 t (_ _________________________)1 1296 1 3924 3860 t (_ _________________________)1 1296 1 3924 3990 t (_ _________________________)1 1296 1 3924 4120 t (_ _________________________)1 1296 1 3924 4249 t (_ _________________________)1 1296 1 3924 4379 t (_ _________________________)1 1296 1 3924 4508 t (_ _________________________)1 1296 1 3924 4638 t (_ _________________________)1 1296 1 3924 4768 t 10 S f (\347)5220 3572 w (\347)5220 3668 w (\347)5220 3768 w (\347)5220 3868 w (\347)5220 3968 w (\347)5220 4068 w (\347)5220 4168 w (\347)5220 4268 w (\347)5220 4368 w (\347)5220 4468 w (\347)5220 4568 w (\347)5220 4668 w (\347)5220 4768 w (\347)5090 3572 w (\347)5090 3668 w (\347)5090 3768 w (\347)5090 3868 w (\347)5090 3968 w (\347)5090 4068 w (\347)5090 4168 w (\347)5090 4268 w (\347)5090 4368 w (\347)5090 4468 w (\347)5090 4568 w (\347)5090 4668 w (\347)5090 4768 w (\347)4960 3572 w (\347)4960 3668 w (\347)4960 3768 w (\347)4960 3868 w (\347)4960 3968 w (\347)4960 4068 w (\347)4960 4168 w (\347)4960 4268 w (\347)4960 4368 w (\347)4960 4468 w (\347)4960 4568 w (\347)4960 4668 w (\347)4960 4768 w (\347)4831 3572 w (\347)4831 3668 w (\347)4831 3768 w (\347)4831 3868 w (\347)4831 3968 w (\347)4831 4068 w (\347)4831 4168 w (\347)4831 4268 w (\347)4831 4368 w (\347)4831 4468 w (\347)4831 4568 w (\347)4831 4668 w (\347)4831 4768 w (\347)4701 3572 w (\347)4701 3668 w (\347)4701 3768 w (\347)4701 3868 w (\347)4701 3968 w (\347)4701 4068 w (\347)4701 4168 w (\347)4701 4268 w (\347)4701 4368 w (\347)4701 4468 w (\347)4701 4568 w (\347)4701 4668 w (\347)4701 4768 w (\347)4572 3572 w (\347)4572 3668 w (\347)4572 3768 w (\347)4572 3868 w (\347)4572 3968 w (\347)4572 4068 w (\347)4572 4168 w (\347)4572 4268 w (\347)4572 4368 w (\347)4572 4468 w (\347)4572 4568 w (\347)4572 4668 w (\347)4572 4768 w (\347)4442 3572 w (\347)4442 3668 w (\347)4442 3768 w (\347)4442 3868 w (\347)4442 3968 w (\347)4442 4068 w (\347)4442 4168 w (\347)4442 4268 w (\347)4442 4368 w (\347)4442 4468 w (\347)4442 4568 w (\347)4442 4668 w (\347)4442 4768 w (\347)4312 3572 w (\347)4312 3668 w (\347)4312 3768 w (\347)4312 3868 w (\347)4312 3968 w (\347)4312 4068 w (\347)4312 4168 w (\347)4312 4268 w (\347)4312 4368 w (\347)4312 4468 w (\347)4312 4568 w (\347)4312 4668 w (\347)4312 4768 w (\347)4183 3572 w (\347)4183 3668 w (\347)4183 3768 w (\347)4183 3868 w (\347)4183 3968 w (\347)4183 4068 w (\347)4183 4168 w (\347)4183 4268 w (\347)4183 4368 w (\347)4183 4468 w (\347)4183 4568 w (\347)4183 4668 w (\347)4183 4768 w (\347)4053 3572 w (\347)4053 3668 w (\347)4053 3768 w (\347)4053 3868 w (\347)4053 3968 w (\347)4053 4068 w (\347)4053 4168 w (\347)4053 4268 w (\347)4053 4368 w (\347)4053 4468 w (\347)4053 4568 w (\347)4053 4668 w (\347)4053 4768 w (\347)3924 3572 w (\347)3924 3668 w (\347)3924 3768 w (\347)3924 3868 w (\347)3924 3968 w (\347)3924 4068 w (\347)3924 4168 w (\347)3924 4268 w (\347)3924 4368 w (\347)3924 4468 w (\347)3924 4568 w (\347)3924 4668 w (\347)3924 4768 w 5114 4689 5155 4764 Dl 5073 4620 5114 4688 Dl 5032 4557 5073 4620 Dl 4991 4499 5032 4557 Dl 4950 4444 4991 4498 Dl 4909 4392 4950 4443 Dl 4867 4344 4908 4392 Dl 4826 4297 4867 4343 Dl 4785 4254 4826 4297 Dl 4744 4212 4785 4253 Dl 4703 4173 4744 4212 Dl 4662 4134 4703 4172 Dl 4621 4098 4662 4134 Dl 4580 4062 4621 4097 Dl 4539 4029 4580 4062 Dl 4498 3996 4539 4028 Dl 4457 3964 4498 3995 Dl 4416 3934 4457 3964 Dl 4375 3905 4416 3934 Dl 4334 3876 4375 3904 Dl 4293 3849 4334 3876 Dl 4252 3823 4293 3849 Dl 4211 3798 4252 3823 Dl 4170 3773 4211 3797 Dl 4129 3749 4170 3772 Dl 4088 3725 4129 3748 Dl 4047 3703 4088 3725 Dl 4006 3682 4047 3703 Dl 3965 3660 4006 3681 Dl 3924 3640 3965 3660 Dl 10 R f (\(c\))4536 4944 w ( in the approximation to the ellipse with)7 1772( Open circles mark square corners)5 1479( \(a\))1 184(Figure 2.)1 385 4 970 5164 t 10 I f (a)970 5284 w 10 S f (=)1069 5284 w 10 I f (b)1173 5284 w 10 S f (=)1272 5284 w 10 R f ( The minimum-residual approximation for)4 1733(4. \(b\))1 251 2 1376 5284 t 10 I f (a)3394 5284 w 10 S f (=)3493 5284 w 10 R f (4 and)1 228 1 3597 5284 t 10 I f (b)3859 5284 w 10 S f (=)3958 5284 w 10 R f (1 differs from the)3 728 1 4062 5284 t ( The)1 223( \(c\))1 203(minimum-displacement approximation at the points marked by open circles.)8 3394 3 970 5404 t ( for)1 185(minimum-distance approximation)1 1403 2 970 5524 t 10 I f (a)2627 5524 w 10 S f (=)2726 5524 w 10 R f (26 and)1 313 1 2830 5524 t 10 I f (b)3212 5524 w 10 S f (=)3311 5524 w 10 R f (18 differs from the minimum-)4 1375 1 3415 5524 t ( \(20,11\) and \(20,12\) the)4 976( At)1 157(displacement approximation at \(20,11\), marked by an open circle.)8 2687 3 970 5644 t ( from the ellipse are about 0.501 and 0.499; the normal distances are)12 2914(vertical displacements)1 906 2 970 5764 t ( dot marks the octant juncture where the slope is)9 1933( A)1 122(about 0.383 and 0.385.)3 916 3 970 5884 t 10 S f (-)3966 5884 w 10 R f (1.)4037 5884 w (A)970 6160 w 10 I f (Freeman approximation)1 983 1 1076 6160 t 10 R f ( where all grid lines are con-)6 1201(is a minimum-displacement approximation)3 1746 2 2093 6160 t (sidered.)720 6280 w 8 R f (3)1041 6248 w 10 R f (The handling of ties between two points on one grid line is left open.)13 2753 1 1106 6280 t ( curve into octants bounded by)5 1274(Most published incremental algorithms for drawing ellipses split the)8 2796 2 970 6436 t ( of the slope is 0, 1, or)7 919(points where the absolute value)4 1279 2 720 6556 t 10 S f (\245)2947 6556 w 10 R f ( compass-point names for a representative)5 1702(. Using)1 318 2 3020 6556 t (outward normal, we speak of the)5 1323 1 720 6676 t 8 R f (NE)2071 6676 w 10 R f (quadrant being divided into a)4 1183 1 2206 6676 t 8 R f (NNE)3417 6676 w 10 R f (and an)1 266 1 3610 6676 t 8 R f (ENE)3904 6676 w 10 R f (octant. The)1 478 1 4088 6676 t 10 I f (juncture)4595 6676 w 10 R f (of)4957 6676 w (the octants is the point with slope)6 1393 1 720 6796 t 10 S f (-)2147 6796 w 10 R f ( 2c shows part of a)5 796(1. Figure)1 395 2 2218 6796 t 8 R f (NE)3443 6796 w 10 R f (quadrant. A dot marks the juncture.)5 1457 1 3583 6796 t (The)720 6916 w 8 R f (NNE)902 6916 w 10 R f (octant lies to the left of the juncture, the)8 1611 1 1094 6916 t 8 R f (ENE)2732 6916 w 10 R f ( directions are also used)4 967( Compass)1 419(octant to the right.)3 739 3 2915 6916 t ( between points; with ``northeast'' referring to any bearing properly between north and)12 3499(to refer to directions)3 821 2 720 7036 t ( is called an)3 507( single point lighted to the northeast of the juncture)9 2136( A)1 132(east, and so forth.)3 737 4 720 7156 t 10 I f (outside point.)1 556 1 4268 7156 t 10 R f (The)4885 7156 w (square corner in the)3 789 1 720 7276 t 8 R f (NE)1534 7276 w 10 R f (quadrant of Figure 2a is an outside point.)7 1643 1 1666 7276 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 48 535 764 %%EndPage: 3 3 %%Page: 4 4 %%PageBoundingBox: (atend) /saveobj save def mark 4 pagesetup 10 R f (- 4 -)2 166 1 2797 480 t ( by Pitteway,)2 535(The published algorithms that I have seen,)6 1719 2 970 840 t 8 R f (4, 5)1 120 1 3232 808 t 10 R f (Wirth,)3382 840 w 8 R f (6)3648 808 w 10 R f (Van Aken,)1 437 1 3718 840 t 8 R f (7, 8)1 120 1 4163 808 t 10 R f (Pratt,)4313 840 w 8 R f (9)4535 808 w 10 R f (DaSilva,)4605 840 w 8 R f (10)4960 808 w 10 R f (and Kappel,)1 492 1 720 960 t 8 R f (11)1220 928 w 10 R f (consider only vertical displacements for the)5 1794 1 1335 960 t 8 R f (NNE)3164 960 w 10 R f ( displacements)1 600(octant and only horizontal)3 1076 2 3364 960 t (for the)1 270 1 720 1080 t 8 R f (ENE)1022 1080 w 10 R f ( convention is certain to yield a thin and connected approximation to each octant)13 3324(octant. This)1 505 2 1211 1080 t (because the slope of the ellipse is bounded to the range [)11 2331 1 720 1200 t 10 S f (-)3067 1200 w 10 R f ( in the)2 264(1 , 0 ])3 182 2 3138 1200 t 8 R f (NNE)3616 1200 w 10 R f ( [)1 64(octant and to)2 530 2 3813 1200 t 10 S f (- \245)1 144 1 4423 1200 t 10 R f (,)4575 1200 w 10 S f (-)4616 1200 w 10 R f ( in the)2 262(1 ])1 91 2 4687 1200 t 8 R f (ENE)720 1320 w 10 R f ( the slope bounds also follows)5 1208(octant. From)1 536 2 901 1320 t 10 I f ( minimum-horizontal-displacement point for the)4 1990( Any)1 222(Lemma 1.)1 411 3 720 1476 t 8 R f (NNE)3385 1476 w 10 I f ( also a minimum-vertical-)3 1089(octant is)1 359 2 3592 1476 t ( similar statement,)2 746( A)1 113( that octant, unless the approximating point is an outside point.)10 2555(displacement point for)2 906 4 720 1596 t (with the roles of horizontal and vertical interchanged, holds for the)10 2698 1 720 1716 t 8 R f (ENE)3443 1716 w 10 I f (octant.)3624 1716 w 8 R f (2)3907 1684 w 10 R f (An outside point may be a minimum-displacement point in both directions, witness configurations)12 3947 1 720 1872 t 10 B f (1)4693 1872 w 10 R f (,)4743 1872 w 10 B f (5)4794 1872 w 10 R f (, and)1 196 1 4844 1872 t 10 B f (6)720 1992 w 10 R f ( it need not be, witness configurations)6 1530(, but)1 181 2 770 1992 t 10 B f (7)2508 1992 w 10 R f (and)2585 1992 w 10 B f (10)2756 1992 w 10 R f ( The)1 207( configurations refer to Appendix 1.)5 1446(. \(Numbered)1 531 3 2856 1992 t ( with the conventions of the diagrams,)6 1619(reader will find it profitable to detour there and become familiar)10 2701 2 720 2112 t (which illuminate nuances of the problem.\))5 1692 1 720 2232 t ( the published algorithms trace are also)6 1678(According to Lemma 1 the one-way approximations that)7 2392 2 970 2388 t ( thinness of the one-way approxi-)5 1364( The)1 210(two-way Freeman approximations\320except possibly at the juncture.)6 2746 3 720 2508 t (mations implies that the Freeman approximation also is thin\320again, except possibly at the juncture.)13 4006 1 720 2628 t 10 B f ( the Freeman approximation)3 1224(2. Generating)1 608 2 720 2868 t 10 R f (We shall develop an analogue of the well known Bresenham algorithm for circles)12 3380 1 970 3024 t 8 R f (1)4358 2992 w 10 R f ( the)1 158(to trace)1 306 2 4433 3024 t 8 R f (NE)4933 3024 w 10 R f ( traced to the point)4 757( the approximation has been)4 1138( Suppose)1 392(quadrant of an ellipse from north to east.)7 1646 4 720 3144 t 10 I f (P)4680 3144 w 10 R f (in Fig-)1 272 1 4768 3144 t ( the)1 148( monotonicity of the ellipse, the next approximating point will be one of the three neighbors to)16 3778( By)1 167(ure 3.)1 227 4 720 3264 t ( Point)1 268( south.)1 273(east, southeast, or)2 724 3 720 3384 t 10 I f (E)2016 3384 w 10 R f (will be chosen if the ellipse meets either of the unit bars)11 2299 1 2108 3384 t 10 I f (EV)4438 3384 w 10 R f (or)4591 3384 w 10 I f (EH)4705 3384 w 10 R f (cen-)4869 3384 w (tered there;)1 451 1 720 3504 t 10 I f (S)1196 3504 w 10 R f (will be chosen if the ellipse meets)6 1354 1 1271 3504 t 10 I f (SV)2650 3504 w 10 R f (or)2786 3504 w 10 I f (SH)2894 3504 w 10 R f (; otherwise)1 441 1 3016 3504 t 10 I f (SE)3482 3504 w 10 R f (will be chosen.)2 602 1 3618 3504 t 10 I f (E)3290 4192 w (SE S)1 -670 1 3290 4912 t (P)2570 4192 w (SV)2359 5100 w (SH)2171 4762 w (EH)3462 4192 w (EV V)1 -820 1 3290 3854 t (H)2196 4192 w 2498 4092 43 43 De 2505 4092 28 28 De 2512 4092 14 14 De 10 S f (\347)3225 3832 w (\347)3225 3930 w (\347)3225 4030 w (\347)3225 4130 w (\347)3225 4230 w (\347)3225 4330 w (\347)3225 4430 w (\347)3254 3832 w (\347)3254 3930 w (\347)3254 4030 w (\347)3254 4130 w (\347)3254 4230 w (\347)3254 4330 w (\347)3254 4430 w (\347)3240 3832 w (\347)3240 3930 w (\347)3240 4030 w (\347)3240 4130 w (\347)3240 4230 w (\347)3240 4330 w (\347)3240 4430 w (\347)2505 3832 w (\347)2505 3930 w (\347)2505 4030 w (\347)2505 4130 w (\347)2505 4230 w (\347)2505 4330 w (\347)2505 4430 w (\347)2534 3832 w (\347)2534 3930 w (\347)2534 4030 w (\347)2534 4130 w (\347)2534 4230 w (\347)2534 4330 w (\347)2534 4430 w (\347)2520 3832 w (\347)2520 3930 w (\347)2520 4030 w (\347)2520 4130 w (\347)2520 4230 w (\347)2520 4330 w (\347)2520 4430 w (\347)3225 4573 w (\347)3225 4671 w (\347)3225 4771 w (\347)3225 4871 w (\347)3225 4971 w (\347)3225 5071 w (\347)3225 5171 w (\347)3254 4573 w (\347)3254 4671 w (\347)3254 4771 w (\347)3254 4871 w (\347)3254 4971 w (\347)3254 5071 w (\347)3254 5171 w (\347)3240 4573 w (\347)3240 4671 w (\347)3240 4771 w (\347)3240 4871 w (\347)3240 4971 w (\347)3240 5071 w (\347)3240 5171 w (\347)2505 4573 w (\347)2505 4671 w (\347)2505 4771 w (\347)2505 4871 w (\347)2505 4971 w (\347)2505 5071 w (\347)2505 5171 w (\347)2534 4573 w (\347)2534 4671 w (\347)2534 4771 w (\347)2534 4871 w (\347)2534 4971 w (\347)2534 5071 w (\347)2534 5171 w (\347)2520 4573 w (\347)2520 4671 w (\347)2520 4771 w (\347)2520 4871 w (\347)2520 4971 w (\347)2520 5071 w (\347)2520 5171 w 10 S1 f (______________)2902 4077 w (______________)2902 4106 w (______________)2902 4092 w (______________)2902 4797 w (______________)2902 4826 w (______________)2902 4812 w (______________)2160 4077 w (______________)2160 4106 w (______________)2160 4092 w (______________)2160 4797 w (______________)2160 4826 w (______________)2160 4812 w 10 R f (Figure 3.)1 361 1 2699 5448 t (If the ellipse meets)3 820 1 970 5724 t 10 I f (SV)1836 5724 w 10 R f ( Lemma 1 it must also meet)6 1243(, then by)2 389 2 1947 5724 t 10 I f (SH)3626 5724 w 10 R f (, for)1 188 1 3748 5724 t 10 I f (S)3983 5724 w 10 R f (is a minimum-vertical-)2 960 1 4080 5724 t (displacement point of the)3 1022 1 720 5844 t 8 R f (ENE)1770 5844 w 10 R f ( latter fact follows from observing that for the ellipse to meet)11 2471(octant. The)1 477 2 1954 5844 t 10 I f (SV)4929 5844 w 10 R f (with)720 5964 w 10 I f (P)927 5964 w 10 R f (lighted it must have average slope less than)7 1762 1 1017 5964 t 10 S f (-)2808 5964 w 10 R f (1 in the region south of)5 950 1 2879 5964 t 10 I f (H)3858 5964 w 10 R f (and north of)2 498 1 3959 5964 t 10 I f (SV)4487 5964 w 10 R f (; hence the)2 442 1 4598 5964 t (juncture lies north of)3 835 1 720 6084 t 10 I f (SV)1580 6084 w 10 R f (.)1691 6084 w ( check whether the ellipse meets)5 1344(To check for a south step, then, it suffices to)9 1853 2 970 6240 t 10 I f (SH)4202 6240 w 10 R f (. Provided)1 446 1 4324 6240 t 10 I f (P)4805 6240 w 10 R f (lies)4901 6240 w (above the)1 386 1 720 6360 t 10 I f (x)1132 6360 w 10 R f (axis, this is equivalent to checking that the right end of)10 2200 1 1202 6360 t 10 I f (SH)3428 6360 w 10 R f ( need not)2 366( We)1 188(lies outside the ellipse.)3 911 3 3575 6360 t (worry about the possibility of a tie, where the ellipse meets)10 2421 1 720 6480 t 10 I f (SH)3172 6480 w 10 R f (and)3325 6480 w 10 I f (SE)3500 6480 w 10 R f (exactly in their common endpoint,)4 1398 1 3642 6480 t ( definiteness in the algorithm, we shall arbitrarily)7 1991( For)1 192( 2.)1 103(for that cannot happen; see Lemma 3 in Appendix)8 2034 4 720 6600 t (break the tie in favor of the outer point, in this case)11 2041 1 720 6720 t 10 I f (SE)2786 6720 w 10 R f (.)2897 6720 w (To check for an east step when)6 1259 1 970 6876 t 10 I f (P)2258 6876 w 10 R f (lies above the)2 557 1 2348 6876 t 10 I f (x)2934 6876 w 10 R f ( the ellipse meets)3 706(axis, we similarly check whether)4 1327 2 3007 6876 t 10 I f (EV)720 6996 w 10 R f (by seeing whether the lower end of)6 1420 1 870 6996 t 10 I f (EV)2318 6996 w 10 R f ( meet)1 221( the ellipse does not)4 806( If)1 119(lies inside or on the ellipse.)5 1109 4 2468 6996 t 10 I f (EV)4750 6996 w 10 R f (, we)1 168 1 4872 6996 t ( it meets)2 343(check whether)1 579 2 720 7116 t 10 I f (EH)1669 7116 w 10 R f (by seeing whether the left end of)6 1320 1 1829 7116 t 10 I f (EH)3176 7116 w 10 R f ( aesthetic)1 376( \(For)1 224(lies inside or on the ellipse.)5 1104 3 3336 7116 t (consistency, ties are again broken in favor of the outer point.\))10 2456 1 720 7236 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 52 514 764 %%EndPage: 4 4 %%Page: 5 5 %%PageBoundingBox: (atend) /saveobj save def mark 5 pagesetup 10 R f (- 5 -)2 166 1 2797 480 t (By unwarranted analogy with the treatment of)6 1841 1 970 840 t 10 I f (SV)2836 840 w 10 R f (, the cited algorithms ignore)4 1128 1 2947 840 t 10 I f (EH)4101 840 w 10 R f ( algorithm thus)2 608(. An)1 198 2 4234 840 t ( not light the square corner in configurations)7 1792(simplified will)1 590 2 720 960 t 10 B f (7)3129 960 w 10 R f (and)3206 960 w 10 B f (20)3377 960 w 10 R f (, although it will light the corner in the)8 1563 1 3477 960 t (transposed configurations)1 1029 1 720 1080 t 10 B f (10)1774 1080 w 10 R f (and)1899 1080 w 10 B f (18)2068 1080 w 10 R f ( the simplification defeats symmetry.)4 1485(. Thus)1 275 2 2168 1080 t (These considerations lead to)3 1134 1 970 1236 t (Algorithm 0.)1 517 1 2621 1392 t 10 I f (x)1080 1572 w 10 R f (:)1165 1572 w 10 S f (=)1209 1572 w 10 R f (0)1313 1572 w 10 I f (y)1080 1692 w 10 R f (:)1165 1692 w 10 S f (=)1209 1692 w 10 I f (b)1313 1692 w 10 CW f (while)1080 1812 w 10 I f (y)1421 1812 w 10 S f (>)1514 1812 w 10 R f (0)1618 1812 w 10 I f (mark)1330 1932 w 10 R f (\()1576 1932 w 10 I f (x)1617 1932 w 10 R f (,)1669 1932 w 10 I f (y)1702 1932 w 10 R f (\))1754 1932 w (if)1330 2052 w 10 I f (meets EV)1 390 1 1432 2052 t 10 CW f (then)1863 2052 w 10 I f (step E)1 263 1 2144 2052 t 10 CW f (else)1330 2172 w 10 R f (if)1611 2172 w 10 I f (meets EH)1 401 1 1713 2172 t 10 CW f (then)2155 2172 w 10 I f (step E)1 263 1 2436 2172 t 10 CW f (else)1330 2292 w 10 R f (if)1611 2292 w 10 I f (meets SH)1 390 1 1713 2292 t 10 CW f (then)2144 2292 w 10 I f (step S)1 252 1 2425 2292 t 10 CW f (else)1330 2412 w 10 I f (step SE)1 313 1 1611 2412 t 10 CW f (while)1080 2532 w 10 I f (x)1421 2532 w 10 S f (\243)1506 2532 w 10 I f (a)1602 2532 w (mark)1330 2652 w 10 R f (\()1576 2652 w 10 I f (x)1617 2652 w 10 R f (,)1669 2652 w 10 I f (y)1702 2652 w 10 R f (\))1754 2652 w 10 I f (step E)1 263 1 1330 2772 t 10 R f (The first loop requires)3 893 1 720 2952 t 10 I f (y)1639 2952 w 10 S f (>)1732 2952 w 10 R f (0 to assure that the)4 753 1 1836 2952 t 10 I f (SH)2615 2952 w 10 R f (and)2763 2952 w 10 I f (EV)2933 2952 w 10 R f (tests are made only when)4 1013 1 3081 2952 t 10 I f (P)4120 2952 w 10 R f (lies above the)2 551 1 4207 2952 t 10 I f (x)4784 2952 w 10 R f (axis.)4854 2952 w (The second loop runs out any remaining steps along the)9 2228 1 720 3072 t 10 I f (x)2973 3072 w 10 R f (axis.)3042 3072 w (To express the predicate)3 972 1 970 3228 t 10 I f (meets)1967 3228 w 10 R f (analytically, we define an error function)5 1601 1 2219 3228 t 10 I f (e)3845 3228 w 10 R f (as)3914 3228 w 10 I f (e)1220 3408 w 10 R f (\()1272 3408 w 10 I f (x)1313 3408 w 10 R f (,)1365 3408 w 10 I f (y)1422 3408 w 10 R f (\))1474 3408 w 10 S f (=)1564 3408 w 10 I f (b)1668 3408 w 7 R f (2)1729 3368 w 10 I f (x)1780 3408 w 7 R f (2)1835 3368 w 10 S f (+)1927 3408 w 10 I f (a)2031 3408 w 7 R f (2)2092 3368 w 10 I f (y)2143 3408 w 7 R f (2)2198 3368 w 10 S f (-)2290 3408 w 10 I f (a)2394 3408 w 7 R f (2)2455 3368 w 10 I f (b)2506 3408 w 7 R f (2)2567 3368 w 10 R f (The equation of the ellipse is)5 1157 1 720 3588 t 10 I f (e)1902 3588 w 10 R f (\()1954 3588 w 10 I f (x)1995 3588 w 10 R f (,)2047 3588 w 10 I f (y)2104 3588 w 10 R f (\))2156 3588 w 10 S f (=)2246 3588 w 10 R f (0 and the meeting conditions are)5 1301 1 2350 3588 t 10 I f (meets SH)1 390 1 1220 4028 t 10 R f (:)1618 4028 w 10 I f (meets EH)1 401 1 1220 3888 t 10 R f (:)1629 3888 w 10 I f (meets EV)1 390 1 1220 3748 t 10 R f (:)1618 3748 w 10 I f (e)1707 4028 w 10 R f (\()1759 4028 w 10 I f (x)1800 4028 w 10 S f (+)1868 4028 w 10 S1 f ()1939 4028 w cleartomark saveobj restore %%BeginGlobal /build_12 { pop /optsize ptsize def /osize size def /ofont font def optsize 2 div dup R exch R f 0 size 2 mul 3 div dup neg exch 0 exch rmoveto (1) show rmoveto optsize R f (\244) show f (2) show optsize ofont f } def %%EndGlobal /saveobj save def mark 10 S1 f 1939 4028 m 75 build_12 2014 4028 m 10 R f (,)2022 4028 w 10 I f (y)2079 4028 w 10 S f (-)2147 4028 w 10 R f (1 \))1 91 1 2218 4028 t 10 S f (>)2366 4028 w 10 R f (0)2470 4028 w 10 I f (e)1707 3888 w 10 R f (\()1759 3888 w 10 I f (x)1800 3888 w 10 S f (+)1868 3888 w 10 S1 f ()1939 3888 w 1939 3888 m 75 build_12 2014 3888 m 10 R f (,)2022 3888 w 10 I f (y)2079 3888 w 10 R f (\))2131 3888 w 10 S f (\243)2213 3888 w 10 R f (0)2309 3888 w 10 I f (e)1707 3748 w 10 R f (\()1759 3748 w 10 I f (x)1800 3748 w 10 S f (+)1868 3748 w 10 R f (1 ,)1 83 1 1939 3748 t 10 I f (y)2054 3748 w 10 S f (-)2122 3748 w 10 S1 f ()2193 3748 w 2193 3748 m 75 build_12 2268 3748 m 10 R f (\))2276 3748 w 10 S f (\243)2358 3748 w 10 R f (0)2454 3748 w ( scaling, or by appropriately)4 1179(The half-integer quantities can be respected in integer calculations by)9 2891 2 970 4224 t ( the considered points, which all lie near)7 1614( At)1 151(rounding the fractional part.)3 1123 3 720 4344 t 10 I f (e)3633 4344 w 10 S f (=)3726 4344 w 10 R f (0, the magnitude of)3 777 1 3830 4344 t 10 I f (e)4632 4344 w 10 R f (is on the)2 339 1 4701 4344 t ( \()1 41(order of max)2 529 2 720 4464 t 10 I f (a)1298 4464 w 7 R f (3)1359 4424 w 10 R f (,)1410 4464 w 10 I f (b)1467 4464 w 7 R f (3)1528 4424 w 10 R f ( \(but not with scaling\) is adequate to cope)8 1734( arithmetic with rounding)3 1040(\). Thirty-two-bit)1 687 3 1579 4464 t (with values of)2 566 1 720 4584 t 10 I f (a)1311 4584 w 10 R f (and)1386 4584 w 10 I f (b)1555 4584 w 10 R f (to just under 900.*)3 750 1 1630 4584 t ( the)1 150(We transform)1 553 2 970 4740 t 10 CW f (if)1701 4740 w 10 R f (tests algebraically to use integer arithmetic and the integer common subexpres-)10 3191 1 1849 4740 t (sion)720 4860 w 10 I f (e)912 4860 w 10 R f (\()964 4860 w 10 I f (x)1005 4860 w 10 S f (+)1073 4860 w 10 S1 f ()1144 4860 w 1144 4860 m 75 build_12 1219 4860 m 10 R f (,)1227 4860 w 10 I f (y)1284 4860 w 10 S f (-)1352 4860 w 10 S1 f ()1423 4860 w 1423 4860 m 75 build_12 1498 4860 m 10 R f (\))1506 4860 w 10 S f (-)1596 4860 w 10 R f (\()1700 4860 w 10 I f (a)1741 4860 w 7 R f (2)1802 4820 w 10 S f (+)1885 4860 w 10 I f (b)1980 4860 w 7 R f (2)2041 4820 w 10 R f (\))2092 4860 w 10 I f (/)2133 4860 w 10 R f (4.)2169 4860 w 10 I f (e)1080 5040 w 10 R f (\()1132 5040 w 10 I f (x)1173 5040 w 10 S f (+)1241 5040 w 10 R f (1 ,)1 83 1 1312 5040 t 10 I f (y)1427 5040 w 10 S f (-)1495 5040 w 10 S1 f ()1566 5040 w 1566 5040 m 75 build_12 1641 5040 m 10 R f (\))1649 5040 w 10 S f (\243)1731 5040 w 10 R f (0 {)1 301 1 1827 5040 t 10 I f (EV)2128 5040 w 10 R f (})2250 5040 w 10 I f (e)1080 5160 w 10 R f (\()1132 5160 w 10 I f (x)1173 5160 w 10 S f (+)1241 5160 w 10 S1 f ()1312 5160 w 1312 5160 m 75 build_12 1387 5160 m 10 R f (,)1395 5160 w 10 I f (y)1452 5160 w 10 S f (-)1520 5160 w 10 S1 f ()1591 5160 w 1591 5160 m 75 build_12 1666 5160 m 10 R f (\))1674 5160 w 10 S f (+)1764 5160 w 10 I f (e)1868 5160 w 10 R f (\()1920 5160 w 10 I f (x)1961 5160 w 10 S f (+)2029 5160 w 10 R f (1 ,)1 83 1 2100 5160 t 10 I f (y)2215 5160 w 10 S f (-)2283 5160 w 10 S1 f ()2354 5160 w 2354 5160 m 75 build_12 2429 5160 m 10 R f (\))2437 5160 w 10 S f (-)2527 5160 w 10 I f (e)2631 5160 w 10 R f (\()2683 5160 w 10 I f (x)2724 5160 w 10 S f (+)2792 5160 w 10 S1 f ()2863 5160 w 2863 5160 m 75 build_12 2938 5160 m 10 R f (,)2946 5160 w 10 I f (y)3003 5160 w 10 S f (-)3071 5160 w 10 S1 f ()3142 5160 w 3142 5160 m 75 build_12 3217 5160 m 10 R f (\))3225 5160 w 10 S f (\243)3307 5160 w 10 R f (0)3403 5160 w 10 I f (e)1080 5280 w 10 R f (\()1132 5280 w 10 I f (x)1173 5280 w 10 S f (+)1241 5280 w 10 S1 f ()1312 5280 w 1312 5280 m 75 build_12 1387 5280 m 10 R f (,)1395 5280 w 10 I f (y)1452 5280 w 10 S f (-)1520 5280 w 10 S1 f ()1591 5280 w 1591 5280 m 75 build_12 1666 5280 m 10 R f (\))1674 5280 w 10 S f (+)1764 5280 w 10 I f (b)1868 5280 w 7 R f (2)1929 5240 w 10 R f (\()1980 5280 w 10 I f (x)2021 5280 w 10 S f (+)2089 5280 w 10 R f (1 \))1 91 1 2160 5280 t 7 R f (2)2256 5240 w 10 S f (-)2348 5280 w 10 I f (b)2452 5280 w 7 R f (2)2513 5240 w 10 R f (\()2564 5280 w 10 I f (x)2605 5280 w 10 S f (+)2673 5280 w 10 S1 f ()2744 5280 w 2744 5280 m 75 build_12 2819 5280 m 10 R f (\))2827 5280 w 7 R f (2)2865 5240 w 10 S f (\243)2949 5280 w 10 R f (0)3045 5280 w 10 I f (e)1080 5400 w 10 R f (\()1132 5400 w 10 I f (x)1173 5400 w 10 S f (+)1241 5400 w 10 S1 f ()1312 5400 w 1312 5400 m 75 build_12 1387 5400 m 10 R f (,)1395 5400 w 10 I f (y)1452 5400 w 10 S f (-)1520 5400 w 10 S1 f ()1591 5400 w 1591 5400 m 75 build_12 1666 5400 m 10 R f (\))1674 5400 w 10 S f (-)1764 5400 w 10 R f (\()1868 5400 w 10 I f (a)1909 5400 w 7 R f (2)1970 5360 w 10 S f (+)2053 5400 w 10 I f (b)2148 5400 w 7 R f (2)2209 5360 w 10 R f (\))2260 5400 w 10 I f (/)2301 5400 w 10 R f (4)2337 5400 w 10 S f (+)2436 5400 w 10 R f (\()2540 5400 w 10 I f (a)2581 5400 w 7 R f (2)2642 5360 w 10 S f (+)2725 5400 w 10 I f (b)2820 5400 w 7 R f (2)2881 5360 w 10 R f (\))2932 5400 w 10 I f (/)2973 5400 w 10 R f (4)3009 5400 w 10 S f (+)3108 5400 w 10 I f (b)3212 5400 w 7 R f (2)3273 5360 w 10 R f (\()3324 5400 w 10 I f (x)3365 5400 w 10 S f (+)3433 5400 w 10 R f (1 \))1 91 1 3504 5400 t 7 R f (2)3600 5360 w 10 S f (-)3692 5400 w 10 I f (b)3796 5400 w 7 R f (2)3857 5360 w 10 R f (\()3908 5400 w 10 I f (x)3949 5400 w 10 S f (+)4017 5400 w 10 S1 f ()4088 5400 w 4088 5400 m 75 build_12 4163 5400 m 10 R f (\))4171 5400 w 7 R f (2)4209 5360 w 10 S f (\243)4293 5400 w 10 R f (0)4389 5400 w 10 I f (e)1080 5520 w 10 R f (\()1132 5520 w 10 I f (x)1173 5520 w 10 S f (+)1241 5520 w 10 S1 f ()1312 5520 w 1312 5520 m 75 build_12 1387 5520 m 10 R f (,)1395 5520 w 10 I f (y)1452 5520 w 10 S f (-)1520 5520 w 10 S1 f ()1591 5520 w 1591 5520 m 75 build_12 1666 5520 m 10 R f (\))1674 5520 w 10 S f (-)1764 5520 w 10 R f (\()1868 5520 w 10 I f (a)1909 5520 w 7 R f (2)1970 5480 w 10 S f (+)2053 5520 w 10 I f (b)2148 5520 w 7 R f (2)2209 5480 w 10 R f (\))2260 5520 w 10 I f (/)2301 5520 w 10 R f (4)2337 5520 w 10 S f (+)2436 5520 w 10 I f (b)2540 5520 w 7 R f (2)2601 5480 w 10 I f (x)2652 5520 w 10 S f (\243 -)1 159 1 2737 5520 t 10 I f (a)2912 5520 w 7 R f (2)2973 5480 w 10 I f (/)3024 5520 w 10 R f (4)3060 5520 w 10 S f (-)3159 5520 w 10 I f (b)3230 5520 w 7 R f (2)3291 5480 w 10 I f (e)1080 5640 w 10 R f (\()1132 5640 w 10 I f (x)1173 5640 w 10 S f (+)1241 5640 w 10 S1 f ()1312 5640 w 1312 5640 m 75 build_12 1387 5640 m 10 R f (,)1395 5640 w 10 I f (y)1452 5640 w 10 S f (-)1520 5640 w 10 S1 f ()1591 5640 w 1591 5640 m 75 build_12 1666 5640 m 10 R f (\))1674 5640 w 10 S f (-)1764 5640 w 10 R f (\()1868 5640 w 10 I f (a)1909 5640 w 7 R f (2)1970 5600 w 10 S f (+)2053 5640 w 10 I f (b)2148 5640 w 7 R f (2)2209 5600 w 10 R f (\))2260 5640 w 10 I f (/)2301 5640 w 10 R f (4)2337 5640 w 10 S f (+)2436 5640 w 10 I f (b)2540 5640 w 7 R f (2)2601 5600 w 10 I f (x)2652 5640 w 10 S f ( \353)1 66(\243 -)1 159 2 2737 5640 t 10 I f (a)2970 5640 w 7 R f (2)3031 5600 w 10 I f (/)3082 5640 w 10 R f (4)3118 5640 w 10 S f (\373 -)1 142 1 3176 5640 t 10 R f (\()3367 5640 w 10 I f (a)3408 5640 w 10 CW f (mod)3490 5640 w 10 R f (2 \))1 91 1 3702 5640 t 10 S f (-)3850 5640 w 10 I f (b)3954 5640 w 7 R f (2)4015 5600 w 10 R f ( Simi-)1 274( side of the last inequality is justified by the integrality of the left hand side.)15 3041(The rounding in the right)4 1005 3 720 5820 t (larly)720 5940 w 10 I f (e)1080 6120 w 10 R f (\()1132 6120 w 10 I f (x)1173 6120 w 10 S f (+)1241 6120 w 10 S1 f ()1312 6120 w 1312 6120 m 75 build_12 1387 6120 m 10 R f (,)1395 6120 w 10 I f (y)1452 6120 w 10 R f (\))1504 6120 w 10 S f (\243)1586 6120 w 10 R f (0 {)1 446 1 1682 6120 t 10 I f (EH)2128 6120 w 10 R f (})2261 6120 w 10 I f (e)1080 6240 w 10 R f (\()1132 6240 w 10 I f (x)1173 6240 w 10 S f (+)1241 6240 w 10 S1 f ()1312 6240 w 1312 6240 m 75 build_12 1387 6240 m 10 R f (,)1395 6240 w 10 I f (y)1452 6240 w 10 S f (-)1520 6240 w 10 S1 f ()1591 6240 w 1591 6240 m 75 build_12 1666 6240 m 10 R f (\))1674 6240 w 10 S f (-)1764 6240 w 10 R f (\()1868 6240 w 10 I f (a)1909 6240 w 7 R f (2)1970 6200 w 10 S f (+)2053 6240 w 10 I f (b)2148 6240 w 7 R f (2)2209 6200 w 10 R f (\))2260 6240 w 10 I f (/)2301 6240 w 10 R f (4)2337 6240 w 10 S f (+)2436 6240 w 10 R f (\()2540 6240 w 10 I f (a)2581 6240 w 7 R f (2)2642 6200 w 10 S f (+)2725 6240 w 10 I f (b)2820 6240 w 7 R f (2)2881 6200 w 10 R f (\))2932 6240 w 10 I f (/)2973 6240 w 10 R f (4)3009 6240 w 10 S f (+)3108 6240 w 10 I f (e)3212 6240 w 10 R f (\()3264 6240 w 10 I f (x)3305 6240 w 10 S f (+)3373 6240 w 10 S1 f ()3444 6240 w 3444 6240 m 75 build_12 3519 6240 m 10 R f (,)3527 6240 w 10 I f (y)3584 6240 w 10 R f (\))3636 6240 w 10 S f (-)3726 6240 w 10 I f (e)3830 6240 w 10 R f (\()3882 6240 w 10 I f (x)3923 6240 w 10 S f (+)3991 6240 w 10 S1 f ()4062 6240 w 4062 6240 m 75 build_12 4137 6240 m 10 R f (,)4145 6240 w 10 I f (y)4202 6240 w 10 S f (-)4270 6240 w 10 S1 f ()4341 6240 w 4341 6240 m 75 build_12 4416 6240 m 10 R f (\))4424 6240 w 10 S f (\243)4506 6240 w 10 R f (0)4602 6240 w 10 I f (e)1080 6360 w 10 R f (\()1132 6360 w 10 I f (x)1173 6360 w 10 S f (+)1241 6360 w 10 S1 f ()1312 6360 w 1312 6360 m 75 build_12 1387 6360 m 10 R f (,)1395 6360 w 10 I f (y)1452 6360 w 10 S f (-)1520 6360 w 10 S1 f ()1591 6360 w 1591 6360 m 75 build_12 1666 6360 m 10 R f (\))1674 6360 w 10 S f (-)1764 6360 w 10 R f (\()1868 6360 w 10 I f (a)1909 6360 w 7 R f (2)1970 6320 w 10 S f (+)2053 6360 w 10 I f (b)2148 6360 w 7 R f (2)2209 6320 w 10 R f (\))2260 6360 w 10 I f (/)2301 6360 w 10 R f (4)2337 6360 w 10 S f (+)2436 6360 w 10 I f (a)2540 6360 w 7 R f (2)2601 6320 w 10 I f (y)2652 6360 w 10 S f (\243 -)1 159 1 2737 6360 t 10 I f (b)2912 6360 w 7 R f (2)2973 6320 w 10 I f (/)3024 6360 w 10 R f (4)3060 6360 w 10 I f (e)1080 6480 w 10 R f (\()1132 6480 w 10 I f (x)1173 6480 w 10 S f (+)1241 6480 w 10 S1 f ()1312 6480 w 1312 6480 m 75 build_12 1387 6480 m 10 R f (,)1395 6480 w 10 I f (y)1452 6480 w 10 S f (-)1520 6480 w 10 S1 f ()1591 6480 w 1591 6480 m 75 build_12 1666 6480 m 10 R f (\))1674 6480 w 10 S f (-)1764 6480 w 10 R f (\()1868 6480 w 10 I f (a)1909 6480 w 7 R f (2)1970 6440 w 10 S f (+)2053 6480 w 10 I f (b)2148 6480 w 7 R f (2)2209 6440 w 10 R f (\))2260 6480 w 10 I f (/)2301 6480 w 10 R f (4)2337 6480 w 10 S f (+)2436 6480 w 10 I f (a)2540 6480 w 7 R f (2)2601 6440 w 10 I f (y)2652 6480 w 10 S f ( \353)1 66(\243 -)1 159 2 2737 6480 t 10 I f (b)2970 6480 w 7 R f (2)3031 6440 w 10 I f (/)3082 6480 w 10 R f (4)3118 6480 w 10 S f (\373 -)1 142 1 3176 6480 t 10 R f (\()3367 6480 w 10 I f (b)3408 6480 w 10 CW f (mod)3490 6480 w 10 R f (2 \))1 91 1 3702 6480 t 8 S1 f (__________________)720 6780 w 8 R f (* Since an approximate point \()5 984 1 720 6880 t 8 I f (x)1710 6880 w 8 R f (,)1751 6880 w 8 I f (y)1777 6880 w 8 R f ( be up to 1/2 unit off the ellipse, a test point, say \()13 1606(\) may)1 184 2 1818 6880 t 8 I f (x)3614 6880 w 8 S f (+)3668 6880 w 8 S1 f ()3725 6880 w 3725 6880 m 60 build_12 3785 6880 m 8 R f (,)3791 6880 w 8 I f (y)3817 6880 w 8 S f (-)3871 6880 w 8 R f ( may be 3/2 unit off.)5 660(1 \),)1 92 2 3928 6880 t (At such a point, with)4 671 1 720 6980 t 8 I f (a)1412 6980 w 8 S f (=)1491 6980 w 8 I f (b)1574 6980 w 8 R f (,)1614 6980 w 8 I f (y)1655 6980 w 8 S f (~)1722 6964 w 8 R f (=)1722 6988 w 8 I f (a)1799 6980 w 8 R f (, and)1 156 1 1839 6980 t 8 I f (x)2016 6980 w 8 S f (~)2083 6964 w 8 R f (=)2083 6988 w (0, we estimate)2 459 1 2160 6980 t 8 S f (\357)2640 6993 w 8 I f (e)2679 6980 w 8 R f (\()2720 6980 w 8 I f (x)2752 6980 w 8 R f (,)2793 6980 w 8 I f (y)2819 6980 w 8 R f (\))2860 6980 w 8 S f (\357)2886 6993 w (~)2951 6964 w 8 R f (=)2951 6988 w 8 S f (\357)3022 6993 w 8 R f (\()3061 6980 w 8 S f (\266)3093 6980 w 8 I f (e /)1 63 1 3138 6980 t 8 S f (\266)3207 6980 w 8 I f (y)3252 6980 w 8 R f (\))3293 6980 w 8 S f (D)3332 6980 w 8 I f (y)3387 6980 w 8 S f (\357)3422 6993 w (~)3487 6964 w 8 R f (=)3487 6988 w (3)3564 6980 w 8 I f (a)3610 6980 w 5 R f (3)3658 6948 w 8 R f (. Thus)1 221 1 3689 6980 t 8 I f (e)3931 6980 w 8 R f (\()3972 6980 w 8 I f (x)4004 6980 w 8 R f (,)4045 6980 w 8 I f (y)4071 6980 w 8 R f (\) is liable to over-)4 568 1 4112 6980 t (flow 32-bit registers at)3 721 1 720 7080 t 8 I f (a)1461 7080 w 8 S f (>)1540 7080 w 8 R f (\( 2)1 72 1 1623 7080 t 5 R f (31)1699 7048 w 8 I f (/)1761 7080 w 8 R f (3 \))1 72 1 1789 7080 t 5 R f (1)1865 7048 w 5 I f (/)1894 7048 w 5 R f (3)1912 7048 w 8 R f (, or)1 106 1 1943 7080 t 8 I f (a)2069 7080 w 8 S f (>)2148 7080 w 8 R f (894.)2231 7080 w cleartomark showpage saveobj restore %%PageBoundingBox: 61 69 514 764 %%EndPage: 5 5 %%Page: 6 6 %%PageBoundingBox: (atend) /saveobj save def mark 6 pagesetup 10 R f (- 6 -)2 166 1 2797 480 t 10 I f (e)1080 900 w 10 R f (\()1132 900 w 10 I f (x)1173 900 w 10 S f (+)1241 900 w 10 S1 f ()1312 900 w 1312 900 m 75 build_12 1387 900 m 10 R f (,)1395 900 w 10 I f (y)1452 900 w 10 S f (-)1520 900 w 10 R f (1 \))1 91 1 1591 900 t 10 S f (>)1739 900 w 10 R f (0 {)1 285 1 1843 900 t 10 I f (SH)2128 900 w 10 R f (})2250 900 w 10 I f (e)1080 1020 w 10 R f (\()1132 1020 w 10 I f (x)1173 1020 w 10 S f (+)1241 1020 w 10 S1 f ()1312 1020 w 1312 1020 m 75 build_12 1387 1020 m 10 R f (,)1395 1020 w 10 I f (y)1452 1020 w 10 S f (-)1520 1020 w 10 S1 f ()1591 1020 w 1591 1020 m 75 build_12 1666 1020 m 10 R f (\))1674 1020 w 10 S f (-)1764 1020 w 10 R f (\()1868 1020 w 10 I f (a)1909 1020 w 7 R f (2)1970 980 w 10 S f (+)2053 1020 w 10 I f (b)2148 1020 w 7 R f (2)2209 980 w 10 R f (\))2260 1020 w 10 I f (/)2301 1020 w 10 R f (4)2337 1020 w 10 S f (+)2436 1020 w 10 R f (\()2540 1020 w 10 I f (a)2581 1020 w 7 R f (2)2642 980 w 10 S f (+)2725 1020 w 10 I f (b)2820 1020 w 7 R f (2)2881 980 w 10 R f (\))2932 1020 w 10 I f (/)2973 1020 w 10 R f (4)3009 1020 w 10 S f (+)3108 1020 w 10 I f (e)3179 1020 w 10 R f (\()3231 1020 w 10 I f (x)3272 1020 w 10 S f (+)3340 1020 w 10 S1 f ()3411 1020 w 3411 1020 m 75 build_12 3486 1020 m 10 R f (,)3494 1020 w 10 I f (y)3551 1020 w 10 S f (-)3619 1020 w 10 R f (1 \))1 91 1 3690 1020 t 10 S f (-)3838 1020 w 10 I f (e)3942 1020 w 10 R f (\()3994 1020 w 10 I f (x)4035 1020 w 10 S f (+)4103 1020 w 10 S1 f ()4174 1020 w 4174 1020 m 75 build_12 4249 1020 m 10 R f (,)4257 1020 w 10 I f (y)4314 1020 w 10 S f (-)4382 1020 w 10 S1 f ()4453 1020 w 4453 1020 m 75 build_12 4528 1020 m 10 R f (\))4536 1020 w 10 S f (>)4626 1020 w 10 R f (0)4730 1020 w 10 I f (e)1080 1140 w 10 R f (\()1132 1140 w 10 I f (x)1173 1140 w 10 S f (+)1241 1140 w 10 S1 f ()1312 1140 w 1312 1140 m 75 build_12 1387 1140 m 10 R f (,)1395 1140 w 10 I f (y)1452 1140 w 10 S f (-)1520 1140 w 10 S1 f ()1591 1140 w 1591 1140 m 75 build_12 1666 1140 m 10 R f (\))1674 1140 w 10 S f (-)1764 1140 w 10 R f (\()1868 1140 w 10 I f (a)1909 1140 w 7 R f (2)1970 1100 w 10 S f (+)2053 1140 w 10 I f (b)2148 1140 w 7 R f (2)2209 1100 w 10 R f (\))2260 1140 w 10 I f (/)2301 1140 w 10 R f (4)2337 1140 w 10 S f (-)2436 1140 w 10 I f (a)2540 1140 w 7 R f (2)2601 1100 w 10 I f (y)2652 1140 w 10 S f (> -)1 167 1 2745 1140 t 10 I f (b)2928 1140 w 7 R f (2)2989 1100 w 10 I f (/)3040 1140 w 10 R f (4)3076 1140 w 10 S f (-)3175 1140 w 10 I f (a)3279 1140 w 7 R f (2)3340 1100 w 10 I f (e)1080 1260 w 10 R f (\()1132 1260 w 10 I f (x)1173 1260 w 10 S f (+)1241 1260 w 10 S1 f ()1312 1260 w 1312 1260 m 75 build_12 1387 1260 m 10 R f (,)1395 1260 w 10 I f (y)1452 1260 w 10 S f (-)1520 1260 w 10 S1 f ()1591 1260 w 1591 1260 m 75 build_12 1666 1260 m 10 R f (\))1674 1260 w 10 S f (-)1764 1260 w 10 R f (\()1868 1260 w 10 I f (a)1909 1260 w 7 R f (2)1970 1220 w 10 S f (+)2053 1260 w 10 I f (b)2148 1260 w 7 R f (2)2209 1220 w 10 R f (\))2260 1260 w 10 I f (/)2301 1260 w 10 R f (4)2337 1260 w 10 S f (-)2436 1260 w 10 I f (a)2540 1260 w 7 R f (2)2601 1220 w 10 I f (y)2652 1260 w 10 S f ( \353)1 66(> -)1 167 2 2745 1260 t 10 I f (b)2986 1260 w 7 R f (2)3047 1220 w 10 I f (/)3098 1260 w 10 R f (4)3134 1260 w 10 S f (\373 -)1 142 1 3192 1260 t 10 R f (\()3383 1260 w 10 I f (b)3424 1260 w 10 CW f (mod)3506 1260 w 10 R f (2 \))1 91 1 3718 1260 t 10 S f (-)3866 1260 w 10 I f (a)3970 1260 w 7 R f (2)4031 1220 w 10 R f ( arranging to calculate)3 895(Installing the transformed tests and)4 1408 2 720 1440 t 10 I f (e)3050 1440 w 10 R f (\()3102 1440 w 10 I f (x)3143 1440 w 10 S f (+)3211 1440 w 10 S1 f ()3282 1440 w 3282 1440 m 75 build_12 3357 1440 m 10 R f (,)3365 1440 w 10 I f (y)3422 1440 w 10 S f (-)3490 1440 w 10 S1 f ()3561 1440 w 3561 1440 m 75 build_12 3636 1440 m 10 R f (\) incrementally, we get the follow-)5 1396 1 3644 1440 t (ing program, for which)3 986 1 720 1560 t 10 I f (t)1751 1560 w 10 S f (=)1828 1560 w 10 I f (e)1932 1560 w 10 R f (\()1984 1560 w 10 I f (x)2025 1560 w 10 S f (+)2093 1560 w 10 S1 f ()2164 1560 w 2164 1560 m 75 build_12 2239 1560 m 10 R f (,)2247 1560 w 10 I f (y)2304 1560 w 10 S f (-)2372 1560 w 10 S1 f ()2443 1560 w 2443 1560 m 75 build_12 2518 1560 m 10 R f (\))2526 1560 w 10 S f (-)2616 1560 w 10 R f (\()2720 1560 w 10 I f (a)2761 1560 w 7 R f (2)2822 1520 w 10 S f (+)2905 1560 w 10 I f (b)3000 1560 w 7 R f (2)3061 1520 w 10 R f (\))3112 1560 w 10 I f (/)3153 1560 w 10 R f ( gives an)2 393( 3)1 95( Appendix)1 464(4 is a loop invariant.)4 899 4 3189 1560 t (implementation in C.)2 848 1 720 1680 t (Algorithm 1.)1 517 1 2801 1860 t 10 I f (x)1080 1980 w 10 R f (:)1165 1980 w 10 S f (=)1209 1980 w 10 R f (0)1313 1980 w 10 I f (y)1080 2100 w 10 R f (:)1165 2100 w 10 S f (=)1209 2100 w 10 I f (b)1313 2100 w (t)1080 2220 w 10 R f (:)1149 2220 w 10 S f (=)1193 2220 w 10 I f (b)1297 2220 w 7 R f (2)1358 2180 w 10 R f (\()1409 2220 w 10 I f (x)1450 2220 w 7 R f (2)1505 2180 w 10 S f (+)1588 2220 w 10 I f (x)1683 2220 w 10 R f (\))1735 2220 w 10 S f (+)1825 2220 w 10 I f (a)1929 2220 w 7 R f (2)1990 2180 w 10 R f (\()2041 2220 w 10 I f (y)2082 2220 w 7 R f (2)2137 2180 w 10 S f (-)2196 2220 w 10 I f (y)2267 2220 w 10 R f (\))2319 2220 w 10 S f (-)2409 2220 w 10 I f (a)2513 2220 w 7 R f (2)2574 2180 w 10 I f (b)2625 2220 w 7 R f (2)2686 2180 w 10 CW f (while)1080 2340 w 10 I f (y)1421 2340 w 10 S f (>)1514 2340 w 10 R f (0)1618 2340 w 10 I f (mark)1330 2460 w 10 R f (\()1543 2460 w 10 I f (x)1584 2460 w 10 R f (,)1636 2460 w 10 I f (y)1669 2460 w 10 R f (\))1721 2460 w 10 CW f (if)1330 2580 w 10 I f (t)1491 2580 w 10 S f (+)1568 2580 w 10 I f (b)1672 2580 w 7 R f (2)1733 2540 w 10 I f (x)1784 2580 w 10 S f ( \353)1 66(\243 -)1 159 2 1869 2580 t 10 I f (a)2102 2580 w 7 R f (2)2163 2540 w 10 I f (/)2214 2580 w 10 R f (4)2250 2580 w 10 S f (\373 -)1 142 1 2308 2580 t 10 R f (\()2499 2580 w 10 I f (a)2540 2580 w 10 CW f (mod)2622 2580 w 10 R f (2 \))1 91 1 2834 2580 t 10 S f (-)2982 2580 w 10 I f (b)3086 2580 w 7 R f (2)3147 2540 w 10 R f ({)3580 2580 w 10 I f (e)3628 2580 w 10 R f (\()3680 2580 w 10 I f (x)3721 2580 w 10 S f (+)3789 2580 w 10 R f (1 ,)1 83 1 3860 2580 t 10 I f (y)3975 2580 w 10 S f (-)4043 2580 w 10 S1 f ()4114 2580 w 4114 2580 m 75 build_12 4189 2580 m 10 R f (\))4197 2580 w 10 S f (\243)4270 2580 w 10 R f (0 ;)1 86 1 4357 2580 t 10 I f (EV)4484 2580 w 10 R f (})4606 2580 w 10 I f (x)1580 2700 w 10 S f (+ =)1 126 1 1673 2700 t 10 R f (1)1848 2700 w 10 I f (t)1580 2820 w 10 S f (+ =)1 126 1 1657 2820 t 10 I f (b)1832 2820 w 7 R f (2)1893 2780 w 10 R f (\( 2)1 91 1 1944 2820 t 10 I f (x)2043 2820 w 10 S f (+)2127 2820 w 10 R f (2 \))1 91 1 2222 2820 t 10 CW f (else if)1 401 1 1330 2940 t 10 I f (t)1772 2940 w 10 S f (+)1849 2940 w 10 I f (a)1953 2940 w 7 R f (2)2014 2900 w 10 I f (y)2065 2940 w 10 S f ( \353)1 66(\243 -)1 159 2 2150 2940 t 10 I f (b)2383 2940 w 7 R f (2)2444 2900 w 10 I f (/)2495 2940 w 10 R f (4)2531 2940 w 10 S f (\373 -)1 142 1 2589 2940 t 10 R f (\()2780 2940 w 10 I f (b)2821 2940 w 10 CW f (mod)2903 2940 w 10 R f ( {)1 422(2 \))1 91 2 3115 2940 t 10 I f (e)3628 2940 w 10 R f (\()3680 2940 w 10 I f (x)3721 2940 w 10 S f (+)3789 2940 w 10 S1 f ()3860 2940 w 3860 2940 m 75 build_12 3935 2940 m 10 R f (,)3943 2940 w 10 I f (y)4000 2940 w 10 R f (\))4052 2940 w 10 S f (\243)4125 2940 w 10 R f (0 ;)1 86 1 4212 2940 t 10 I f (EH)4339 2940 w 10 R f (})4472 2940 w 10 I f (x)1580 3060 w 10 S f (+ =)1 126 1 1673 3060 t 10 R f (1)1848 3060 w 10 I f (t)1580 3180 w 10 S f (+ =)1 126 1 1657 3180 t 10 I f (b)1832 3180 w 7 R f (2)1893 3140 w 10 R f (\( 2)1 91 1 1944 3180 t 10 I f (x)2043 3180 w 10 S f (+)2127 3180 w 10 R f (2 \))1 91 1 2222 3180 t 10 CW f (else if)1 401 1 1330 3300 t 10 I f (t)1772 3300 w 10 S f (-)1849 3300 w 10 I f (a)1953 3300 w 7 R f (2)2014 3260 w 10 I f (y)2065 3300 w 10 S f ( \353)1 66(> -)1 167 2 2158 3300 t 10 I f (b)2399 3300 w 7 R f (2)2460 3260 w 10 I f (/)2511 3300 w 10 R f (4)2547 3300 w 10 S f (\373 -)1 142 1 2605 3300 t 10 R f (\()2796 3300 w 10 I f (b)2837 3300 w 10 CW f (mod)2919 3300 w 10 R f (2 \))1 91 1 3131 3300 t 10 S f (-)3279 3300 w 10 I f (a)3383 3300 w 7 R f (2)3444 3260 w 10 R f ({)3580 3300 w 10 I f (e)3628 3300 w 10 R f (\()3680 3300 w 10 I f (x)3721 3300 w 10 S f (+)3789 3300 w 10 S1 f ()3860 3300 w 3860 3300 m 75 build_12 3935 3300 m 10 R f (,)3943 3300 w 10 I f (y)4000 3300 w 10 S f (-)4068 3300 w 10 R f (1 \))1 91 1 4139 3300 t 10 S f (>)4278 3300 w 10 R f (0 ;)1 86 1 4373 3300 t 10 I f (SH)4500 3300 w 10 R f (})4622 3300 w 10 I f (y)1580 3420 w 10 S f (- =)1 126 1 1673 3420 t 10 R f (1)1848 3420 w 10 I f (t)1580 3540 w 10 S f (+ =)1 126 1 1657 3540 t 10 I f (a)1832 3540 w 7 R f (2)1893 3500 w 10 R f (\()1944 3540 w 10 S f (-)1993 3540 w 10 R f (2)2064 3540 w 10 I f (y)2122 3540 w 10 S f (+)2190 3540 w 10 R f (2 \))1 91 1 2261 3540 t 10 CW f (else)1330 3660 w 10 I f (x)1580 3780 w 10 S f (+ =)1 126 1 1673 3780 t 10 R f (1)1848 3780 w 10 I f (y)1580 3900 w 10 S f (- =)1 126 1 1673 3900 t 10 R f (1)1848 3900 w 10 I f (t)1580 4020 w 10 S f (+ =)1 126 1 1657 4020 t 10 I f (b)1832 4020 w 7 R f (2)1893 3980 w 10 R f (\( 2)1 91 1 1944 4020 t 10 I f (x)2043 4020 w 10 S f (+)2127 4020 w 10 R f (2 \))1 91 1 2222 4020 t 10 S f (+)2370 4020 w 10 I f (a)2474 4020 w 7 R f (2)2535 3980 w 10 R f (\()2586 4020 w 10 S f (-)2635 4020 w 10 R f (2)2706 4020 w 10 I f (y)2764 4020 w 10 S f (+)2832 4020 w 10 R f (2 \))1 91 1 2903 4020 t 10 CW f (while)1080 4140 w 10 I f (x)1421 4140 w 10 S f (\243)1506 4140 w 10 I f (a)1602 4140 w (mark)1330 4260 w 10 R f (\()1543 4260 w 10 I f (x)1584 4260 w 10 R f (,)1636 4260 w 10 I f (y)1669 4260 w 10 R f (\))1721 4260 w 10 I f (x)1330 4380 w 10 S f (+ =)1 126 1 1423 4380 t 10 R f (1)1598 4380 w (It is a straightforward matter to check that the program works in degenerate cases where)14 3672 1 970 4596 t 10 I f (a)4679 4596 w 10 R f (or)4766 4596 w 10 I f (b)4886 4596 w 10 R f (is)4973 4596 w (zero.)720 4716 w (On the)1 272 1 970 4872 t 10 I f (x)1270 4872 w 10 R f (axis the)1 311 1 1342 4872 t 10 I f (EH)1681 4872 w 10 R f (test would evaluate to true at points inside the ellipse, or at any point if)14 2890 1 1843 4872 t 10 I f (b)4762 4872 w 10 S f (=)4861 4872 w 10 R f (0.)4965 4872 w (By modifying the loop condition so the)6 1633 1 720 4992 t 10 I f (EH)2388 4992 w 10 R f (test gets performed with)3 998 1 2556 4992 t 10 I f (y)3588 4992 w 10 S f (=)3681 4992 w 10 R f (0 when there is a tail, we may)7 1255 1 3785 4992 t ( 3 also explains how to)5 929( Appendix)1 446( Appendix 3 incorporates this idea.)5 1408( program in)2 468( The)1 206(drop the second loop.)3 863 6 720 5112 t (gain speed by exploiting the fact that not all of the tests are needed in all parts of the quadrant.)19 3766 1 720 5232 t ( be started at any)4 776(To trace an elliptic arc that spans only part of a quadrant, Algorithm 1 can)14 3294 2 970 5388 t ( possible sticking point, overflow during the initialization of)8 2464( One)1 223(minimum-displacement point.)1 1213 3 720 5508 t 10 I f (t)4652 5508 w 10 R f ( more)1 236(, is)1 124 2 4680 5508 t ( If)1 118(apparent than real.)2 743 2 720 5628 t 10 I f (t)1608 5628 w 10 R f ( arithmetic without regard to overflow to)6 1649(is in range, it can be computed in unsigned)8 1728 2 1663 5628 t (yield a correct twos-complement result.)4 1583 1 720 5748 t 10 B f (3. Discussion)1 576 1 720 5988 t 10 R f (Rescuing)970 6144 w 10 I f (EH)1367 6144 w 10 R f ( approximation, yet)2 782(from unjustified oblivion, Algorithm 1 generates a genuine Freeman)8 2733 2 1525 6144 t ( that yield ad hoc approximations; witness)6 1790(can be implemented as compactly as comparable algorithms)7 2530 2 720 6264 t ( the derivation has been long, informal, and riddled with case)10 2456( I have confidence in it,)5 942( Although)1 428(Appendix 3.)1 494 4 720 6384 t ( proof outline exists,)3 822(analysis. A)1 469 2 720 6504 t 8 R f (12)2019 6472 w 10 R f (but it would be reassuring to have a formal proof.)9 1980 1 2124 6504 t ( the possibility)2 598(The Freeman approximation satisfies the six desiderata set forth at the outset, save for)13 3472 2 970 6660 t ( although I don't know a way to do so)9 1617( corners could be sheared off,)5 1239( The)1 217(of having four square corners.)4 1247 4 720 6780 t ( lost:)1 201( may be)2 320( Accuracy)1 433( square corners would entail other difficulties as well.)8 2159( Rejecting)1 435(without extra code.)2 772 6 720 6900 t (in configuration)1 645 1 720 7020 t 10 B f (6)1394 7020 w 10 R f ( example, the ``bad'' corner point is noticeably closer to the ellipse than are either of)15 3426(, for)1 170 2 1444 7020 t ( line will be bought at the expense of roundness of)10 2124( visual terms, uniformity of)4 1137( In)1 143(its ``good'' neighbors.)2 916 4 720 7140 t ( tellingly, the algorithm's usefulness for drawing elliptic arcs would be)10 2943( Most)1 267( Figure 2a shows.)3 738(shape, as)1 372 4 720 7260 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 50 514 764 %%EndPage: 6 6 %%Page: 7 7 %%PageBoundingBox: (atend) /saveobj save def mark 7 pagesetup 10 R f (- 7 -)2 166 1 2797 480 t ( certain result)2 555( would have to be drawn differently, with the almost)9 2142( Arcs)1 243(sacrificed, as we shall see shortly.)5 1380 4 720 840 t (that an arc ending at the juncture would not coincide with the underlying ellipse there.)14 3442 1 720 960 t ( switch between the)3 796( They)1 257(The cited algorithms differ primarily in their treatment of the juncture.)10 2825 3 970 1116 t 8 R f (NNE)4875 1116 w 10 R f (and)720 1236 w 8 R f (ENE)891 1236 w 10 R f ( result)1 248( a)1 71( As)1 163(octants according to various heuristic criteria that defeat one or more of the desiderata.)13 3484 4 1074 1236 t ( can,)1 192( All)1 182( mathematically definable without reference to the algorithm.)7 2479(none yields an approximation that is)5 1467 4 720 1356 t ( can stray; in configuration)4 1080( DaSilva's)1 445( corners.)1 344(but do not necessarily, shear off square)6 1572 4 720 1476 t 10 B f (5)4187 1476 w 10 R f (DaSilva visits \(9,5\))2 777 1 4263 1476 t (rather than \(9,6\).)2 670 1 720 1596 t 8 R f (10)1398 1564 w 10 R f (Tails bedevil most of the algorithms; only Pratt is careful about them.)11 2947 1 970 1752 t 8 R f (9)3925 1720 w 10 R f ( paper)1 262(Pitteway's original)1 773 2 4005 1752 t ( does not handle them;)4 917(mentions tails but)2 720 2 720 1872 t 8 R f (4)2365 1840 w 10 R f (his later paper tries to cope by backing off to 4-connected \(rook-)11 2607 1 2433 1872 t (move\) approximations.)1 928 1 720 1992 t 8 R f (5)1656 1960 w 10 R f ( catastrophic tracking)2 863(Typically algorithms designed without regard to tails suffer a)8 2455 2 1722 1992 t (failure when a bar in Figure 3 stretches clear across the ellipse instead of cutting just one branch.)17 3966 1 720 2112 t 8 R f (9, 10)1 160 1 4694 2080 t 10 R f (The)4885 2112 w ( 4, for example, arise from such a tracking failure in Wirth's algorithm.)12 3003(fishy tails of Figure)3 822 2 720 2232 t 8 R f (6)4553 2200 w 10 R f (\(In partial)1 409 1 4631 2232 t ( the major and)3 613(redemption, Wirth's is the only algorithm that respects symmetry\320by virtue of handling)11 3707 2 720 2352 t (minor axes unsymmetrically!\))2 1204 1 720 2472 t 1584 2748 50 50 De 1592 2748 33 33 De 1600 2748 16 16 De 1584 3087 50 50 De 1592 3087 33 33 De 1600 3087 16 16 De 4125 2748 50 50 De 4133 2748 33 33 De 4142 2748 16 16 De 4125 3087 50 50 De 4133 3087 33 33 De 4142 3087 16 16 De 1668 2833 50 50 De 1677 2833 33 33 De 1685 2833 16 16 De 1668 3002 50 50 De 1677 3002 33 33 De 1685 3002 16 16 De 4040 2833 50 50 De 4048 2833 33 33 De 4057 2833 16 16 De 4040 3002 50 50 De 4048 3002 33 33 De 4057 3002 16 16 De 1753 2918 50 50 De 1761 2918 33 33 De 1770 2918 16 16 De 1753 2918 50 50 De 1761 2918 33 33 De 1770 2918 16 16 De 3955 2918 50 50 De 3964 2918 33 33 De 3972 2918 16 16 De 3955 2918 50 50 De 3964 2918 33 33 De 3972 2918 16 16 De 1838 3002 50 50 De 1846 3002 33 33 De 1855 3002 16 16 De 1838 2833 50 50 De 1846 2833 33 33 De 1855 2833 16 16 De 3871 3002 50 50 De 3879 3002 33 33 De 3888 3002 16 16 De 3871 2833 50 50 De 3879 2833 33 33 De 3888 2833 16 16 De 1922 3002 50 50 De 1931 3002 33 33 De 1939 3002 16 16 De 1922 2833 50 50 De 1931 2833 33 33 De 1939 2833 16 16 De 3786 3002 50 50 De 3794 3002 33 33 De 3803 3002 16 16 De 3786 2833 50 50 De 3794 2833 33 33 De 3803 2833 16 16 De 2007 3002 50 50 De 2016 3002 33 33 De 2024 3002 16 16 De 2007 2833 50 50 De 2016 2833 33 33 De 2024 2833 16 16 De 3701 3002 50 50 De 3710 3002 33 33 De 3718 3002 16 16 De 3701 2833 50 50 De 3710 2833 33 33 De 3718 2833 16 16 De 2092 3002 50 50 De 2100 3002 33 33 De 2109 3002 16 16 De 2092 2833 50 50 De 2100 2833 33 33 De 2109 2833 16 16 De 3616 3002 50 50 De 3625 3002 33 33 De 3633 3002 16 16 De 3616 2833 50 50 De 3625 2833 33 33 De 3633 2833 16 16 De 2176 3002 50 50 De 2185 3002 33 33 De 2193 3002 16 16 De 2176 2833 50 50 De 2185 2833 33 33 De 2193 2833 16 16 De 3532 3002 50 50 De 3540 3002 33 33 De 3549 3002 16 16 De 3532 2833 50 50 De 3540 2833 33 33 De 3549 2833 16 16 De 2261 3002 50 50 De 2270 3002 33 33 De 2278 3002 16 16 De 2261 2833 50 50 De 2270 2833 33 33 De 2278 2833 16 16 De 3447 3002 50 50 De 3456 3002 33 33 De 3464 3002 16 16 De 3447 2833 50 50 De 3456 2833 33 33 De 3464 2833 16 16 De 2346 3002 50 50 De 2354 3002 33 33 De 2363 3002 16 16 De 2346 2833 50 50 De 2354 2833 33 33 De 2363 2833 16 16 De 3362 3002 50 50 De 3371 3002 33 33 De 3379 3002 16 16 De 3362 2833 50 50 De 3371 2833 33 33 De 3379 2833 16 16 De 2431 3002 50 50 De 2439 3002 33 33 De 2448 3002 16 16 De 2431 2833 50 50 De 2439 2833 33 33 De 2448 2833 16 16 De 3278 3002 50 50 De 3286 3002 33 33 De 3295 3002 16 16 De 3278 2833 50 50 De 3286 2833 33 33 De 3295 2833 16 16 De 2515 3002 50 50 De 2524 3002 33 33 De 2532 3002 16 16 De 2515 2833 50 50 De 2524 2833 33 33 De 2532 2833 16 16 De 3193 3002 50 50 De 3201 3002 33 33 De 3210 3002 16 16 De 3193 2833 50 50 De 3201 2833 33 33 De 3210 2833 16 16 De 2600 3002 50 50 De 2608 3002 33 33 De 2617 3002 16 16 De 2600 2833 50 50 De 2608 2833 33 33 De 2617 2833 16 16 De 3108 3002 50 50 De 3117 3002 33 33 De 3125 3002 16 16 De 3108 2833 50 50 De 3117 2833 33 33 De 3125 2833 16 16 De 2685 3002 50 50 De 2693 3002 33 33 De 2702 3002 16 16 De 2685 2833 50 50 De 2693 2833 33 33 De 2702 2833 16 16 De 3024 3002 50 50 De 3032 3002 33 33 De 3040 3002 16 16 De 3024 2833 50 50 De 3032 2833 33 33 De 3040 2833 16 16 De 2769 3002 50 50 De 2778 3002 33 33 De 2786 3002 16 16 De 2769 2833 50 50 De 2778 2833 33 33 De 2786 2833 16 16 De 2939 3002 50 50 De 2947 3002 33 33 De 2956 3002 16 16 De 2939 2833 50 50 De 2947 2833 33 33 De 2956 2833 16 16 De 2854 3002 50 50 De 2863 3002 33 33 De 2871 3002 16 16 De 2854 2833 50 50 De 2863 2833 33 33 De 2871 2833 16 16 De 2854 3002 50 50 De 2863 3002 33 33 De 2871 3002 16 16 De 2854 2833 50 50 De 2863 2833 33 33 De 2871 2833 16 16 De 10 S1 f (_ _________________________________________________________)1 2880 1 1440 2664 t (_ _________________________________________________________)1 2880 1 1440 2748 t (_ _________________________________________________________)1 2880 1 1440 2833 t (_ _________________________________________________________)1 2880 1 1440 2918 t (_ _________________________________________________________)1 2880 1 1440 3002 t (_ _________________________________________________________)1 2880 1 1440 3087 t (_ _________________________________________________________)1 2880 1 1440 3172 t 10 S f (\347)4320 2764 w (\347)4320 2772 w (\347)4320 2872 w (\347)4320 2972 w (\347)4320 3072 w (\347)4320 3172 w (\347)4235 2764 w (\347)4235 2772 w (\347)4235 2872 w (\347)4235 2972 w (\347)4235 3072 w (\347)4235 3172 w (\347)4150 2764 w (\347)4150 2772 w (\347)4150 2872 w (\347)4150 2972 w (\347)4150 3072 w (\347)4150 3172 w (\347)4065 2764 w (\347)4065 2772 w (\347)4065 2872 w (\347)4065 2972 w (\347)4065 3072 w (\347)4065 3172 w (\347)3981 2764 w (\347)3981 2772 w (\347)3981 2872 w (\347)3981 2972 w (\347)3981 3072 w (\347)3981 3172 w (\347)3896 2764 w (\347)3896 2772 w (\347)3896 2872 w (\347)3896 2972 w (\347)3896 3072 w (\347)3896 3172 w (\347)3811 2764 w (\347)3811 2772 w (\347)3811 2872 w (\347)3811 2972 w (\347)3811 3072 w (\347)3811 3172 w (\347)3727 2764 w (\347)3727 2772 w (\347)3727 2872 w (\347)3727 2972 w (\347)3727 3072 w (\347)3727 3172 w (\347)3642 2764 w (\347)3642 2772 w (\347)3642 2872 w (\347)3642 2972 w (\347)3642 3072 w (\347)3642 3172 w (\347)3557 2764 w (\347)3557 2772 w (\347)3557 2872 w (\347)3557 2972 w (\347)3557 3072 w (\347)3557 3172 w (\347)3472 2764 w (\347)3472 2772 w (\347)3472 2872 w (\347)3472 2972 w (\347)3472 3072 w (\347)3472 3172 w (\347)3388 2764 w (\347)3388 2772 w (\347)3388 2872 w (\347)3388 2972 w (\347)3388 3072 w (\347)3388 3172 w (\347)3303 2764 w (\347)3303 2772 w (\347)3303 2872 w (\347)3303 2972 w (\347)3303 3072 w (\347)3303 3172 w (\347)3218 2764 w (\347)3218 2772 w (\347)3218 2872 w (\347)3218 2972 w (\347)3218 3072 w (\347)3218 3172 w (\347)3134 2764 w (\347)3134 2772 w (\347)3134 2872 w (\347)3134 2972 w (\347)3134 3072 w (\347)3134 3172 w (\347)3049 2764 w (\347)3049 2772 w (\347)3049 2872 w (\347)3049 2972 w (\347)3049 3072 w (\347)3049 3172 w (\347)2964 2764 w (\347)2964 2772 w (\347)2964 2872 w (\347)2964 2972 w (\347)2964 3072 w (\347)2964 3172 w (\347)2880 2764 w (\347)2880 2772 w (\347)2880 2872 w (\347)2880 2972 w (\347)2880 3072 w (\347)2880 3172 w (\347)2795 2764 w (\347)2795 2772 w (\347)2795 2872 w (\347)2795 2972 w (\347)2795 3072 w (\347)2795 3172 w (\347)2710 2764 w (\347)2710 2772 w (\347)2710 2872 w (\347)2710 2972 w (\347)2710 3072 w (\347)2710 3172 w (\347)2625 2764 w (\347)2625 2772 w (\347)2625 2872 w (\347)2625 2972 w (\347)2625 3072 w (\347)2625 3172 w (\347)2541 2764 w (\347)2541 2772 w (\347)2541 2872 w (\347)2541 2972 w (\347)2541 3072 w (\347)2541 3172 w (\347)2456 2764 w (\347)2456 2772 w (\347)2456 2872 w (\347)2456 2972 w (\347)2456 3072 w (\347)2456 3172 w (\347)2371 2764 w (\347)2371 2772 w (\347)2371 2872 w (\347)2371 2972 w (\347)2371 3072 w (\347)2371 3172 w (\347)2287 2764 w (\347)2287 2772 w (\347)2287 2872 w (\347)2287 2972 w (\347)2287 3072 w (\347)2287 3172 w (\347)2202 2764 w (\347)2202 2772 w (\347)2202 2872 w (\347)2202 2972 w (\347)2202 3072 w (\347)2202 3172 w (\347)2117 2764 w (\347)2117 2772 w (\347)2117 2872 w (\347)2117 2972 w (\347)2117 3072 w (\347)2117 3172 w (\347)2032 2764 w (\347)2032 2772 w (\347)2032 2872 w (\347)2032 2972 w (\347)2032 3072 w (\347)2032 3172 w (\347)1948 2764 w (\347)1948 2772 w (\347)1948 2872 w (\347)1948 2972 w (\347)1948 3072 w (\347)1948 3172 w (\347)1863 2764 w (\347)1863 2772 w (\347)1863 2872 w (\347)1863 2972 w (\347)1863 3072 w (\347)1863 3172 w (\347)1778 2764 w (\347)1778 2772 w (\347)1778 2872 w (\347)1778 2972 w (\347)1778 3072 w (\347)1778 3172 w (\347)1694 2764 w (\347)1694 2772 w (\347)1694 2872 w (\347)1694 2972 w (\347)1694 3072 w (\347)1694 3172 w (\347)1609 2764 w (\347)1609 2772 w (\347)1609 2872 w (\347)1609 2972 w (\347)1609 3072 w (\347)1609 3172 w (\347)1524 2764 w (\347)1524 2772 w (\347)1524 2872 w (\347)1524 2972 w (\347)1524 3072 w (\347)1524 3172 w (\347)1440 2764 w (\347)1440 2772 w (\347)1440 2872 w (\347)1440 2972 w (\347)1440 3072 w (\347)1440 3172 w 1638 2900 1609 2918 Dl 4121 2936 4150 2918 Dl 1638 2936 1609 2918 Dl 4121 2900 4150 2918 Dl 1668 2892 1639 2899 Dl 4091 2943 4120 2936 Dl 1668 2943 1639 2936 Dl 4091 2892 4120 2899 Dl 1697 2887 1668 2892 Dl 4062 2948 4091 2943 Dl 1697 2948 1668 2943 Dl 4062 2887 4091 2892 Dl 1727 2882 1698 2886 Dl 4032 2953 4061 2949 Dl 1727 2953 1698 2949 Dl 4032 2882 4061 2886 Dl 1757 2879 1728 2882 Dl 4003 2956 4032 2953 Dl 1757 2956 1728 2953 Dl 4003 2879 4032 2882 Dl 1786 2875 1757 2878 Dl 3973 2960 4002 2957 Dl 1786 2960 1757 2957 Dl 3973 2875 4002 2878 Dl 1816 2871 1787 2874 Dl 3943 2964 3972 2961 Dl 1816 2964 1787 2961 Dl 3943 2871 3972 2874 Dl 1845 2869 1816 2871 Dl 3914 2966 3943 2964 Dl 1845 2966 1816 2964 Dl 3914 2869 3943 2871 Dl 1875 2866 1846 2868 Dl 3884 2969 3913 2967 Dl 1875 2969 1846 2967 Dl 3884 2866 3913 2868 Dl 1905 2864 1876 2866 Dl 3854 2972 3883 2970 Dl 1905 2972 1876 2970 Dl 3854 2864 3883 2866 Dl 1934 2861 1905 2863 Dl 3825 2974 3854 2972 Dl 1934 2974 1905 2972 Dl 3825 2861 3854 2863 Dl 1964 2859 1935 2861 Dl 3795 2976 3824 2974 Dl 1964 2976 1935 2974 Dl 3795 2859 3824 2861 Dl 1994 2858 1965 2859 Dl 3765 2977 3794 2976 Dl 1994 2977 1965 2976 Dl 3765 2858 3794 2859 Dl 2023 2856 1994 2857 Dl 3736 2979 3765 2978 Dl 2023 2979 1994 2978 Dl 3736 2856 3765 2857 Dl 2053 2854 2024 2855 Dl 3706 2981 3735 2980 Dl 2053 2981 2024 2980 Dl 3706 2854 3735 2855 Dl 2083 2852 2054 2853 Dl 3676 2983 3705 2982 Dl 2083 2983 2054 2982 Dl 3676 2852 3705 2853 Dl 2112 2851 2083 2852 Dl 3647 2985 3676 2984 Dl 2112 2985 2083 2984 Dl 3647 2851 3676 2852 Dl 2142 2849 2113 2850 Dl 3617 2986 3646 2985 Dl 2142 2986 2113 2985 Dl 3617 2849 3646 2850 Dl 2172 2848 2143 2849 Dl 3587 2988 3616 2987 Dl 2172 2988 2143 2987 Dl 3587 2848 3616 2849 Dl 2201 2846 2172 2847 Dl 3558 2989 3587 2988 Dl 2201 2989 2172 2988 Dl 3558 2846 3587 2847 Dl 2231 2845 2202 2846 Dl 3528 2990 3557 2989 Dl 2231 2990 2202 2989 Dl 3528 2845 3557 2846 Dl 2261 2844 2232 2845 Dl 3499 2991 3528 2990 Dl 2261 2991 2232 2990 Dl 3499 2844 3528 2845 Dl 2290 2843 2261 2844 Dl 3469 2993 3498 2992 Dl 2290 2993 2261 2992 Dl 3469 2843 3498 2844 Dl 2320 2842 2291 2843 Dl 3439 2994 3468 2993 Dl 2320 2994 2291 2993 Dl 3439 2842 3468 2843 Dl 2349 2842 2320 2842 Dl 3410 2994 3439 2994 Dl 2349 2994 2320 2994 Dl 3410 2842 3439 2842 Dl 2379 2841 2350 2841 Dl 3380 2995 3409 2995 Dl 2379 2995 2350 2995 Dl 3380 2841 3409 2841 Dl 2409 2840 2380 2840 Dl 3350 2995 3379 2995 Dl 2409 2995 2380 2995 Dl 3350 2840 3379 2840 Dl 2438 2839 2409 2839 Dl 3321 2996 3350 2996 Dl 2438 2996 2409 2996 Dl 3321 2839 3350 2839 Dl 2468 2838 2439 2838 Dl 3291 2997 3320 2997 Dl 2468 2997 2439 2997 Dl 3291 2838 3320 2838 Dl 2498 2837 2469 2837 Dl 3261 2998 3290 2998 Dl 2498 2998 2469 2998 Dl 3261 2837 3290 2837 Dl 2527 2837 2498 2837 Dl 3232 2998 3261 2998 Dl 2527 2998 2498 2998 Dl 3232 2837 3261 2837 Dl 2557 2836 2528 2836 Dl 3202 2999 3231 2999 Dl 2557 2999 2528 2999 Dl 3202 2836 3231 2836 Dl 2587 2836 2558 2836 Dl 3172 3000 3201 3000 Dl 2587 3000 2558 3000 Dl 3172 2836 3201 2836 Dl 2616 2835 2587 2835 Dl 3143 3000 3172 3000 Dl 2616 3000 2587 3000 Dl 3143 2835 3172 2835 Dl 2646 2835 2617 2835 Dl 3113 3000 3142 3000 Dl 2646 3000 2617 3000 Dl 3113 2835 3142 2835 Dl 2676 2834 2647 2834 Dl 3083 3001 3112 3001 Dl 2676 3001 2647 3001 Dl 3083 2834 3112 2834 Dl 2705 2834 2676 2834 Dl 3054 3001 3083 3001 Dl 2705 3001 2676 3001 Dl 3054 2834 3083 2834 Dl 2735 2834 2706 2834 Dl 3024 3002 3053 3002 Dl 2735 3002 2706 3002 Dl 3024 2834 3053 2834 Dl 2765 2833 2736 2833 Dl 2995 3002 3024 3002 Dl 2765 3002 2736 3002 Dl 2995 2833 3024 2833 Dl 2794 2833 2765 2833 Dl 2965 3002 2994 3002 Dl 2794 3002 2765 3002 Dl 2965 2833 2994 2833 Dl 10 R f ( ellipse is the same as)5 873( The)1 207( typical mistake in drawing tails.)5 1315( A)1 123(Figure 4.)1 362 5 1440 3448 t (that in Figure 1:)3 642 1 1440 3568 t 10 I f (a)2107 3568 w 10 S f (=)2197 3568 w 10 R f (15,)2292 3568 w 10 I f (b)2442 3568 w 10 S f (=)2532 3568 w 10 R f (1.)2627 3568 w ( the Freeman approximation in favor of ad hoc criteria?)9 2232(Why have the published algorithms eschewed)5 1838 2 970 3724 t ( the Pitteway-)2 567( When)1 294( the reason.)2 469(Optimistic imitation of the best algorithms for circles is doubtless part of)11 2990 4 720 3844 t ( as precise determination of)4 1127(like algorithms were seen to produce visually satisfactory ellipses, details such)10 3193 2 720 3964 t ( the Freeman approximation has been)5 1500( Perhaps)1 366( simply forgotten.)2 714(the juncture and respect for symmetry were)6 1740 4 720 4084 t ( in view of Lemma 1\) worry about the possibility)9 2004(overlooked also because of an unspoken \(and groundless,)7 2316 2 720 4204 t ( certainly the possibility of configurations such as)7 2046( Almost)1 353(of excessive square corners.)3 1142 3 720 4324 t 10 B f (7)4294 4324 w 10 R f (and)4377 4324 w 10 B f (18)4554 4324 w 10 R f (has been)1 353 1 4687 4324 t (overlooked.)720 4444 w ( loop)1 216(Most of all, though, a desire for a fast)8 1599 2 970 4600 t 8 R f (9)2793 4568 w 10 R f (has probably upstaged other considerations: the pro-)6 2169 1 2871 4600 t ( ellipses, where four points are plotted for)7 1669( least for drawing full)4 869( At)1 151(grams have been optimized prematurely.)4 1631 4 720 4720 t (each one that is calculated, the price of one extra test to get the Freeman approximation is negligible.)17 4029 1 720 4840 t ( approximations are indescribable, the published algorithms cannot easily be modified)10 3523(Because their)1 547 2 970 4996 t ( to be a minimum-)4 783( when a proposed endpoint is verified)6 1575( Even)1 267(to draw elliptic arcs given the endpoints.)6 1695 4 720 5116 t ( corner that)2 466( could, for example, be a square)6 1304( It)1 116(displacement point, it may not belong to the approximation.)8 2434 4 720 5236 t ( con-)1 208( In)1 139( a point.)2 337( infinite loop can result from testing for termination against such)10 2651( An)1 179(the algorithm skips.)2 806 6 720 5356 t ( because)1 348(trast, an arc-tracing program based on the Freeman approximation can be made accurate and safe)14 3972 2 720 5476 t (the question of whether a point belongs to the approximation can be quickly decided.*)13 3451 1 720 5596 t (Some open questions: Is the uncertain configuration)6 2093 1 970 5752 t 10 B f (13)3090 5752 w 10 R f ( there a simpler way to find the)7 1266(realizable? Is)1 556 2 3218 5752 t ( general conic sections be handled as easily?)7 1776( Can)1 211(Freeman approximation?)1 1001 3 720 5872 t ( stimulated it,)2 552(I wish to acknowledge Rob Pike, who requested the program, the reference that)12 3188 2 970 6028 t 8 R f (6)4718 5996 w 10 R f (fellow)4785 6028 w ( their insights into program devel-)5 1383(members of IFIP Working Group 2.3 on Programming Methodology for)9 2937 2 720 6148 t (opment, which helped shape it, and conscientious referees, who helped polish it.)11 3209 1 720 6268 t 8 S1 f (__________________)720 6880 w 8 R f (* Solve \(1\) for)3 469 1 720 6980 t 8 I f (y)1210 6980 w 8 R f (at integer)1 299 1 1267 6980 t 8 I f (x)1588 6980 w 8 R f (\(or vice versa\) and round, or check for a sign difference in the the error function)15 2573 1 1645 6980 t 8 I f (e)4240 6980 w 8 R f (evaluated at)1 383 1 4297 6980 t (the ends of bars)3 501 1 720 7080 t 8 I f (V)1241 7080 w 8 R f (and)1310 7080 w 8 I f (H)1445 7080 w 8 R f (in Figure 3.)2 370 1 1523 7080 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 69 514 764 %%EndPage: 7 7 %%Page: 8 8 %%PageBoundingBox: (atend) /saveobj save def mark 8 pagesetup 10 R f (- 8 -)2 166 1 2797 480 t 10 B f (References)720 840 w 10 R f ( ``A linear algorithm for incremental digital display of circular arcs,'')10 2814( J.,)1 117([1] Bresenham,)1 665 3 889 1056 t 10 I f (Comm. ACM)1 526 1 4514 1056 t 10 B f (20)1080 1176 w 10 R f (, pp. 100-106 \(1977\).)3 849 1 1180 1176 t ( D., ``Best approximate circles on integer grids,'')7 2073( M.)1 153([2] McIlroy,)1 543 3 889 1356 t 10 I f (ACM Trans. on Graphics)3 1062 1 3698 1356 t 10 B f (2)4800 1356 w 10 R f (, pp.)1 190 1 4850 1356 t (237-264 \(Oct. 1983\).)2 843 1 1080 1476 t ( H., ``Computer processing of line-drawing images,'')6 2132([3] Freeman,)1 565 2 889 1656 t 10 I f (Computing Surveys)1 780 1 3611 1656 t 10 B f (6)4416 1656 w 10 R f (, p. 63 \(1974\).)3 566 1 4466 1656 t ( M. L. V., ``Algorithms for drawing ellipses or hyperbolae with a digital plotter,'')13 3333([4] Pitteway,)1 566 2 889 1836 t 10 I f (Com-)4818 1836 w (puter J.)1 305 1 1080 1956 t 10 B f (10)1410 1956 w 10 R f (, pp. 282-289 \(1967\).)3 849 1 1510 1956 t ( of conic generation,'' pp. 219-237 in)6 1557( M. L. V., ``Algorithms)4 980([5] Pitteway,)1 566 3 889 2136 t 10 I f (Fundamental Algorithms)1 1013 1 4027 2136 t (for Computer Graphics)2 939 1 1080 2256 t 10 R f (, Earnshaw, R. A. \(Ed.\), Springer-Verlag, Heidelberg \(1985\).)7 2441 1 2019 2256 t ( N., ``Drawing lines, circles and ellipses in a raster,'' pp. 427-434 in)12 2737([6] Wirth,)1 449 2 889 2436 t 10 I f (Beauty is our Business)3 914 1 4101 2436 t 10 R f (,)5015 2436 w ( D., and Misra, J. \(Eds.\), Springer-Verlag, New)7 1974(Feijen, W. H. J., van Gasteren, A. J. M., Gries,)9 1986 2 1080 2556 t (York \(1990\).)1 521 1 1080 2676 t ( Aken, J. R., ``An efficient ellipse-drawing algorithm,'')7 2253([7] Van)1 357 2 889 2856 t 10 I f ( Graphics and Appli-)3 865(IEEE Computer)1 646 2 3529 2856 t (cations)1080 2976 w 10 B f (4)1394 2976 w 10 R f (\(9\), pp. 24-35 \(1984\).)3 865 1 1444 2976 t ( displays,'')1 447( Aken, J. and Novak, M., ``Curve-drawing algorithms for raster)9 2557([8] Van)1 357 3 889 3156 t 10 I f (ACM Transactions)1 762 1 4278 3156 t (on Graphics)1 497 1 1080 3276 t 10 B f (4)1602 3276 w 10 R f (\(2\), pp. 147-169 \(1985\).)3 965 1 1652 3276 t ( V., ``Techniques for conic splines,'')5 1504([9] Pratt,)1 405 2 889 3456 t 10 I f (Computer Graphics)1 803 1 2829 3456 t 10 B f (19)3663 3456 w 10 R f ( A.)1 129(\(3\), pp. 151-159, Barsky, B.)4 1148 2 3763 3456 t ( '85 Conference Proceedings.)3 1180( SIGGRAPH)1 550(\(Ed.\), ACM \(1985\).)2 796 3 1080 3576 t ( S. K., and Hughes, J. F.,)6 1057( J. D., Van Dam, A., Feiner,)6 1177([10] Foley,)1 494 3 839 3756 t 10 I f (Computer Graphics Principles and)3 1438 1 3602 3756 t (Practice)1080 3876 w 10 R f (, Addison-Wesley \(1990\).)2 1037 1 1418 3876 t ( in)1 107( M. A., ``An Ellipse-Drawing Algorithm for Raster Displays,'' pp. 257-280)10 3051([11] Kappel,)1 554 3 839 4056 t 10 I f (Fundamen-)4580 4056 w (tal Algorithms for Computer Graphics)4 1660 1 1080 4176 t 10 R f ( R. A. \(Ed.\), Springer-Verlag, Heidelberg)5 1802(, Earnshaw,)1 498 2 2740 4176 t (\(1985\).)1080 4296 w ( trilogy on raster ellipses and programming)6 1738( M. D., ``There is no royal road to programs: a)10 1870([12] McIlroy,)1 593 3 839 4476 t (methodology,'' Computing Science Technical Report 155, AT&T Bell Laboratories \(1990\).)9 3679 1 1080 4596 t ( I.,)1 108([13] Lakatos,)1 582 2 839 4776 t 10 I f (Proofs and Refutations: the Logic of Mathematical Discovery)7 2468 1 1554 4776 t 10 R f (, Cambridge \(1976\).)2 810 1 4022 4776 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 298 514 764 %%EndPage: 8 8 %%Page: 9 9 %%PageBoundingBox: (atend) /saveobj save def mark 9 pagesetup 10 R f (- 9 -)2 166 1 2797 480 t 10 B f ( of configurations at the juncture)5 1407( Inventory)1 472(Appendix 1.)1 518 3 720 840 t 10 R f ( distinct ways that the ellipse can cross grid lines in the neighbor-)12 2674(The diagrams below enumerate all)4 1396 2 970 996 t (hood of the juncture, where the slope is)7 1589 1 720 1116 t 10 S f (-)2336 1116 w 10 R f ( diagram shows the four grid lines that surround the junc-)10 2307(1. Each)1 326 2 2407 1116 t ( a bar will be lighted if)6 928( midpoint of)2 499( The)1 207( indicate intervals of equivalent crossings as in Figure 3.)9 2271(ture. Bars)1 415 5 720 1236 t ( bar missing from the top \(or right\) line is understood to be somewhere out)14 3042( A)1 126( the bar.)2 332(the ellipse intersects)2 820 4 720 1356 t ( dot marks the juncture.)4 946( A)1 122(of sight to the left \(or bottom\).)6 1219 3 720 1476 t ( coordinates of a grid point for an ellipse that realizes the con-)12 2514(Numbers show the dimensions and the)5 1556 2 970 1632 t (figuration with)1 599 1 720 1752 t 10 I f (a)1346 1752 w 10 S f (\263)1437 1752 w 10 I f (b)1533 1752 w 10 R f ( Con-)1 251( X are impossible; see Lemmas 4 and 5 in Appendix 2.)11 2203( marked)1 326(. Configurations)1 677 4 1583 1752 t (figuration)720 1872 w 10 B f (13)1140 1872 w 10 R f ( representative with)2 791(, marked ?, has no)4 730 2 1240 1872 t 10 I f (a)2788 1872 w 10 S f (\243)2879 1872 w 10 R f (1000; its possibility is a number-theoretic question.)6 2065 1 2975 1872 t (Note, though, that configuration)3 1285 1 720 1992 t 10 B f (13)2030 1992 w 10 R f (is the transpose of)3 724 1 2155 1992 t 10 B f (19)2904 1992 w 10 R f (, and thus can be realized with)6 1211 1 3004 1992 t 10 I f (a)4240 1992 w 10 S f (<)4339 1992 w 10 I f (b)4443 1992 w 10 R f (.)4493 1992 w ( bar on the left, top, right,)6 1078(The inventory is ordered lexicographically by decreasing positions of the)9 2992 2 970 2148 t ( Nei-)1 231( inconsistent with a monotone decreasing curve are not shown.)9 2543( Configurations)1 654( lines.)1 243(and bottom grid)2 649 5 720 2268 t ( that would violate slope requirements in grid squares that do not contain the junc-)14 3403(ther are configurations)2 917 2 720 2388 t ( across a square must be at least)7 1279(ture: the average slope of the ellipse)6 1453 2 720 2508 t 10 S f (-)3478 2508 w 10 R f (1 if only the)3 489 1 3549 2508 t 8 R f (NNE)4064 2508 w 10 R f (octant enters it, and)3 785 1 4255 2508 t (at most)1 292 1 720 2628 t 10 S f (-)1037 2628 w 10 R f (1 if only the)3 486 1 1108 2628 t 8 R f (ENE)1619 2628 w 10 R f (octant.)1800 2628 w 10 S1 f (_ _____)1 259 1 1829 3882 t (_ _____)1 259 1 1829 3903 t (_ _____)1 259 1 1829 3892 t 10 S f (\347)1559 3215 w (\347)1559 3233 w (\347)1559 3333 w (\347)1559 3433 w (\347)1559 3533 w (\347)1559 3633 w (\347)1579 3215 w (\347)1579 3233 w (\347)1579 3333 w (\347)1579 3433 w (\347)1579 3533 w (\347)1579 3633 w (\347)1569 3215 w (\347)1569 3233 w (\347)1569 3333 w (\347)1569 3433 w (\347)1569 3533 w (\347)1569 3633 w 10 S1 f (_ __________)1 518 1 1310 3364 t (_ __________)1 518 1 1310 3384 t (_ __________)1 518 1 1310 3374 t 10 S f (\347)1040 2956 w (\347)1040 3015 w (\347)1040 3115 w (\347)1061 2956 w (\347)1061 3015 w (\347)1061 3115 w (\347)1051 2956 w (\347)1051 3015 w (\347)1051 3115 w 1828 3917 1828 3866 Dl 1544 3633 1595 3633 Dl 1310 3399 1310 3348 Dl 1026 3633 1077 3633 Dl 1310 3917 1310 3866 Dl 1310 3399 1310 3348 Dl 1026 3115 1077 3115 Dl 10 S1 f (_ ____________________)1 1036 1 792 3374 t 10 S f (\347)1569 3215 w (\347)1569 3251 w (\347)1569 3351 w (\347)1569 3451 w (\347)1569 3551 w (\347)1569 3651 w (\347)1569 3751 w (\347)1569 3851 w (\347)1569 3951 w (\347)1569 4051 w (\347)1569 4151 w 1072 2902 1051 2856 Dl 1050 2856 1029 2902 Dl (\347)1051 2956 w (\347)1051 3052 w (\347)1051 3152 w (\347)1051 3252 w (\347)1051 3352 w (\347)1051 3452 w (\347)1051 3552 w (\347)1051 3652 w (\347)1051 3752 w (\347)1051 3852 w (\347)1051 3952 w (\347)1051 4052 w (\347)1051 4152 w 2042 3913 2088 3892 Dl 2087 3891 2041 3870 Dl 10 S1 f (_ _________________________)1 1296 1 792 3892 t 10 B f (1)1285 4152 w 1520 3400 46 46 De 1527 3400 31 31 De 1535 3400 15 15 De 1972 3916 2025 3996 Dl 1918 3839 1971 3915 Dl 1864 3767 1917 3839 Dl 1810 3698 1863 3766 Dl 1756 3632 1809 3697 Dl 1702 3569 1755 3631 Dl 1648 3510 1701 3569 Dl 1594 3452 1647 3509 Dl 1541 3398 1594 3452 Dl 1487 3345 1540 3397 Dl 1433 3294 1486 3344 Dl 1379 3245 1432 3293 Dl 1325 3199 1378 3245 Dl 1271 3154 1324 3198 Dl 1217 3111 1270 3154 Dl 1163 3069 1216 3110 Dl 1110 3030 1163 3069 Dl 1056 2991 1109 3029 Dl 1002 2955 1055 2991 Dl 948 2919 1001 2954 Dl 10 I f (b)1076 2906 w 10 R f (= 7)1 131 1 1151 2906 t 10 I f (a)1882 3842 w 10 R f (= 7)1 131 1 1957 3842 t (\(4,4\))835 3992 w 10 S1 f (_ __________)1 518 1 2750 3882 t (_ __________)1 518 1 2750 3903 t (_ __________)1 518 1 2750 3892 t 10 S f (\347)2999 3215 w (\347)2999 3233 w (\347)2999 3333 w (\347)2999 3433 w (\347)2999 3533 w (\347)2999 3633 w (\347)3019 3215 w (\347)3019 3233 w (\347)3019 3333 w (\347)3019 3433 w (\347)3019 3533 w (\347)3019 3633 w (\347)3009 3215 w (\347)3009 3233 w (\347)3009 3333 w (\347)3009 3433 w (\347)3009 3533 w (\347)3009 3633 w 10 S1 f (_ __________)1 518 1 2750 3364 t (_ __________)1 518 1 2750 3384 t (_ __________)1 518 1 2750 3374 t 10 S f (\347)2480 2956 w (\347)2480 3015 w (\347)2480 3115 w (\347)2501 2956 w (\347)2501 3015 w (\347)2501 3115 w (\347)2491 2956 w (\347)2491 3015 w (\347)2491 3115 w 3268 3917 3268 3866 Dl 2984 3633 3035 3633 Dl 2750 3399 2750 3348 Dl 2466 3633 2517 3633 Dl 2750 3917 2750 3866 Dl 2750 3399 2750 3348 Dl 2466 3115 2517 3115 Dl 10 S1 f (_ ____________________)1 1036 1 2232 3374 t 10 S f (\347)3009 3215 w (\347)3009 3251 w (\347)3009 3351 w (\347)3009 3451 w (\347)3009 3551 w (\347)3009 3651 w (\347)3009 3751 w (\347)3009 3851 w (\347)3009 3951 w (\347)3009 4051 w (\347)3009 4151 w 2512 2902 2491 2856 Dl 2490 2856 2469 2902 Dl (\347)2491 2956 w (\347)2491 3052 w (\347)2491 3152 w (\347)2491 3252 w (\347)2491 3352 w (\347)2491 3452 w (\347)2491 3552 w (\347)2491 3652 w (\347)2491 3752 w (\347)2491 3852 w (\347)2491 3952 w (\347)2491 4052 w (\347)2491 4152 w 3482 3913 3528 3892 Dl 3527 3891 3481 3870 Dl 10 S1 f (_ _________________________)1 1296 1 2232 3892 t 10 B f (2)2725 4152 w 2848 3448 46 46 De 2856 3448 31 31 De 2864 3448 15 15 De 3299 3933 3346 3996 Dl 3251 3871 3298 3932 Dl 3203 3812 3250 3871 Dl 3155 3755 3202 3812 Dl 3107 3699 3154 3754 Dl 3059 3646 3106 3699 Dl 3011 3593 3058 3645 Dl 2963 3541 3010 3592 Dl 2915 3492 2962 3541 Dl 2867 3444 2914 3492 Dl 2819 3396 2866 3443 Dl 2772 3350 2819 3396 Dl 2724 3305 2771 3350 Dl 2676 3261 2723 3305 Dl 2628 3218 2675 3261 Dl 2580 3176 2627 3218 Dl 2532 3135 2579 3176 Dl 2484 3094 2531 3134 Dl 2436 3055 2483 3094 Dl 2388 3016 2435 3054 Dl 10 I f (b)2516 2906 w 10 R f (= 10)1 181 1 2591 2906 t 10 I f (a)3272 3842 w 10 R f (= 18)1 181 1 3347 3842 t (\(15,4\))2225 3992 w 10 S1 f (_ __________)1 518 1 4190 3882 t (_ __________)1 518 1 4190 3903 t (_ __________)1 518 1 4190 3892 t 10 S f (\347)4439 3733 w (\347)4439 3751 w (\347)4439 3851 w (\347)4439 3951 w (\347)4439 4051 w (\347)4439 4151 w (\347)4459 3733 w (\347)4459 3751 w (\347)4459 3851 w (\347)4459 3951 w (\347)4459 4051 w (\347)4459 4151 w (\347)4449 3733 w (\347)4449 3751 w (\347)4449 3851 w (\347)4449 3951 w (\347)4449 4051 w (\347)4449 4151 w 10 S1 f (_ __________)1 518 1 4190 3364 t (_ __________)1 518 1 4190 3384 t (_ __________)1 518 1 4190 3374 t 10 S f (\347)3920 2956 w (\347)3920 3015 w (\347)3920 3115 w (\347)3941 2956 w (\347)3941 3015 w (\347)3941 3115 w (\347)3931 2956 w (\347)3931 3015 w (\347)3931 3115 w 4708 3917 4708 3866 Dl 4424 3633 4475 3633 Dl 4190 3399 4190 3348 Dl 3906 3633 3957 3633 Dl 4190 3917 4190 3866 Dl 4190 3399 4190 3348 Dl 3906 3115 3957 3115 Dl 10 S1 f (_ ____________________)1 1036 1 3672 3374 t 10 S f (\347)4449 3215 w (\347)4449 3251 w (\347)4449 3351 w (\347)4449 3451 w (\347)4449 3551 w (\347)4449 3651 w (\347)4449 3751 w (\347)4449 3851 w (\347)4449 3951 w (\347)4449 4051 w (\347)4449 4151 w 3952 2902 3931 2856 Dl 3930 2856 3909 2902 Dl (\347)3931 2956 w (\347)3931 3052 w (\347)3931 3152 w (\347)3931 3252 w (\347)3931 3352 w (\347)3931 3452 w (\347)3931 3552 w (\347)3931 3652 w (\347)3931 3752 w (\347)3931 3852 w (\347)3931 3952 w (\347)3931 4052 w (\347)3931 4152 w 4922 3913 4968 3892 Dl 4967 3891 4921 3870 Dl 10 S1 f (_ _________________________)1 1296 1 3672 3892 t 10 B f (3)4165 4152 w 14 R f (X)4140 3703 w 10 S1 f (_ __________)1 518 1 1310 5454 t (_ __________)1 518 1 1310 5475 t (_ __________)1 518 1 1310 5464 t (_ __________)1 518 1 1310 4936 t (_ __________)1 518 1 1310 4956 t (_ __________)1 518 1 1310 4946 t 10 S f (\347)1040 4528 w (\347)1040 4587 w (\347)1040 4687 w (\347)1061 4528 w (\347)1061 4587 w (\347)1061 4687 w (\347)1051 4528 w (\347)1051 4587 w (\347)1051 4687 w 1828 5489 1828 5438 Dl 1544 5205 1595 5205 Dl 1310 4971 1310 4920 Dl 1026 5205 1077 5205 Dl 1310 5489 1310 5438 Dl 1310 4971 1310 4920 Dl 1026 4687 1077 4687 Dl 10 S1 f (_ ____________________)1 1036 1 792 4946 t 10 S f (\347)1569 4787 w (\347)1569 4823 w (\347)1569 4923 w (\347)1569 5023 w (\347)1569 5123 w (\347)1569 5223 w (\347)1569 5323 w (\347)1569 5423 w (\347)1569 5523 w (\347)1569 5623 w (\347)1569 5723 w 1072 4474 1051 4428 Dl 1050 4428 1029 4474 Dl (\347)1051 4528 w (\347)1051 4624 w (\347)1051 4724 w (\347)1051 4824 w (\347)1051 4924 w (\347)1051 5024 w (\347)1051 5124 w (\347)1051 5224 w (\347)1051 5324 w (\347)1051 5424 w (\347)1051 5524 w (\347)1051 5624 w (\347)1051 5724 w 2042 5485 2088 5464 Dl 2087 5463 2041 5442 Dl 10 S1 f (_ _________________________)1 1296 1 792 5464 t 10 B f (4)1285 5724 w 14 R f (X)1260 5275 w 10 S1 f (_ _____)1 259 1 3269 5454 t (_ _____)1 259 1 3269 5475 t (_ _____)1 259 1 3269 5464 t 10 S f (\347)2999 4787 w (\347)2999 4805 w (\347)2999 4905 w (\347)2999 5005 w (\347)2999 5105 w (\347)2999 5205 w (\347)3019 4787 w (\347)3019 4805 w (\347)3019 4905 w (\347)3019 5005 w (\347)3019 5105 w (\347)3019 5205 w (\347)3009 4787 w (\347)3009 4805 w (\347)3009 4905 w (\347)3009 5005 w (\347)3009 5105 w (\347)3009 5205 w 10 S1 f (_ __________)1 518 1 2750 4936 t (_ __________)1 518 1 2750 4956 t (_ __________)1 518 1 2750 4946 t 10 S f (\347)2480 4787 w (\347)2480 4805 w (\347)2480 4905 w (\347)2480 5005 w (\347)2480 5105 w (\347)2480 5205 w (\347)2501 4787 w (\347)2501 4805 w (\347)2501 4905 w (\347)2501 5005 w (\347)2501 5105 w (\347)2501 5205 w (\347)2491 4787 w (\347)2491 4805 w (\347)2491 4905 w (\347)2491 5005 w (\347)2491 5105 w (\347)2491 5205 w 3268 5489 3268 5438 Dl 2984 5205 3035 5205 Dl 2750 4971 2750 4920 Dl 2466 5205 2517 5205 Dl 2750 5489 2750 5438 Dl 2750 4971 2750 4920 Dl 2466 4687 2517 4687 Dl 10 S1 f (_ ____________________)1 1036 1 2232 4946 t 10 S f (\347)3009 4787 w (\347)3009 4823 w (\347)3009 4923 w (\347)3009 5023 w (\347)3009 5123 w (\347)3009 5223 w (\347)3009 5323 w (\347)3009 5423 w (\347)3009 5523 w (\347)3009 5623 w (\347)3009 5723 w 2512 4474 2491 4428 Dl 2490 4428 2469 4474 Dl (\347)2491 4528 w (\347)2491 4624 w (\347)2491 4724 w (\347)2491 4824 w (\347)2491 4924 w (\347)2491 5024 w (\347)2491 5124 w (\347)2491 5224 w (\347)2491 5324 w (\347)2491 5424 w (\347)2491 5524 w (\347)2491 5624 w (\347)2491 5724 w 3482 5485 3528 5464 Dl 3527 5463 3481 5442 Dl 10 S1 f (_ _________________________)1 1296 1 2232 5464 t 10 B f (5)2725 5724 w 2932 5099 46 46 De 2940 5099 31 31 De 2947 5099 15 15 De 3314 5504 3362 5568 Dl 3265 5443 3313 5504 Dl 3216 5383 3264 5442 Dl 3167 5326 3215 5383 Dl 3119 5270 3167 5325 Dl 3070 5217 3118 5270 Dl 3021 5166 3069 5217 Dl 2972 5115 3020 5165 Dl 2924 5067 2972 5115 Dl 2875 5020 2923 5067 Dl 2826 4975 2874 5020 Dl 2778 4930 2826 4974 Dl 2729 4887 2777 4930 Dl 2680 4844 2728 4886 Dl 2631 4804 2679 4844 Dl 2583 4764 2631 4803 Dl 2534 4725 2582 4763 Dl 2485 4688 2533 4725 Dl 2436 4651 2484 4687 Dl 2388 4614 2436 4650 Dl 10 I f (b)2516 4478 w 10 R f (= 8)1 131 1 2591 4478 t 10 I f (a)3272 5414 w 10 R f (= 11)1 181 1 3347 5414 t (\(8,4\))2275 5564 w 10 S1 f (_ __________)1 518 1 4190 5454 t (_ __________)1 518 1 4190 5475 t (_ __________)1 518 1 4190 5464 t 10 S f (\347)4439 4787 w (\347)4439 4805 w (\347)4439 4905 w (\347)4439 5005 w (\347)4439 5105 w (\347)4439 5205 w (\347)4459 4787 w (\347)4459 4805 w (\347)4459 4905 w (\347)4459 5005 w (\347)4459 5105 w (\347)4459 5205 w (\347)4449 4787 w (\347)4449 4805 w (\347)4449 4905 w (\347)4449 5005 w (\347)4449 5105 w (\347)4449 5205 w 10 S1 f (_ __________)1 518 1 4190 4936 t (_ __________)1 518 1 4190 4956 t (_ __________)1 518 1 4190 4946 t 10 S f (\347)3920 4787 w (\347)3920 4805 w (\347)3920 4905 w (\347)3920 5005 w (\347)3920 5105 w (\347)3920 5205 w (\347)3941 4787 w (\347)3941 4805 w (\347)3941 4905 w (\347)3941 5005 w (\347)3941 5105 w (\347)3941 5205 w (\347)3931 4787 w (\347)3931 4805 w (\347)3931 4905 w (\347)3931 5005 w (\347)3931 5105 w (\347)3931 5205 w 4708 5489 4708 5438 Dl 4424 5205 4475 5205 Dl 4190 4971 4190 4920 Dl 3906 5205 3957 5205 Dl 4190 5489 4190 5438 Dl 4190 4971 4190 4920 Dl 3906 4687 3957 4687 Dl 10 S1 f (_ ____________________)1 1036 1 3672 4946 t 10 S f (\347)4449 4787 w (\347)4449 4823 w (\347)4449 4923 w (\347)4449 5023 w (\347)4449 5123 w (\347)4449 5223 w (\347)4449 5323 w (\347)4449 5423 w (\347)4449 5523 w (\347)4449 5623 w (\347)4449 5723 w 3952 4474 3931 4428 Dl 3930 4428 3909 4474 Dl (\347)3931 4528 w (\347)3931 4624 w (\347)3931 4724 w (\347)3931 4824 w (\347)3931 4924 w (\347)3931 5024 w (\347)3931 5124 w (\347)3931 5224 w (\347)3931 5324 w (\347)3931 5424 w (\347)3931 5524 w (\347)3931 5624 w (\347)3931 5724 w 4922 5485 4968 5464 Dl 4967 5463 4921 5442 Dl 10 S1 f (_ _________________________)1 1296 1 3672 5464 t 10 B f (6)4165 5724 w 4337 5035 46 46 De 4345 5035 31 31 De 4352 5035 15 15 De 4701 5482 4746 5568 Dl 4655 5405 4700 5482 Dl 4609 5335 4654 5405 Dl 4563 5270 4608 5334 Dl 4517 5210 4562 5269 Dl 4471 5154 4516 5209 Dl 4425 5103 4470 5154 Dl 4379 5054 4424 5102 Dl 4333 5008 4378 5053 Dl 4287 4965 4332 5007 Dl 4241 4925 4286 4965 Dl 4195 4886 4240 4924 Dl 4149 4851 4194 4886 Dl 4104 4817 4149 4850 Dl 4058 4784 4103 4816 Dl 4012 4754 4057 4784 Dl 3966 4726 4011 4754 Dl 3920 4699 3965 4725 Dl 3874 4674 3919 4699 Dl 3828 4650 3873 4673 Dl 10 I f (b)3956 4478 w 10 R f (= 4)1 131 1 4031 4478 t 10 I f (a)4762 5414 w 10 R f (= 4)1 131 1 4837 5414 t (\(2,2\))3715 5564 w cleartomark showpage saveobj restore %%PageBoundingBox: 61 204 514 764 %%EndPage: 9 9 %%Page: 10 10 %%PageBoundingBox: (atend) /saveobj save def mark 10 pagesetup 10 R f (- 10 -)2 216 1 2772 480 t 10 S1 f (_ __________)1 518 1 1310 1866 t (_ __________)1 518 1 1310 1887 t (_ __________)1 518 1 1310 1876 t 10 S f (\347)1559 1717 w (\347)1559 1735 w (\347)1559 1835 w (\347)1559 1935 w (\347)1559 2035 w (\347)1559 2135 w (\347)1579 1717 w (\347)1579 1735 w (\347)1579 1835 w (\347)1579 1935 w (\347)1579 2035 w (\347)1579 2135 w (\347)1569 1717 w (\347)1569 1735 w (\347)1569 1835 w (\347)1569 1935 w (\347)1569 2035 w (\347)1569 2135 w 10 S1 f (_ __________)1 518 1 1310 1348 t (_ __________)1 518 1 1310 1368 t (_ __________)1 518 1 1310 1358 t 10 S f (\347)1040 1199 w (\347)1040 1217 w (\347)1040 1317 w (\347)1040 1417 w (\347)1040 1517 w (\347)1040 1617 w (\347)1061 1199 w (\347)1061 1217 w (\347)1061 1317 w (\347)1061 1417 w (\347)1061 1517 w (\347)1061 1617 w (\347)1051 1199 w (\347)1051 1217 w (\347)1051 1317 w (\347)1051 1417 w (\347)1051 1517 w (\347)1051 1617 w 1828 1901 1828 1850 Dl 1544 1617 1595 1617 Dl 1310 1383 1310 1332 Dl 1026 1617 1077 1617 Dl 1310 1901 1310 1850 Dl 1310 1383 1310 1332 Dl 1026 1099 1077 1099 Dl 10 S1 f (_ ____________________)1 1036 1 792 1358 t 10 S f (\347)1569 1199 w (\347)1569 1235 w (\347)1569 1335 w (\347)1569 1435 w (\347)1569 1535 w (\347)1569 1635 w (\347)1569 1735 w (\347)1569 1835 w (\347)1569 1935 w (\347)1569 2035 w (\347)1569 2135 w 1072 886 1051 840 Dl 1050 840 1029 886 Dl (\347)1051 940 w (\347)1051 1036 w (\347)1051 1136 w (\347)1051 1236 w (\347)1051 1336 w (\347)1051 1436 w (\347)1051 1536 w (\347)1051 1636 w (\347)1051 1736 w (\347)1051 1836 w (\347)1051 1936 w (\347)1051 2036 w (\347)1051 2136 w 2042 1897 2088 1876 Dl 2087 1875 2041 1854 Dl 10 S1 f (_ _________________________)1 1296 1 792 1876 t 10 B f (7)1285 2136 w 1368 1424 46 46 De 1376 1424 31 31 De 1383 1424 15 15 De 1782 1908 1825 1980 Dl 1738 1840 1781 1907 Dl 1694 1778 1737 1840 Dl 1650 1718 1693 1777 Dl 1606 1662 1649 1717 Dl 1562 1608 1605 1661 Dl 1518 1558 1561 1608 Dl 1475 1510 1518 1558 Dl 1431 1463 1474 1509 Dl 1387 1419 1430 1463 Dl 1343 1376 1386 1418 Dl 1299 1335 1342 1376 Dl 1255 1295 1298 1334 Dl 1211 1256 1254 1294 Dl 1167 1219 1210 1256 Dl 1123 1182 1166 1218 Dl 1080 1147 1123 1182 Dl 1036 1113 1079 1147 Dl 992 1079 1035 1112 Dl 948 1047 991 1079 Dl 10 I f (b)1076 890 w 10 R f (= 6)1 131 1 1151 890 t 10 I f (a)1832 1826 w 10 R f (= 11)1 181 1 1907 1826 t (\(9,2\))835 1976 w 10 S1 f (_ __________)1 518 1 2750 1866 t (_ __________)1 518 1 2750 1887 t (_ __________)1 518 1 2750 1876 t (_ __________)1 518 1 2750 1348 t (_ __________)1 518 1 2750 1368 t (_ __________)1 518 1 2750 1358 t 10 S f (\347)2480 1199 w (\347)2480 1217 w (\347)2480 1317 w (\347)2480 1417 w (\347)2480 1517 w (\347)2480 1617 w (\347)2501 1199 w (\347)2501 1217 w (\347)2501 1317 w (\347)2501 1417 w (\347)2501 1517 w (\347)2501 1617 w (\347)2491 1199 w (\347)2491 1217 w (\347)2491 1317 w (\347)2491 1417 w (\347)2491 1517 w (\347)2491 1617 w 3268 1901 3268 1850 Dl 2984 1617 3035 1617 Dl 2750 1383 2750 1332 Dl 2466 1617 2517 1617 Dl 2750 1901 2750 1850 Dl 2750 1383 2750 1332 Dl 2466 1099 2517 1099 Dl 10 S1 f (_ ____________________)1 1036 1 2232 1358 t 10 S f (\347)3009 1199 w (\347)3009 1235 w (\347)3009 1335 w (\347)3009 1435 w (\347)3009 1535 w (\347)3009 1635 w (\347)3009 1735 w (\347)3009 1835 w (\347)3009 1935 w (\347)3009 2035 w (\347)3009 2135 w 2512 886 2491 840 Dl 2490 840 2469 886 Dl (\347)2491 940 w (\347)2491 1036 w (\347)2491 1136 w (\347)2491 1236 w (\347)2491 1336 w (\347)2491 1436 w (\347)2491 1536 w (\347)2491 1636 w (\347)2491 1736 w (\347)2491 1836 w (\347)2491 1936 w (\347)2491 2036 w (\347)2491 2136 w 3482 1897 3528 1876 Dl 3527 1875 3481 1854 Dl 10 S1 f (_ _________________________)1 1296 1 2232 1876 t 10 B f (8)2725 2136 w 14 R f (X)2700 1687 w 10 S1 f (_ _____)1 259 1 4709 1866 t (_ _____)1 259 1 4709 1887 t (_ _____)1 259 1 4709 1876 t 10 S f (\347)4439 1199 w (\347)4439 1217 w (\347)4439 1317 w (\347)4439 1417 w (\347)4439 1517 w (\347)4439 1617 w (\347)4459 1199 w (\347)4459 1217 w (\347)4459 1317 w (\347)4459 1417 w (\347)4459 1517 w (\347)4459 1617 w (\347)4449 1199 w (\347)4449 1217 w (\347)4449 1317 w (\347)4449 1417 w (\347)4449 1517 w (\347)4449 1617 w 10 S1 f (_ __________)1 518 1 3672 1348 t (_ __________)1 518 1 3672 1368 t (_ __________)1 518 1 3672 1358 t 10 S f (\347)3920 1199 w (\347)3920 1217 w (\347)3920 1317 w (\347)3920 1417 w (\347)3920 1517 w (\347)3920 1617 w (\347)3941 1199 w (\347)3941 1217 w (\347)3941 1317 w (\347)3941 1417 w (\347)3941 1517 w (\347)3941 1617 w (\347)3931 1199 w (\347)3931 1217 w (\347)3931 1317 w (\347)3931 1417 w (\347)3931 1517 w (\347)3931 1617 w 4708 1901 4708 1850 Dl 4424 1617 4475 1617 Dl 4190 1383 4190 1332 Dl 3906 1617 3957 1617 Dl 4190 1901 4190 1850 Dl 4190 1383 4190 1332 Dl 3906 1099 3957 1099 Dl 10 S1 f (_ ____________________)1 1036 1 3672 1358 t 10 S f (\347)4449 1199 w (\347)4449 1235 w (\347)4449 1335 w (\347)4449 1435 w (\347)4449 1535 w (\347)4449 1635 w (\347)4449 1735 w (\347)4449 1835 w (\347)4449 1935 w (\347)4449 2035 w (\347)4449 2135 w 3952 886 3931 840 Dl 3930 840 3909 886 Dl (\347)3931 940 w (\347)3931 1036 w (\347)3931 1136 w (\347)3931 1236 w (\347)3931 1336 w (\347)3931 1436 w (\347)3931 1536 w (\347)3931 1636 w (\347)3931 1736 w (\347)3931 1836 w (\347)3931 1936 w (\347)3931 2036 w (\347)3931 2136 w 4922 1897 4968 1876 Dl 4967 1875 4921 1854 Dl 10 S1 f (_ _________________________)1 1296 1 3672 1876 t 10 B f (9)4165 2136 w 14 R f (X)4140 1687 w 10 S1 f (_ __________)1 518 1 1310 3438 t (_ __________)1 518 1 1310 3459 t (_ __________)1 518 1 1310 3448 t 10 S f (\347)1559 2771 w (\347)1559 2789 w (\347)1559 2889 w (\347)1559 2989 w (\347)1559 3089 w (\347)1559 3189 w (\347)1579 2771 w (\347)1579 2789 w (\347)1579 2889 w (\347)1579 2989 w (\347)1579 3089 w (\347)1579 3189 w (\347)1569 2771 w (\347)1569 2789 w (\347)1569 2889 w (\347)1569 2989 w (\347)1569 3089 w (\347)1569 3189 w 10 S1 f (_ __________)1 518 1 792 2920 t (_ __________)1 518 1 792 2940 t (_ __________)1 518 1 792 2930 t 10 S f (\347)1040 2771 w (\347)1040 2789 w (\347)1040 2889 w (\347)1040 2989 w (\347)1040 3089 w (\347)1040 3189 w (\347)1061 2771 w (\347)1061 2789 w (\347)1061 2889 w (\347)1061 2989 w (\347)1061 3089 w (\347)1061 3189 w (\347)1051 2771 w (\347)1051 2789 w (\347)1051 2889 w (\347)1051 2989 w (\347)1051 3089 w (\347)1051 3189 w 1828 3473 1828 3422 Dl 1544 3189 1595 3189 Dl 1310 2955 1310 2904 Dl 1026 3189 1077 3189 Dl 1310 3473 1310 3422 Dl 1310 2955 1310 2904 Dl 1026 2671 1077 2671 Dl 10 S1 f (_ ____________________)1 1036 1 792 2930 t 10 S f (\347)1569 2771 w (\347)1569 2807 w (\347)1569 2907 w (\347)1569 3007 w (\347)1569 3107 w (\347)1569 3207 w (\347)1569 3307 w (\347)1569 3407 w (\347)1569 3507 w (\347)1569 3607 w (\347)1569 3707 w 1072 2458 1051 2412 Dl 1050 2412 1029 2458 Dl (\347)1051 2512 w (\347)1051 2608 w (\347)1051 2708 w (\347)1051 2808 w (\347)1051 2908 w (\347)1051 3008 w (\347)1051 3108 w (\347)1051 3208 w (\347)1051 3308 w (\347)1051 3408 w (\347)1051 3508 w (\347)1051 3608 w (\347)1051 3708 w 2042 3469 2088 3448 Dl 2087 3447 2041 3426 Dl 10 S1 f (_ _________________________)1 1296 1 792 3448 t 10 B f (10)1260 3708 w 1528 3162 46 46 De 1536 3162 31 31 De 1544 3162 15 15 De 1867 3497 1915 3552 Dl 1818 3443 1866 3497 Dl 1770 3389 1818 3442 Dl 1721 3337 1769 3389 Dl 1673 3286 1721 3337 Dl 1625 3236 1673 3286 Dl 1576 3187 1624 3236 Dl 1528 3139 1576 3187 Dl 1480 3091 1528 3138 Dl 1431 3045 1479 3091 Dl 1383 2999 1431 3044 Dl 1334 2953 1382 2998 Dl 1286 2909 1334 2953 Dl 1238 2866 1286 2909 Dl 1189 2823 1237 2865 Dl 1141 2781 1189 2823 Dl 1093 2740 1141 2781 Dl 1044 2699 1092 2739 Dl 996 2660 1044 2699 Dl 947 2620 995 2659 Dl 10 I f (b)1076 2462 w 10 R f (= 14)1 181 1 1151 2462 t 10 I f (a)1832 3398 w 10 R f (= 15)1 181 1 1907 3398 t (\(10,9\))785 3548 w 10 S1 f (_ __________)1 518 1 2750 3438 t (_ __________)1 518 1 2750 3459 t (_ __________)1 518 1 2750 3448 t 10 S f (\347)2999 3289 w (\347)2999 3307 w (\347)2999 3407 w (\347)2999 3507 w (\347)2999 3607 w (\347)2999 3707 w (\347)3019 3289 w (\347)3019 3307 w (\347)3019 3407 w (\347)3019 3507 w (\347)3019 3607 w (\347)3019 3707 w (\347)3009 3289 w (\347)3009 3307 w (\347)3009 3407 w (\347)3009 3507 w (\347)3009 3607 w (\347)3009 3707 w 10 S1 f (_ __________)1 518 1 2232 2920 t (_ __________)1 518 1 2232 2940 t (_ __________)1 518 1 2232 2930 t 10 S f (\347)2480 2771 w (\347)2480 2789 w (\347)2480 2889 w (\347)2480 2989 w (\347)2480 3089 w (\347)2480 3189 w (\347)2501 2771 w (\347)2501 2789 w (\347)2501 2889 w (\347)2501 2989 w (\347)2501 3089 w (\347)2501 3189 w (\347)2491 2771 w (\347)2491 2789 w (\347)2491 2889 w (\347)2491 2989 w (\347)2491 3089 w (\347)2491 3189 w 3268 3473 3268 3422 Dl 2984 3189 3035 3189 Dl 2750 2955 2750 2904 Dl 2466 3189 2517 3189 Dl 2750 3473 2750 3422 Dl 2750 2955 2750 2904 Dl 2466 2671 2517 2671 Dl 10 S1 f (_ ____________________)1 1036 1 2232 2930 t 10 S f (\347)3009 2771 w (\347)3009 2807 w (\347)3009 2907 w (\347)3009 3007 w (\347)3009 3107 w (\347)3009 3207 w (\347)3009 3307 w (\347)3009 3407 w (\347)3009 3507 w (\347)3009 3607 w (\347)3009 3707 w 2512 2458 2491 2412 Dl 2490 2412 2469 2458 Dl (\347)2491 2512 w (\347)2491 2608 w (\347)2491 2708 w (\347)2491 2808 w (\347)2491 2908 w (\347)2491 3008 w (\347)2491 3108 w (\347)2491 3208 w (\347)2491 3308 w (\347)2491 3408 w (\347)2491 3508 w (\347)2491 3608 w (\347)2491 3708 w 3482 3469 3528 3448 Dl 3527 3447 3481 3426 Dl 10 S1 f (_ _________________________)1 1296 1 2232 3448 t 10 B f (11)2700 3708 w 2834 3082 46 46 De 2842 3082 31 31 De 2849 3082 15 15 De 2984 3287 3009 3448 Dl 2958 3222 2983 3286 Dl 2932 3175 2957 3222 Dl 2906 3138 2931 3175 Dl 2880 3106 2905 3137 Dl 2855 3078 2880 3105 Dl 2829 3055 2854 3078 Dl 2803 3034 2828 3054 Dl 2777 3016 2802 3034 Dl 2751 3000 2776 3015 Dl 2725 2986 2750 2999 Dl 2699 2973 2724 2985 Dl 2673 2963 2698 2973 Dl 2647 2955 2672 2963 Dl 2621 2947 2646 2954 Dl 2595 2941 2620 2946 Dl 2569 2936 2594 2940 Dl 2543 2933 2568 2936 Dl 2518 2931 2543 2932 Dl 2492 2931 2517 2931 Dl 10 I f (b)2516 2462 w 10 R f (= 1)1 131 1 2591 2462 t 10 I f (a)3322 3398 w 10 R f (= 1)1 131 1 3397 3398 t (\(0,0\))2275 3548 w 10 S1 f (_ __________)1 518 1 4190 3438 t (_ __________)1 518 1 4190 3459 t (_ __________)1 518 1 4190 3448 t (_ __________)1 518 1 3672 2920 t (_ __________)1 518 1 3672 2940 t (_ __________)1 518 1 3672 2930 t 10 S f (\347)3920 2771 w (\347)3920 2789 w (\347)3920 2889 w (\347)3920 2989 w (\347)3920 3089 w (\347)3920 3189 w (\347)3941 2771 w (\347)3941 2789 w (\347)3941 2889 w (\347)3941 2989 w (\347)3941 3089 w (\347)3941 3189 w (\347)3931 2771 w (\347)3931 2789 w (\347)3931 2889 w (\347)3931 2989 w (\347)3931 3089 w (\347)3931 3189 w 4708 3473 4708 3422 Dl 4424 3189 4475 3189 Dl 4190 2955 4190 2904 Dl 3906 3189 3957 3189 Dl 4190 3473 4190 3422 Dl 4190 2955 4190 2904 Dl 3906 2671 3957 2671 Dl 10 S1 f (_ ____________________)1 1036 1 3672 2930 t 10 S f (\347)4449 2771 w (\347)4449 2807 w (\347)4449 2907 w (\347)4449 3007 w (\347)4449 3107 w (\347)4449 3207 w (\347)4449 3307 w (\347)4449 3407 w (\347)4449 3507 w (\347)4449 3607 w (\347)4449 3707 w 3952 2458 3931 2412 Dl 3930 2412 3909 2458 Dl (\347)3931 2512 w (\347)3931 2608 w (\347)3931 2708 w (\347)3931 2808 w (\347)3931 2908 w (\347)3931 3008 w (\347)3931 3108 w (\347)3931 3208 w (\347)3931 3308 w (\347)3931 3408 w (\347)3931 3508 w (\347)3931 3608 w (\347)3931 3708 w 4922 3469 4968 3448 Dl 4967 3447 4921 3426 Dl 10 S1 f (_ _________________________)1 1296 1 3672 3448 t 10 B f (12)4140 3708 w 4011 3034 46 46 De 4019 3034 31 31 De 4027 3034 15 15 De 4391 3599 4420 3707 Dl 4361 3518 4390 3599 Dl 4332 3450 4361 3518 Dl 4302 3391 4331 3450 Dl 4272 3338 4301 3390 Dl 4243 3290 4272 3338 Dl 4213 3245 4242 3289 Dl 4184 3205 4213 3245 Dl 4154 3166 4183 3204 Dl 4124 3131 4153 3166 Dl 4095 3097 4124 3130 Dl 4065 3064 4094 3096 Dl 4035 3034 4064 3064 Dl 4006 3006 4035 3034 Dl 3976 2978 4005 3005 Dl 3946 2951 3975 2977 Dl 3917 2926 3946 2951 Dl 3887 2902 3916 2926 Dl 3857 2878 3886 2901 Dl 3828 2856 3857 2878 Dl 10 I f (b)3956 2462 w 10 R f (= 3)1 131 1 4031 2462 t 10 I f (a)4762 3398 w 10 R f (= 4)1 131 1 4837 3398 t (\(3,1\))3715 3548 w 10 S1 f (_ __________)1 518 1 792 5010 t (_ __________)1 518 1 792 5031 t (_ __________)1 518 1 792 5020 t (_ __________)1 518 1 792 4492 t (_ __________)1 518 1 792 4512 t (_ __________)1 518 1 792 4502 t 10 S f (\347)1040 4343 w (\347)1040 4361 w (\347)1040 4461 w (\347)1040 4561 w (\347)1040 4661 w (\347)1040 4761 w (\347)1061 4343 w (\347)1061 4361 w (\347)1061 4461 w (\347)1061 4561 w (\347)1061 4661 w (\347)1061 4761 w (\347)1051 4343 w (\347)1051 4361 w (\347)1051 4461 w (\347)1051 4561 w (\347)1051 4661 w (\347)1051 4761 w 1828 5045 1828 4994 Dl 1544 4761 1595 4761 Dl 1310 4527 1310 4476 Dl 1026 4761 1077 4761 Dl 1310 5045 1310 4994 Dl 1310 4527 1310 4476 Dl 1026 4243 1077 4243 Dl 10 S1 f (_ ____________________)1 1036 1 792 4502 t 10 S f (\347)1569 4343 w (\347)1569 4379 w (\347)1569 4479 w (\347)1569 4579 w (\347)1569 4679 w (\347)1569 4779 w (\347)1569 4879 w (\347)1569 4979 w (\347)1569 5079 w (\347)1569 5179 w (\347)1569 5279 w 1072 4030 1051 3984 Dl 1050 3984 1029 4030 Dl (\347)1051 4084 w (\347)1051 4180 w (\347)1051 4280 w (\347)1051 4380 w (\347)1051 4480 w (\347)1051 4580 w (\347)1051 4680 w (\347)1051 4780 w (\347)1051 4880 w (\347)1051 4980 w (\347)1051 5080 w (\347)1051 5180 w (\347)1051 5280 w 2042 5041 2088 5020 Dl 2087 5019 2041 4998 Dl 10 S1 f (_ _________________________)1 1296 1 792 5020 t 10 B f (13)1260 5280 w 14 R f (?)1279 4831 w 10 S1 f (_ _____)1 259 1 3269 5010 t (_ _____)1 259 1 3269 5031 t (_ _____)1 259 1 3269 5020 t 10 S f (\347)2999 4343 w (\347)2999 4361 w (\347)2999 4461 w (\347)2999 4561 w (\347)2999 4661 w (\347)2999 4761 w (\347)3019 4343 w (\347)3019 4361 w (\347)3019 4461 w (\347)3019 4561 w (\347)3019 4661 w (\347)3019 4761 w (\347)3009 4343 w (\347)3009 4361 w (\347)3009 4461 w (\347)3009 4561 w (\347)3009 4661 w (\347)3009 4761 w (\347)2480 4343 w (\347)2480 4361 w (\347)2480 4461 w (\347)2480 4561 w (\347)2480 4661 w (\347)2480 4761 w (\347)2501 4343 w (\347)2501 4361 w (\347)2501 4461 w (\347)2501 4561 w (\347)2501 4661 w (\347)2501 4761 w (\347)2491 4343 w (\347)2491 4361 w (\347)2491 4461 w (\347)2491 4561 w (\347)2491 4661 w (\347)2491 4761 w 3268 5045 3268 4994 Dl 2984 4761 3035 4761 Dl 2750 4527 2750 4476 Dl 2466 4761 2517 4761 Dl 2750 5045 2750 4994 Dl 2750 4527 2750 4476 Dl 2466 4243 2517 4243 Dl 10 S1 f (_ ____________________)1 1036 1 2232 4502 t 10 S f (\347)3009 4343 w (\347)3009 4379 w (\347)3009 4479 w (\347)3009 4579 w (\347)3009 4679 w (\347)3009 4779 w (\347)3009 4879 w (\347)3009 4979 w (\347)3009 5079 w (\347)3009 5179 w (\347)3009 5279 w 2512 4030 2491 3984 Dl 2490 3984 2469 4030 Dl (\347)2491 4084 w (\347)2491 4180 w (\347)2491 4280 w (\347)2491 4380 w (\347)2491 4480 w (\347)2491 4580 w (\347)2491 4680 w (\347)2491 4780 w (\347)2491 4880 w (\347)2491 4980 w (\347)2491 5080 w (\347)2491 5180 w (\347)2491 5280 w 3482 5041 3528 5020 Dl 3527 5019 3481 4998 Dl 10 S1 f (_ _________________________)1 1296 1 2232 5020 t 10 B f (14)2700 5280 w 14 R f (X)2700 4831 w 10 S1 f (_ __________)1 518 1 4190 5010 t (_ __________)1 518 1 4190 5031 t (_ __________)1 518 1 4190 5020 t 10 S f (\347)4439 4343 w (\347)4439 4361 w (\347)4439 4461 w (\347)4439 4561 w (\347)4439 4661 w (\347)4439 4761 w (\347)4459 4343 w (\347)4459 4361 w (\347)4459 4461 w (\347)4459 4561 w (\347)4459 4661 w (\347)4459 4761 w (\347)4449 4343 w (\347)4449 4361 w (\347)4449 4461 w (\347)4449 4561 w (\347)4449 4661 w (\347)4449 4761 w (\347)3920 4343 w (\347)3920 4361 w (\347)3920 4461 w (\347)3920 4561 w (\347)3920 4661 w (\347)3920 4761 w (\347)3941 4343 w (\347)3941 4361 w (\347)3941 4461 w (\347)3941 4561 w (\347)3941 4661 w (\347)3941 4761 w (\347)3931 4343 w (\347)3931 4361 w (\347)3931 4461 w (\347)3931 4561 w (\347)3931 4661 w (\347)3931 4761 w 4708 5045 4708 4994 Dl 4424 4761 4475 4761 Dl 4190 4527 4190 4476 Dl 3906 4761 3957 4761 Dl 4190 5045 4190 4994 Dl 4190 4527 4190 4476 Dl 3906 4243 3957 4243 Dl 10 S1 f (_ ____________________)1 1036 1 3672 4502 t 10 S f (\347)4449 4343 w (\347)4449 4379 w (\347)4449 4479 w (\347)4449 4579 w (\347)4449 4679 w (\347)4449 4779 w (\347)4449 4879 w (\347)4449 4979 w (\347)4449 5079 w (\347)4449 5179 w (\347)4449 5279 w 3952 4030 3931 3984 Dl 3930 3984 3909 4030 Dl (\347)3931 4084 w (\347)3931 4180 w (\347)3931 4280 w (\347)3931 4380 w (\347)3931 4480 w (\347)3931 4580 w (\347)3931 4680 w (\347)3931 4780 w (\347)3931 4880 w (\347)3931 4980 w (\347)3931 5080 w (\347)3931 5180 w (\347)3931 5280 w 4922 5041 4968 5020 Dl 4967 5019 4921 4998 Dl 10 S1 f (_ _________________________)1 1296 1 3672 5020 t 10 B f (15)4140 5280 w 14 R f (X)4140 4831 w 10 S1 f (_ __________)1 518 1 1310 6582 t (_ __________)1 518 1 1310 6603 t (_ __________)1 518 1 1310 6592 t 10 S f (\347)1559 6433 w (\347)1559 6451 w (\347)1559 6551 w (\347)1559 6651 w (\347)1559 6751 w (\347)1559 6851 w (\347)1579 6433 w (\347)1579 6451 w (\347)1579 6551 w (\347)1579 6651 w (\347)1579 6751 w (\347)1579 6851 w (\347)1569 6433 w (\347)1569 6451 w (\347)1569 6551 w (\347)1569 6651 w (\347)1569 6751 w (\347)1569 6851 w (\347)1040 5915 w (\347)1040 5933 w (\347)1040 6033 w (\347)1040 6133 w (\347)1040 6233 w (\347)1040 6333 w (\347)1061 5915 w (\347)1061 5933 w (\347)1061 6033 w (\347)1061 6133 w (\347)1061 6233 w (\347)1061 6333 w (\347)1051 5915 w (\347)1051 5933 w (\347)1051 6033 w (\347)1051 6133 w (\347)1051 6233 w (\347)1051 6333 w 1828 6617 1828 6566 Dl 1544 6333 1595 6333 Dl 1310 6099 1310 6048 Dl 1026 6333 1077 6333 Dl 1310 6617 1310 6566 Dl 1310 6099 1310 6048 Dl 1026 5815 1077 5815 Dl 10 S1 f (_ ____________________)1 1036 1 792 6074 t 10 S f (\347)1569 5915 w (\347)1569 5951 w (\347)1569 6051 w (\347)1569 6151 w (\347)1569 6251 w (\347)1569 6351 w (\347)1569 6451 w (\347)1569 6551 w (\347)1569 6651 w (\347)1569 6751 w (\347)1569 6851 w 1072 5602 1051 5556 Dl 1050 5556 1029 5602 Dl (\347)1051 5656 w (\347)1051 5752 w (\347)1051 5852 w (\347)1051 5952 w (\347)1051 6052 w (\347)1051 6152 w (\347)1051 6252 w (\347)1051 6352 w (\347)1051 6452 w (\347)1051 6552 w (\347)1051 6652 w (\347)1051 6752 w (\347)1051 6852 w 2042 6613 2088 6592 Dl 2087 6591 2041 6570 Dl 10 S1 f (_ _________________________)1 1296 1 792 6592 t 10 B f (16)1260 6852 w 1436 6360 46 46 De 1444 6360 31 31 De 1452 6360 15 15 De 1531 6452 1569 6592 Dl 1492 6396 1530 6452 Dl 1453 6354 1491 6395 Dl 1414 6319 1452 6353 Dl 1376 6290 1414 6319 Dl 1337 6265 1375 6290 Dl 1298 6243 1336 6265 Dl 1259 6222 1297 6242 Dl 1220 6204 1258 6222 Dl 1181 6188 1219 6204 Dl 1142 6174 1180 6188 Dl 1103 6160 1141 6173 Dl 1065 6147 1103 6159 Dl 1026 6136 1064 6147 Dl 987 6127 1025 6136 Dl 948 6118 986 6126 Dl 909 6109 947 6117 Dl 870 6102 908 6109 Dl 831 6096 869 6102 Dl 792 6091 830 6096 Dl 10 I f (b)1076 5606 w 10 R f (= 1)1 131 1 1151 5606 t 10 I f (a)1882 6542 w 10 R f (= 2)1 131 1 1957 6542 t (\(1,0\))835 6692 w 10 S1 f (_ __________)1 518 1 2750 6582 t (_ __________)1 518 1 2750 6603 t (_ __________)1 518 1 2750 6592 t 10 S f (\347)2480 5915 w (\347)2480 5933 w (\347)2480 6033 w (\347)2480 6133 w (\347)2480 6233 w (\347)2480 6333 w (\347)2501 5915 w (\347)2501 5933 w (\347)2501 6033 w (\347)2501 6133 w (\347)2501 6233 w (\347)2501 6333 w (\347)2491 5915 w (\347)2491 5933 w (\347)2491 6033 w (\347)2491 6133 w (\347)2491 6233 w (\347)2491 6333 w 3268 6617 3268 6566 Dl 2984 6333 3035 6333 Dl 2750 6099 2750 6048 Dl 2466 6333 2517 6333 Dl 2750 6617 2750 6566 Dl 2750 6099 2750 6048 Dl 2466 5815 2517 5815 Dl 10 S1 f (_ ____________________)1 1036 1 2232 6074 t 10 S f (\347)3009 5915 w (\347)3009 5951 w (\347)3009 6051 w (\347)3009 6151 w (\347)3009 6251 w (\347)3009 6351 w (\347)3009 6451 w (\347)3009 6551 w (\347)3009 6651 w (\347)3009 6751 w (\347)3009 6851 w 2512 5602 2491 5556 Dl 2490 5556 2469 5602 Dl (\347)2491 5656 w (\347)2491 5752 w (\347)2491 5852 w (\347)2491 5952 w (\347)2491 6052 w (\347)2491 6152 w (\347)2491 6252 w (\347)2491 6352 w (\347)2491 6452 w (\347)2491 6552 w (\347)2491 6652 w (\347)2491 6752 w (\347)2491 6852 w 3482 6613 3528 6592 Dl 3527 6591 3481 6570 Dl 10 S1 f (_ _________________________)1 1296 1 2232 6592 t 10 B f (17)2700 6852 w 2682 6378 46 46 De 2690 6378 31 31 De 2698 6378 15 15 De 2939 6737 2976 6852 Dl 2902 6652 2939 6736 Dl 2865 6583 2902 6651 Dl 2827 6525 2864 6583 Dl 2790 6474 2827 6524 Dl 2753 6428 2790 6473 Dl 2716 6388 2753 6428 Dl 2679 6352 2716 6388 Dl 2641 6319 2678 6351 Dl 2604 6289 2641 6318 Dl 2567 6261 2604 6288 Dl 2530 6237 2567 6261 Dl 2492 6214 2529 6236 Dl 2455 6194 2492 6214 Dl 2418 6175 2455 6193 Dl 2381 6158 2418 6174 Dl 2343 6143 2380 6158 Dl 2306 6130 2343 6143 Dl 2269 6118 2306 6129 Dl 2232 6107 2269 6117 Dl 10 I f (b)2516 5606 w 10 R f (= 2)1 131 1 2591 5606 t 10 I f (a)3322 6542 w 10 R f (= 2)1 131 1 3397 6542 t (\(1,1\))2275 6692 w 10 S1 f (_ __________)1 518 1 3672 6582 t (_ __________)1 518 1 3672 6603 t (_ __________)1 518 1 3672 6592 t 10 S f (\347)3920 5915 w (\347)3920 5933 w (\347)3920 6033 w (\347)3920 6133 w (\347)3920 6233 w (\347)3920 6333 w (\347)3941 5915 w (\347)3941 5933 w (\347)3941 6033 w (\347)3941 6133 w (\347)3941 6233 w (\347)3941 6333 w (\347)3931 5915 w (\347)3931 5933 w (\347)3931 6033 w (\347)3931 6133 w (\347)3931 6233 w (\347)3931 6333 w 4708 6617 4708 6566 Dl 4424 6333 4475 6333 Dl 4190 6099 4190 6048 Dl 3906 6333 3957 6333 Dl 4190 6617 4190 6566 Dl 4190 6099 4190 6048 Dl 3906 5815 3957 5815 Dl 10 S1 f (_ ____________________)1 1036 1 3672 6074 t 10 S f (\347)4449 5915 w (\347)4449 5951 w (\347)4449 6051 w (\347)4449 6151 w (\347)4449 6251 w (\347)4449 6351 w (\347)4449 6451 w (\347)4449 6551 w (\347)4449 6651 w (\347)4449 6751 w (\347)4449 6851 w 3952 5602 3931 5556 Dl 3930 5556 3909 5602 Dl (\347)3931 5656 w (\347)3931 5752 w (\347)3931 5852 w (\347)3931 5952 w (\347)3931 6052 w (\347)3931 6152 w (\347)3931 6252 w (\347)3931 6352 w (\347)3931 6452 w (\347)3931 6552 w (\347)3931 6652 w (\347)3931 6752 w (\347)3931 6852 w 4922 6613 4968 6592 Dl 4967 6591 4921 6570 Dl 10 S1 f (_ _________________________)1 1296 1 3672 6592 t 10 B f (18)4140 6852 w 3909 6331 46 46 De 3917 6331 31 31 De 3925 6331 15 15 De 4386 6810 4423 6852 Dl 4348 6768 4385 6809 Dl 4311 6727 4348 6768 Dl 4273 6686 4310 6726 Dl 4236 6645 4273 6685 Dl 4198 6605 4235 6645 Dl 4160 6566 4197 6605 Dl 4123 6526 4160 6565 Dl 4085 6487 4122 6526 Dl 4048 6449 4085 6487 Dl 4010 6410 4047 6448 Dl 3973 6371 4010 6409 Dl 3935 6334 3972 6371 Dl 3897 6297 3934 6334 Dl 3860 6259 3897 6296 Dl 3822 6223 3859 6259 Dl 3785 6186 3822 6222 Dl 3747 6150 3784 6186 Dl 3710 6114 3747 6149 Dl 3672 6079 3709 6114 Dl 10 I f (b)3956 5606 w 10 R f (= 25)1 181 1 4031 5606 t 10 I f (a)4712 6542 w 10 R f (= 54)1 181 1 4787 6542 t (\(49,10\))3615 6692 w cleartomark showpage saveobj restore %%PageBoundingBox: 68 91 507 764 %%EndPage: 10 10 %%Page: 11 11 %%PageBoundingBox: (atend) /saveobj save def mark 11 pagesetup 10 R f (- 11 -)2 216 1 2772 480 t 10 S1 f (_ __________)1 518 1 1310 1866 t (_ __________)1 518 1 1310 1887 t (_ __________)1 518 1 1310 1876 t 10 S f (\347)1559 1717 w (\347)1559 1735 w (\347)1559 1835 w (\347)1559 1935 w (\347)1559 2035 w (\347)1559 2135 w (\347)1579 1717 w (\347)1579 1735 w (\347)1579 1835 w (\347)1579 1935 w (\347)1579 2035 w (\347)1579 2135 w (\347)1569 1717 w (\347)1569 1735 w (\347)1569 1835 w (\347)1569 1935 w (\347)1569 2035 w (\347)1569 2135 w (\347)1040 1717 w (\347)1040 1735 w (\347)1040 1835 w (\347)1040 1935 w (\347)1040 2035 w (\347)1040 2135 w (\347)1061 1717 w (\347)1061 1735 w (\347)1061 1835 w (\347)1061 1935 w (\347)1061 2035 w (\347)1061 2135 w (\347)1051 1717 w (\347)1051 1735 w (\347)1051 1835 w (\347)1051 1935 w (\347)1051 2035 w (\347)1051 2135 w 1828 1901 1828 1850 Dl 1544 1617 1595 1617 Dl 1310 1383 1310 1332 Dl 1026 1617 1077 1617 Dl 1310 1901 1310 1850 Dl 1310 1383 1310 1332 Dl 1026 1099 1077 1099 Dl 10 S1 f (_ ____________________)1 1036 1 792 1358 t 10 S f (\347)1569 1199 w (\347)1569 1235 w (\347)1569 1335 w (\347)1569 1435 w (\347)1569 1535 w (\347)1569 1635 w (\347)1569 1735 w (\347)1569 1835 w (\347)1569 1935 w (\347)1569 2035 w (\347)1569 2135 w 1072 886 1051 840 Dl 1050 840 1029 886 Dl (\347)1051 940 w (\347)1051 1036 w (\347)1051 1136 w (\347)1051 1236 w (\347)1051 1336 w (\347)1051 1436 w (\347)1051 1536 w (\347)1051 1636 w (\347)1051 1736 w (\347)1051 1836 w (\347)1051 1936 w (\347)1051 2036 w (\347)1051 2136 w 2042 1897 2088 1876 Dl 2087 1875 2041 1854 Dl 10 S1 f (_ _________________________)1 1296 1 792 1876 t 10 B f (19)1260 2136 w 1514 1812 46 46 De 1522 1812 31 31 De 1529 1812 15 15 De 1531 1806 1569 1876 Dl 1492 1776 1530 1805 Dl 1453 1754 1491 1776 Dl 1414 1736 1452 1754 Dl 1376 1720 1414 1736 Dl 1337 1705 1375 1719 Dl 1298 1692 1336 1705 Dl 1259 1680 1297 1692 Dl 1220 1668 1258 1679 Dl 1181 1658 1219 1668 Dl 1142 1647 1180 1657 Dl 1103 1638 1141 1647 Dl 1065 1628 1103 1637 Dl 1026 1620 1064 1628 Dl 987 1612 1025 1620 Dl 948 1603 986 1611 Dl 909 1596 947 1603 Dl 870 1589 908 1596 Dl 831 1581 869 1588 Dl 792 1575 830 1581 Dl 10 I f (b)1076 890 w 10 R f (= 1)1 131 1 1151 890 t 10 I f (a)1882 1826 w 10 R f (= 8)1 131 1 1957 1826 t (\(7,0\))835 1976 w 10 S1 f (_ __________)1 518 1 2750 1866 t (_ __________)1 518 1 2750 1887 t (_ __________)1 518 1 2750 1876 t 10 S f (\347)2480 1717 w (\347)2480 1735 w (\347)2480 1835 w (\347)2480 1935 w (\347)2480 2035 w (\347)2480 2135 w (\347)2501 1717 w (\347)2501 1735 w (\347)2501 1835 w (\347)2501 1935 w (\347)2501 2035 w (\347)2501 2135 w (\347)2491 1717 w (\347)2491 1735 w (\347)2491 1835 w (\347)2491 1935 w (\347)2491 2035 w (\347)2491 2135 w 3268 1901 3268 1850 Dl 2984 1617 3035 1617 Dl 2750 1383 2750 1332 Dl 2466 1617 2517 1617 Dl 2750 1901 2750 1850 Dl 2750 1383 2750 1332 Dl 2466 1099 2517 1099 Dl 10 S1 f (_ ____________________)1 1036 1 2232 1358 t 10 S f (\347)3009 1199 w (\347)3009 1235 w (\347)3009 1335 w (\347)3009 1435 w (\347)3009 1535 w (\347)3009 1635 w (\347)3009 1735 w (\347)3009 1835 w (\347)3009 1935 w (\347)3009 2035 w (\347)3009 2135 w 2512 886 2491 840 Dl 2490 840 2469 886 Dl (\347)2491 940 w (\347)2491 1036 w (\347)2491 1136 w (\347)2491 1236 w (\347)2491 1336 w (\347)2491 1436 w (\347)2491 1536 w (\347)2491 1636 w (\347)2491 1736 w (\347)2491 1836 w (\347)2491 1936 w (\347)2491 2036 w (\347)2491 2136 w 3482 1897 3528 1876 Dl 3527 1875 3481 1854 Dl 10 S1 f (_ _________________________)1 1296 1 2232 1876 t 10 B f (20)2700 2136 w 2725 1820 46 46 De 2732 1820 31 31 De 2740 1820 15 15 De 2924 2056 2960 2136 Dl 2887 1992 2923 2055 Dl 2851 1939 2887 1992 Dl 2814 1893 2850 1939 Dl 2778 1851 2814 1892 Dl 2742 1814 2778 1851 Dl 2705 1779 2741 1813 Dl 2669 1748 2705 1779 Dl 2632 1718 2668 1747 Dl 2596 1691 2632 1718 Dl 2560 1666 2596 1691 Dl 2523 1642 2559 1665 Dl 2487 1620 2523 1642 Dl 2450 1599 2486 1619 Dl 2414 1579 2450 1598 Dl 2378 1561 2414 1579 Dl 2341 1543 2377 1560 Dl 2305 1527 2341 1543 Dl 2268 1512 2304 1527 Dl 2232 1497 2268 1511 Dl 10 I f (b)2516 890 w 10 R f (= 2)1 131 1 2591 890 t 10 I f (a)3322 1826 w 10 R f (= 3)1 131 1 3397 1826 t (\(2,1\))2275 1976 w 10 S1 f (_ __________)1 518 1 3672 1866 t (_ __________)1 518 1 3672 1887 t (_ __________)1 518 1 3672 1876 t 10 S f (\347)3920 1717 w (\347)3920 1735 w (\347)3920 1835 w (\347)3920 1935 w (\347)3920 2035 w (\347)3920 2135 w (\347)3941 1717 w (\347)3941 1735 w (\347)3941 1835 w (\347)3941 1935 w (\347)3941 2035 w (\347)3941 2135 w (\347)3931 1717 w (\347)3931 1735 w (\347)3931 1835 w (\347)3931 1935 w (\347)3931 2035 w (\347)3931 2135 w 4708 1901 4708 1850 Dl 4424 1617 4475 1617 Dl 4190 1383 4190 1332 Dl 3906 1617 3957 1617 Dl 4190 1901 4190 1850 Dl 4190 1383 4190 1332 Dl 3906 1099 3957 1099 Dl 10 S1 f (_ ____________________)1 1036 1 3672 1358 t 10 S f (\347)4449 1199 w (\347)4449 1235 w (\347)4449 1335 w (\347)4449 1435 w (\347)4449 1535 w (\347)4449 1635 w (\347)4449 1735 w (\347)4449 1835 w (\347)4449 1935 w (\347)4449 2035 w (\347)4449 2135 w 3952 886 3931 840 Dl 3930 840 3909 886 Dl (\347)3931 940 w (\347)3931 1036 w (\347)3931 1136 w (\347)3931 1236 w (\347)3931 1336 w (\347)3931 1436 w (\347)3931 1536 w (\347)3931 1636 w (\347)3931 1736 w (\347)3931 1836 w (\347)3931 1936 w (\347)3931 2036 w (\347)3931 2136 w 4922 1897 4968 1876 Dl 4967 1875 4921 1854 Dl 10 S1 f (_ _________________________)1 1296 1 3672 1876 t 10 B f (21)4140 2136 w 3970 1813 46 46 De 3978 1813 31 31 De 3986 1813 15 15 De 4213 2089 4241 2136 Dl 4184 2045 4212 2088 Dl 4156 2004 4184 2044 Dl 4127 1966 4155 2004 Dl 4099 1930 4127 1965 Dl 4070 1896 4098 1929 Dl 4042 1865 4070 1896 Dl 4014 1834 4042 1864 Dl 3985 1805 4013 1833 Dl 3957 1777 3985 1804 Dl 3928 1752 3956 1777 Dl 3900 1727 3928 1751 Dl 3871 1703 3899 1726 Dl 3843 1681 3871 1703 Dl 3814 1660 3842 1681 Dl 3786 1639 3814 1659 Dl 3757 1620 3785 1639 Dl 3729 1601 3757 1619 Dl 3700 1584 3728 1601 Dl 3672 1567 3700 1583 Dl 10 I f (b)3956 890 w 10 R f (= 3)1 131 1 4031 890 t 10 I f (a)4762 1826 w 10 R f (= 3)1 131 1 4837 1826 t (\(2,2\))3715 1976 w 10 B f ( lemmas)1 352( Supporting)1 535(Appendix 2.)1 518 3 720 2568 t 10 I f ( approximate ellipse with a)4 1089( An)1 161(Lemma 2.)1 394 3 720 2724 t 10 S f (\263)2405 2724 w 10 I f (b has tails if and only if a)7 1021 1 2501 2724 t 10 S f (\263)3563 2724 w 10 R f (8)3659 2724 w 10 I f (b)3717 2724 w 7 R f (2)3778 2684 w 10 I f (and a)1 225 1 3846 2724 t 10 S f (>)4120 2724 w 10 R f (0)4224 2724 w 10 I f (.)4274 2724 w 10 R f (A tail occurs if \()4 706 1 970 2880 t 10 I f (a)1684 2880 w 10 S f (-)1758 2880 w 10 R f ( if and only if the ordinate of the ellipse at)10 1821( is lighted, or in other words)6 1208(1 , 0 \))3 182 3 1829 2880 t 10 I f (x)720 3000 w 10 S f (=)813 3000 w 10 I f (a)917 3000 w 10 S f (-)991 3000 w 10 R f (1 is less than 1)4 589 1 1062 3000 t 10 I f (/)1659 3000 w 10 R f (2. Thus)1 325 1 1695 3000 t 10 I f (a)2045 3000 w 10 S f (>)2144 3000 w 10 R f (0 and)1 219 1 2248 3000 t 10 I f (y)2167 3180 w 7 R f (2)2222 3140 w 10 S f (=)2314 3180 w 10 I f (b)2418 3180 w 7 R f (2)2479 3140 w 10 R f (\( 1)1 91 1 2530 3180 t 10 S f (-)2670 3180 w 10 R f (\()2774 3180 w 10 I f (a)2815 3180 w 10 S f (-)2889 3180 w 10 R f (1 \))1 91 1 2960 3180 t 7 R f (2)3056 3140 w 10 I f (/ a)1 86 1 3107 3180 t 7 R f (2)3204 3140 w 10 R f (\))3255 3180 w 10 S f (<)3345 3180 w 10 R f (1)3449 3180 w 10 I f (/)3507 3180 w 10 R f (4)3543 3180 w (Expand and clear of fractions:)4 1202 1 720 3360 t 10 I f (a)2386 3540 w 7 R f (2)2447 3500 w 10 S f (-)2539 3540 w 10 R f (8)2643 3540 w 10 I f (ab)2701 3540 w 7 R f (2)2812 3500 w 10 S f (+)2904 3540 w 10 R f (4)3008 3540 w 10 I f (b)3066 3540 w 7 R f (2)3127 3500 w 10 S f (>)3219 3540 w 10 R f (0)3323 3540 w (Now)720 3720 w 10 I f (a)939 3720 w 10 R f ( smaller root is less than 1.\))6 1102( \(The)1 238(must exceed the larger root of the associated quadratic equation.)9 2566 3 1014 3720 t 10 I f (a)2325 3918 w 10 S f (>)2399 3918 w 10 R f (4)2503 3918 w 10 I f (b)2561 3918 w 7 R f (2)2622 3878 w 10 S f (+)2714 3918 w 12 S f (\326` `````````)1 609 1 2818 3918 t 10 R f (16)2892 3918 w 10 I f (b)3000 3918 w 7 R f (4)3061 3878 w 10 S f (-)3153 3918 w 10 R f (4)3257 3918 w 10 I f (b)3315 3918 w 7 R f (2)3376 3878 w 10 R f (If)720 4098 w 10 I f (b)811 4098 w 10 S f (>)910 4098 w 10 R f (0, this is equivalent to)4 881 1 1014 4098 t 10 I f (a)2246 4377 w 10 S f (>)2345 4377 w 10 R f (4)2449 4377 w 10 I f (b)2507 4377 w 7 R f (2)2568 4337 w 10 S f (+)2660 4377 w 10 R f (4)2764 4377 w 10 I f (b)2822 4377 w 7 R f (2)2883 4337 w 27 S f (\326` ```)1 580 1 2926 4457 t 10 R f (1)3083 4377 w 10 S f (-)3182 4377 w 10 R f (4)3311 4457 w 10 I f (b)3369 4457 w 7 R f (2)3430 4417 w 10 R f (1)3367 4317 w 10 S1 f (_ ___)1 192 1 3296 4347 t 10 R f (By Taylor's theorem with remainder)4 1464 1 720 4617 t 10 I f (a)2442 4797 w 10 S f (>)2541 4797 w 10 R f (8)2645 4797 w 10 I f (b)2703 4797 w 7 R f (2)2764 4757 w 10 S f (-)2856 4797 w 10 R f (1)2960 4797 w 10 I f (/)3018 4797 w 10 R f (2)3054 4797 w 10 S f (+)3153 4797 w 10 I f (R)3257 4797 w 10 R f (where)720 4977 w (0)2115 5242 w 10 S f (<)2214 5242 w 10 I f (R)2318 5242 w 10 S f (\243)2403 5242 w 10 R f (8)2532 5312 w (1)2532 5182 w 10 S1 f (_ _)1 80 1 2517 5212 t 10 S f (\354)2615 5155 w (\357)2615 5255 w (\356)2615 5355 w 10 R f (4)2689 5322 w 10 I f (b)2747 5322 w 7 R f (2)2808 5282 w 10 R f (1)2745 5182 w 10 S1 f (_ ___)1 192 1 2674 5212 t 10 S f (\374)2876 5155 w (\357)2876 5255 w (\376)2876 5355 w 7 R f (2)2930 5093 w 10 S f (\354)2981 5155 w (\357)2981 5255 w (\356)2981 5355 w 10 R f (1)3030 5242 w 10 S f (-)3120 5242 w 10 R f (4)3240 5322 w 10 I f (b)3298 5322 w 7 R f (2)3359 5282 w 10 R f (1)3296 5182 w 10 S1 f (_ ___)1 192 1 3225 5212 t 10 S f (\374)3427 5155 w (\357)3427 5255 w (\376)3427 5355 w 7 S f (-)3487 5093 w 7 R f (3)3537 5093 w 7 I f (/)3577 5093 w 7 R f (2)3602 5093 w 10 R f (Since)720 5527 w 10 I f (b)970 5527 w 10 R f (is a positive integer,)3 814 1 1048 5527 t 10 I f (R)1890 5527 w 10 R f (is surely less than 1)4 799 1 1979 5527 t 10 I f (/)2786 5527 w 10 R f ( the fact that)3 508(2. Using)1 368 2 2822 5527 t 10 I f (a)3727 5527 w 10 R f (is an integer, we find)4 856 1 3806 5527 t 10 I f (a)4691 5527 w 10 S f (\263)4782 5527 w 10 R f (8)4878 5527 w 10 I f (b)4936 5527 w 7 R f (2)4997 5487 w 10 R f (for positive)1 464 1 720 5647 t 10 I f (b)1215 5647 w 10 R f ( result also holds for)4 840(. The)1 236 2 1265 5647 t 10 I f (b)2372 5647 w 10 S f (=)2471 5647 w 10 R f (0 and)1 224 1 2575 5647 t 10 I f (a)2829 5647 w 10 S f (>)2928 5647 w 10 R f (0, in which case the approximate ellipse degener-)7 2008 1 3032 5647 t (ates to a line segment\320all tail.)5 1238 1 720 5767 t 10 I f ( ellipse of equation)3 836( The)1 224(Lemma 3.)1 418 3 720 5923 t 10 R f (\( 1 \))2 132 1 2247 5923 t 10 I f ( b does not pass through any point, one)8 1775(with integral a and)3 837 2 2428 5923 t (coordinate of which is an integer, and the other half an odd integer.)12 2722 1 720 6043 t 10 R f (Suppose that the ellipse does pass through such a point, \()10 2299 1 970 6199 t 10 I f (x)3277 6199 w 10 R f (,)3329 6199 w 10 I f (y)3362 6199 w 10 R f ( let)1 128( loss of generality,)3 744(\). Without)1 438 3 3414 6199 t 10 I f (x)4752 6199 w 10 R f (be an)1 216 1 4824 6199 t (integer and)1 459 1 720 6319 t 10 I f (y)1217 6319 w 10 S f (=)1310 6319 w 10 I f (z /)1 75 1 1414 6319 t 10 R f (2, where)1 356 1 1497 6319 t 10 I f (z)1891 6319 w 10 R f ( may assume that)3 727( We)1 200( integer.)1 339(is an odd)2 387 4 1968 6319 t 10 I f (gcd)3658 6319 w 10 R f (\()3810 6319 w 10 I f (x)3851 6319 w 10 R f (,)3903 6319 w 10 I f (a)3936 6319 w 10 R f (\))3994 6319 w 10 S f (=)4084 6319 w 10 I f (gcd)4188 6319 w 10 R f (\()4340 6319 w 10 I f (z)4381 6319 w 10 R f (, 2)1 83 1 4428 6319 t 10 I f (b)4519 6319 w 10 R f (\))4577 6319 w 10 S f (=)4667 6319 w 10 R f (1; if it)2 269 1 4771 6319 t (were not, we could reduce the fractions in the defining equation)10 2547 1 720 6439 t 10 I f (a)2543 6749 w 7 R f (2)2604 6709 w 10 I f (x)2546 6609 w 7 R f (2)2601 6569 w 10 S1 f (_ __)1 134 1 2528 6639 t 10 S f (+)2721 6669 w 10 R f (4)2850 6749 w 10 I f (b)2908 6749 w 7 R f (2)2969 6709 w 10 I f (z)2885 6609 w 7 R f (2)2935 6569 w 10 S1 f (_ ___)1 192 1 2836 6639 t 10 S f (=)3087 6669 w 10 R f (1)3191 6669 w ( sum of two fractions in)5 972( The)1 209( the assumption does hold.)4 1080(to get a counterexample of the same form in which)9 2059 4 720 6909 t ( Hence)1 312( their denominators are the same.)5 1365(lowest terms can be 1 only if)6 1208 3 720 7029 t 10 I f (a)3638 7029 w 10 S f (=)3737 7029 w 10 R f (2)3841 7029 w 10 I f (b)3899 7029 w 10 R f (. Because)1 415 1 3949 7029 t 10 I f (x / a)2 138 1 4397 7029 t 10 R f (is in lowest)2 472 1 4568 7029 t (terms,)720 7149 w 10 I f (x)992 7149 w 10 R f ( we have a triple \()5 717( Consequently)1 600(must be odd.)2 514 3 1061 7149 t 10 I f (x)2900 7149 w 10 R f (,)2952 7149 w 10 I f (z)2985 7149 w 10 R f (,)3032 7149 w 10 I f (a)3065 7149 w 10 R f (\) with parities \(odd,odd,even\) that satisfies)5 1706 1 3123 7149 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 61 514 764 %%EndPage: 11 11 %%Page: 12 12 %%PageBoundingBox: (atend) /saveobj save def mark 12 pagesetup 10 R f (- 12 -)2 216 1 2772 480 t 10 I f (x)2579 840 w 7 R f (2)2634 800 w 10 S f (+)2726 840 w 10 I f (z)2830 840 w 7 R f (2)2880 800 w 10 S f (=)2972 840 w 10 I f (a)3076 840 w 7 R f (2)3137 800 w 10 R f (But as is well known, no such triple exists, for that would imply)12 2561 1 720 1020 t (1)2099 1200 w 10 S f (+)2198 1200 w 10 R f (1)2302 1200 w 10 S f (\272)2393 1200 w 10 I f (x)2489 1200 w 7 R f (2)2544 1160 w 10 S f (+)2636 1200 w 10 I f (z)2740 1200 w 7 R f (2)2790 1160 w 10 S f (\272)2874 1200 w 10 I f (a)2970 1200 w 7 R f (2)3031 1160 w 10 S f (\272)3115 1200 w 10 R f ( \).)1 66( 4)1 82( mod)1 186(0 \()1 115 4 3211 1200 t 10 I f ( Configurations)1 657(Lemma 4.)1 394 2 720 1416 t 10 B f (4)1796 1416 w 10 I f (,)1846 1416 w 10 B f (8)1896 1416 w 10 I f (,)1946 1416 w 10 B f (14)1996 1416 w 10 I f (, and)1 200 1 2096 1416 t 10 B f (15)2321 1416 w 10 I f (are impossible.)1 611 1 2446 1416 t 10 R f (The coordinates of the juncture, \()5 1330 1 970 1572 t 10 I f (X)2308 1572 w 10 R f (,)2377 1572 w 10 I f (Y)2410 1572 w 10 R f (\), must satisfy both the equation of the ellipse)8 1824 1 2474 1572 t 10 I f (a)2573 1882 w 7 R f (2)2634 1842 w 10 I f (x)2576 1742 w 7 R f (2)2631 1702 w 10 S1 f (_ __)1 134 1 2558 1772 t 10 S f (+)2751 1802 w 10 I f (b)2880 1882 w 7 R f (2)2941 1842 w 10 I f (y)2883 1742 w 7 R f (2)2938 1702 w 10 S1 f (_ __)1 134 1 2865 1772 t 10 S f (=)3058 1802 w 10 R f (1 \(1\))1 1878 1 3162 1802 t (and its derivative)2 688 1 720 2042 t 10 I f (a)2501 2342 w 7 R f (2)2562 2302 w 10 R f (2)2502 2202 w 10 I f (x)2560 2202 w 10 S1 f (_ __)1 134 1 2486 2232 t 10 S f (+)2679 2262 w 10 I f (b)2808 2342 w 7 R f (2)2869 2302 w 10 R f (2)2809 2202 w 10 I f (y)2867 2202 w 10 S1 f (_ __)1 134 1 2793 2232 t 10 I f (dx)2962 2332 w (dy)2962 2202 w 10 S1 f (_ __)1 124 1 2947 2232 t 10 S f (=)3130 2262 w 10 R f (0)3234 2262 w (with)720 2502 w 10 I f (dy / dx)2 232 1 923 2502 t 10 S f (= -)1 167 1 1204 2502 t 10 R f ( simultaneously, we find)3 983(1. Solving)1 437 2 1387 2502 t 10 I f (X)2131 2732 w 10 S f (=)2241 2732 w 12 S f (\326``````)2370 2840 w 10 I f (a)2444 2840 w 7 R f (2)2505 2800 w 10 S f (+)2588 2840 w 10 I f (b)2683 2840 w 7 R f (2)2744 2800 w 10 I f (a)2535 2672 w 7 R f (2)2596 2632 w 10 S1 f (_ _________)1 463 1 2356 2702 t 10 R f (,)2837 2732 w 10 I f (Y)2936 2732 w 10 S f (=)3041 2732 w 12 S f (\326``````)3170 2840 w 10 I f (a)3244 2840 w 7 R f (2)3305 2800 w 10 S f (+)3388 2840 w 10 I f (b)3483 2840 w 7 R f (2)3544 2800 w 10 I f (b)3335 2672 w 7 R f (2)3396 2632 w 10 S1 f (_ _________)1 463 1 3156 2702 t 10 R f (\(2\))4924 2732 w 2966 3468 43 43 De 2973 3468 28 28 De 2980 3468 14 14 De (\()2923 3310 w 10 I f (x)2964 3310 w 7 R f (1)3019 3330 w 10 R f (,)3070 3310 w 10 I f (y)3103 3310 w 7 R f (1)3158 3330 w 10 R f (\))3209 3310 w (\()2723 3518 w 10 I f (X)2764 3518 w 10 R f (,)2833 3518 w 10 I f (Y)2866 3518 w 10 R f (\))2930 3518 w (\()2896 4620 w 10 I f (x)2937 4620 w 7 R f (2)2992 4640 w 10 R f (,)3043 4620 w 10 I f (y)3076 4620 w 7 R f (2)3131 4640 w 10 R f (\))3182 4620 w 2736 3144 2736 3144 2844 3252 Ds 2736 3144 2844 3252 2923 3360 Ds 2844 3252 2923 3360 2987 3468 Ds 2923 3360 2987 3468 3167 4080 Ds 2987 3468 3167 4080 3239 4620 Ds 3167 4080 3239 4620 3239 4620 Ds 2880 4116 2880 4044 Dl 2880 3395 2880 3324 Dl 3204 3720 3276 3720 Dl 3204 4440 3276 4440 Dl 10 S f (\347)3240 3100 w (\347)3240 3120 w (\347)3240 3220 w (\347)3240 3320 w (\347)3240 3420 w (\347)3240 3520 w (\347)3240 3620 w (\347)3240 3720 w (\347)3240 3820 w (\347)3240 3920 w (\347)3240 4020 w (\347)3240 4120 w (\347)3240 4220 w (\347)3240 4320 w (\347)3240 4420 w (\347)3240 4520 w (\347)3240 4620 w (\347)2520 3100 w (\347)2520 3120 w (\347)2520 3220 w (\347)2520 3320 w (\347)2520 3420 w (\347)2520 3520 w (\347)2520 3620 w (\347)2520 3720 w (\347)2520 3820 w (\347)2520 3920 w (\347)2520 4020 w (\347)2520 4120 w (\347)2520 4220 w (\347)2520 4320 w (\347)2520 4420 w (\347)2520 4520 w (\347)2520 4620 w 10 S1 f (_ ________________________)1 1224 1 2160 3360 t (_ ____________________________)1 1440 1 2160 4080 t 10 R f (Figure 5.)1 361 1 2699 4896 t (Configurations)970 5172 w 10 B f (4)1611 5172 w 10 R f (and)1702 5172 w 10 B f (8)1887 5172 w 10 R f ( ellipse meets the top grid line at)7 1421( The)1 222( Figure 5.)2 420(are both exemplified by)3 999 4 1978 5172 t (\()720 5292 w 10 I f (x)761 5292 w 7 R f (1)816 5312 w 10 R f (,)867 5292 w 10 I f (y)900 5292 w 7 R f (1)955 5312 w 10 R f (\) and the right grid line at \()7 1079 1 1006 5292 t 10 I f (x)2093 5292 w 7 R f (2)2148 5312 w 10 R f (,)2199 5292 w 10 I f (y)2232 5292 w 7 R f (2)2287 5312 w 10 R f ( the figure we see that)5 878(\). From)1 325 2 2338 5292 t (1)2200 5472 w 10 I f (/)2258 5472 w 10 R f (2)2294 5472 w 10 S f (\243)2385 5472 w 10 I f (x)2481 5472 w 7 R f (1)2536 5492 w 10 S f (\243)2620 5472 w 10 I f (X)2716 5472 w 10 S f (<)2826 5472 w 10 I f (x)2930 5472 w 7 R f (2)2985 5492 w 10 S f (\243)3069 5472 w 10 I f (x)3165 5472 w 7 R f (1)3220 5492 w 10 S f (+)3312 5472 w 10 R f (1)3416 5472 w 10 I f (/)3474 5472 w 10 R f (2 \(3\))1 1530 1 3510 5472 t (0)2241 5652 w 10 S f (\243)2332 5652 w 10 I f (y)2428 5652 w 7 R f (2)2483 5672 w 10 S f (<)2575 5652 w 10 I f (y)2679 5652 w 7 R f (1)2734 5672 w 10 S f (-)2826 5652 w 10 R f (3)2930 5652 w 10 I f (/)2988 5652 w 10 R f (2)3024 5652 w 10 S f (<)3123 5652 w 10 I f (Y)3227 5652 w 10 S f (\243)3324 5652 w 10 I f (y)3420 5652 w 7 R f (1)3475 5672 w 10 R f (\(4\))4924 5652 w (From the equation of the ellipse \(1\),)6 1440 1 720 5832 t (\()2380 6012 w 10 I f (x)2421 6012 w 7 R f (1)2476 6032 w 10 I f (/ a)1 86 1 2527 6012 t 10 R f (\))2621 6012 w 7 R f (2)2659 5972 w 10 S f (+)2751 6012 w 10 R f (\()2855 6012 w 10 I f (y)2896 6012 w 7 R f (1)2951 6032 w 10 I f (/ b)1 86 1 3002 6012 t 10 R f (\))3096 6012 w 7 R f (2)3134 5972 w 10 S f (=)3226 6012 w 10 R f (1)3330 6012 w (\()2380 6192 w 10 I f (x)2421 6192 w 7 R f (2)2476 6212 w 10 I f (/ a)1 86 1 2527 6192 t 10 R f (\))2621 6192 w 7 R f (2)2659 6152 w 10 S f (+)2751 6192 w 10 R f (\()2855 6192 w 10 I f (y)2896 6192 w 7 R f (2)2951 6212 w 10 I f (/ b)1 86 1 3002 6192 t 10 R f (\))3096 6192 w 7 R f (2)3134 6152 w 10 S f (=)3226 6192 w 10 R f (1)3330 6192 w (Subtract and factor.)2 784 1 720 6372 t (\()1821 6552 w 10 I f (y)1862 6552 w 7 R f (1)1917 6572 w 10 S f (-)2000 6552 w 10 I f (y)2095 6552 w 7 R f (2)2150 6572 w 10 R f (\) \()1 74 1 2201 6552 t 10 I f (y)2283 6552 w 7 R f (1)2338 6572 w 10 S f (+)2421 6552 w 10 I f (y)2516 6552 w 7 R f (2)2571 6572 w 10 R f (\))2622 6552 w 10 I f (/ b)1 86 1 2663 6552 t 7 R f (2)2760 6512 w 10 S f (=)2852 6552 w 10 R f (\()2956 6552 w 10 I f (x)2997 6552 w 7 R f (2)3052 6572 w 10 S f (-)3135 6552 w 10 I f (x)3230 6552 w 7 R f (1)3285 6572 w 10 R f (\) \()1 74 1 3336 6552 t 10 I f (x)3418 6552 w 7 R f (2)3473 6572 w 10 S f (+)3556 6552 w 10 I f (x)3651 6552 w 7 R f (1)3706 6572 w 10 R f (\))3757 6552 w 10 I f (/ a)1 86 1 3798 6552 t 7 R f (2)3895 6512 w 10 R f (From \(3\))1 358 1 720 6732 t 10 I f (x)1103 6732 w 7 R f (2)1158 6752 w 10 S f (-)1250 6732 w 10 I f (x)1354 6732 w 7 R f (1)1409 6752 w 10 S f (\243)1493 6732 w 10 R f (1)1589 6732 w 10 I f (/)1647 6732 w 10 R f ( \(3\), the second factor on the right is at most 2)11 1853( from)1 219(2. Also)1 314 3 1683 6732 t 10 I f (x)4077 6732 w 7 R f (1)4132 6752 w 10 S f (+)4215 6732 w 10 R f (1)4310 6732 w 10 I f (/)4368 6732 w 10 R f (2, which in turn)3 636 1 4404 6732 t (is at most 2)3 459 1 720 6852 t 10 I f (X)1187 6852 w 10 S f (+)1288 6852 w 10 R f (1)1383 6852 w 10 I f (/)1441 6852 w 10 R f (2. Hence)1 379 1 1477 6852 t 10 I f (b)2177 7162 w 7 R f (2)2238 7122 w 10 I f (a)2177 7022 w 7 R f (2)2238 6982 w 10 S1 f (_ __)1 134 1 2162 7052 t 10 R f (\()2314 7082 w 10 I f (y)2355 7082 w 7 R f (1)2410 7102 w 10 S f (-)2493 7082 w 10 I f (y)2588 7082 w 7 R f (2)2643 7102 w 10 R f (\) \()1 74 1 2694 7082 t 10 I f (y)2776 7082 w 7 R f (1)2831 7102 w 10 S f (+)2914 7082 w 10 I f (y)3009 7082 w 7 R f (2)3064 7102 w 10 R f (\))3115 7082 w 10 S f (\243)3197 7082 w 10 I f (X)3293 7082 w 10 S f (+)3403 7082 w 10 R f (4)3532 7152 w (1)3532 7022 w 10 S1 f (_ _)1 80 1 3517 7052 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 60 514 764 %%EndPage: 12 12 %%Page: 13 13 %%PageBoundingBox: (atend) /saveobj save def mark 13 pagesetup 10 R f (- 13 -)2 216 1 2772 480 t (From \(4\),)1 383 1 720 840 t 10 I f (y)1128 840 w 7 R f (1)1183 860 w 10 S f (-)1266 840 w 10 I f (y)1361 840 w 7 R f (2)1416 860 w 10 S f (>)1508 840 w 10 R f (3)1612 840 w 10 I f (/)1670 840 w 10 R f (2 and)1 219 1 1706 840 t 10 I f (y)1950 840 w 7 R f (1)2005 860 w 10 S f (\263)2089 840 w 10 I f (Y)2185 840 w 10 R f (:)2241 840 w (2)2335 1140 w (3)2335 1010 w 10 S1 f (_ _)1 80 1 2320 1040 t 10 I f (b)2467 1150 w 7 R f (2)2528 1110 w 10 I f (a)2467 1010 w 7 R f (2)2528 970 w 10 S1 f (_ __)1 134 1 2452 1040 t 10 R f (\()2604 1070 w 10 I f (Y)2645 1070 w 10 S f (+)2741 1070 w 10 I f (y)2836 1070 w 7 R f (2)2891 1090 w 10 R f (\))2942 1070 w 10 S f (<)3032 1070 w 10 I f (X)3136 1070 w 10 S f (+)3246 1070 w 10 R f (4)3375 1140 w (1)3375 1010 w 10 S1 f (_ _)1 80 1 3360 1040 t 10 R f (According to \(2\),)2 690 1 720 1310 t 10 I f (a)1435 1310 w 7 R f (2)1496 1270 w 10 I f (Y / b)2 150 1 1547 1310 t 7 R f (2)1708 1270 w 10 S f (=)1800 1310 w 10 I f (X)1904 1310 w 10 R f ( and simplifying, we find)4 1008(. Substituting)1 560 2 1965 1310 t 10 I f (X)2487 1540 w 10 S f (<)2597 1540 w 10 R f (2)2726 1610 w (1)2726 1480 w 10 S1 f (_ _)1 80 1 2711 1510 t 10 S f (-)2850 1540 w 10 R f (3)2954 1540 w 10 I f (b)3037 1620 w 7 R f (2)3098 1580 w 10 I f (a)3037 1480 w 7 R f (2)3098 1440 w 10 S1 f (_ __)1 134 1 3022 1510 t 10 I f (y)3174 1540 w 7 R f (2)3229 1560 w 10 R f (From \(4\))1 360 1 720 1780 t 10 I f (y)1107 1780 w 7 R f (2)1162 1800 w 10 R f ( the last inequality implies)4 1064( Thus)1 252(is nonnegative.)1 607 3 1232 1780 t 10 I f (X)3182 1780 w 10 S f (<)3292 1780 w 10 R f (1)3396 1780 w 10 I f (/)3454 1780 w 10 R f ( figure is)2 361( The)1 208( contradicts \(3\).)2 635(2, which)1 346 4 3490 1780 t (impossible, as are its instances)4 1268 1 720 1900 t 10 B f (4)2024 1900 w 10 R f (and)2110 1900 w 10 B f (8.)2290 1900 w 10 R f (The derivation has not used the assumption)6 1798 1 2426 1900 t 10 I f (a)4260 1900 w 10 S f (\263)4351 1900 w 10 I f (b)4447 1900 w 10 R f ( similar)1 313(, so a)2 230 2 4497 1900 t (argument proves the impossibility of the transposed configurations)7 2673 1 720 2020 t 10 B f (14)3418 2020 w 10 R f (and)3543 2020 w 10 B f (15)3712 2020 w 10 R f (.)3812 2020 w 10 I f ( Configurations)1 657(Lemma 5.)1 394 2 720 2176 t 10 B f (3)1796 2176 w 10 I f (and)1871 2176 w 10 B f (9)2046 2176 w 10 I f (are impossible.)1 611 1 2121 2176 t 10 R f (Figure 6a illustrates configuration)3 1393 1 970 2332 t 10 B f (3)2400 2332 w 10 R f ( ordinates of points)3 805(. The)1 242 2 2450 2332 t 10 I f (A)3534 2332 w 10 R f (and)3633 2332 w 10 I f (B)3815 2332 w 10 R f ( We)1 201(differ by more than 1.)4 925 2 3914 2332 t ( points)1 273( There)1 285(shall show this is impossible by considering Figure 6b.)8 2222 3 720 2452 t 10 I f (A)3527 2452 w 10 R f (and)3615 2452 w 10 I f (B)3786 2452 w 10 S f (\242)3858 2447 w 10 R f (are intersections of adjacent)3 1122 1 3918 2452 t ( ordinates of)2 515( The)1 212(grid lines with an ellipse in standard position.)7 1871 3 720 2572 t 10 I f (A)3351 2572 w 10 R f (and)3445 2572 w 10 I f (B)3622 2572 w 10 S f (\242)3694 2567 w 10 R f ( Rolle's)1 322( By)1 175(differ by exactly 1.)3 783 3 3760 2572 t (theorem,* the juncture \()3 974 1 720 2692 t 10 I f (X)1702 2692 w 10 S f (\242)1774 2687 w 10 R f (,)1815 2692 w 10 I f (Y)1848 2692 w 10 S f (\242)1915 2687 w 10 R f (\), where the slope is)4 821 1 1956 2692 t 10 S f (-)2808 2692 w 10 R f ( point \()2 301( Let)1 189(1, must lie between the grid lines.)6 1385 3 2879 2692 t 10 I f (x)4762 2692 w 7 R f (0)4817 2712 w 10 R f (,)4868 2692 w 10 I f (y)4901 2692 w 7 R f (0)4956 2712 w 10 R f (\))5007 2692 w (be the midpoint of)3 742 1 720 2812 t 10 I f (AB)1489 2812 w 10 S f (\242)1622 2807 w 10 R f ( let the lengths of the semiaxes of the ellipse be)10 1902(, and)1 196 2 1655 2812 t 10 I f (a)3779 2812 w 10 S f (\242)3840 2807 w 10 R f (and)3899 2812 w 10 I f (b)4069 2812 w 10 S f (\242)4130 2807 w 10 R f ( of)1 109(. \(None)1 325 2 4163 2812 t 10 I f (x)4623 2812 w 7 R f (0)4678 2832 w 10 R f (,)4721 2812 w 10 I f (y)4772 2812 w 7 R f (0)4827 2832 w 10 R f (,)4870 2812 w 10 I f (a)4921 2812 w 10 S f (\242)4982 2807 w 10 R f (,)5015 2812 w (or)720 2932 w 10 I f (b)828 2932 w 10 S f (\242)889 2927 w 10 R f ( both)1 203( Since)1 272(is constrained to be integral.\))4 1162 3 947 2932 t 10 I f (A)2609 2932 w 10 R f (and)2695 2932 w 10 I f (B)2864 2932 w 10 S f (\242)2936 2927 w 10 R f (lie on the ellipse,)3 683 1 2994 2932 t (\(a\))1385 4600 w 10 I f (B)1789 4052 w (A)1030 3469 w 1828 4124 1828 4124 1764 4052 Ds 1828 4124 1764 4052 1473 3736 Ds 1764 4052 1473 3736 1117 3420 Ds 1473 3736 1117 3420 1053 3369 Ds 1117 3420 1053 3369 1053 3369 Ds 10 R f (\()1239 3864 w 10 I f (X)1280 3864 w 10 R f (,)1349 3864 w 10 I f (Y)1382 3864 w 10 R f (\))1446 3864 w 1485 3764 38 38 De 1491 3764 25 25 De 1498 3764 12 12 De 10 S1 f (_____________)1440 3724 w (_____________)1440 3747 w (_____________)1440 3736 w 10 S f (\347)1103 3260 w (\347)1103 3348 w (\347)1103 3448 w (\347)1128 3260 w (\347)1128 3348 w (\347)1128 3448 w (\347)1116 3260 w (\347)1116 3348 w (\347)1116 3448 w (\347)1751 4124 w (\347)1751 4200 w (\347)1751 4300 w (\347)1751 4400 w (\347)1751 4500 w (\347)1751 4600 w (\347)1776 4124 w (\347)1776 4200 w (\347)1776 4300 w (\347)1776 4400 w (\347)1776 4500 w (\347)1776 4600 w (\347)1764 4124 w (\347)1764 4200 w (\347)1764 4300 w (\347)1764 4400 w (\347)1764 4500 w (\347)1764 4600 w (\347)1764 3548 w (\347)1764 3600 w (\347)1764 3700 w (\347)1764 3800 w (\347)1764 3900 w (\347)1764 4000 w (\347)1764 4100 w (\347)1764 4200 w (\347)1764 4300 w (\347)1764 4400 w (\347)1764 4500 w (\347)1764 4600 w 10 S1 f (_ _________________________)1 1296 1 792 3736 t (_ _________________________)1 1296 1 792 4312 t 10 S f (\347)1116 3260 w (\347)1116 3300 w (\347)1116 3400 w (\347)1116 3500 w (\347)1116 3600 w (\347)1116 3700 w (\347)1116 3800 w (\347)1116 3900 w (\347)1116 4000 w (\347)1116 4100 w (\347)1116 4200 w (\347)1116 4300 w (\347)1116 4400 w (\347)1116 4500 w (\347)1116 4600 w 10 I f (y)2973 3880 w 7 R f (0)3028 3900 w 10 R f (\()4135 3990 w 10 I f (X)4176 3990 w 10 R f (,)4245 3990 w 10 I f (Y)4278 3990 w 10 R f (\))4342 3990 w 10 I f (B)4309 4182 w 4062 3990 45 45 De 4070 3990 30 30 De 4077 3990 15 15 De 4276 4174 4338 4240 Dl 4213 4111 4275 4173 Dl 4151 4051 4213 4110 Dl 4089 3995 4151 4051 Dl 4027 3940 4089 3994 Dl 3965 3889 4027 3940 Dl 3903 3839 3965 3888 Dl 3841 3792 3903 3839 Dl 3779 3746 3841 3791 Dl 3717 3703 3779 3746 Dl 3655 3661 3717 3702 Dl 3592 3621 3654 3661 Dl 3530 3582 3592 3620 Dl 3468 3544 3530 3581 Dl 3406 3509 3468 3544 Dl 3344 3474 3406 3508 Dl 3282 3440 3344 3473 Dl 3220 3408 3282 3440 Dl 3158 3377 3220 3408 Dl 3096 3347 3158 3377 Dl 10 R f (\(b\))3686 4600 w 3788 3840 45 45 De 3795 3840 30 30 De 3803 3840 15 15 De (\()3860 3840 w 10 I f (X)3901 3840 w 10 S f (\242)3973 3835 w 10 R f (,)4014 3840 w 10 I f (Y)4047 3840 w 10 S f (\242)4114 3835 w 10 R f (\))4155 3840 w 4276 4349 4338 4437 Dl 4213 4266 4275 4348 Dl 4151 4190 4213 4266 Dl 4089 4119 4151 4190 Dl 4027 4051 4089 4118 Dl 3965 3986 4027 4050 Dl 3903 3926 3965 3986 Dl 3841 3868 3903 3925 Dl 3779 3812 3841 3867 Dl 3717 3760 3779 3812 Dl 3655 3709 3717 3759 Dl 3592 3661 3654 3709 Dl 3530 3615 3592 3661 Dl 3468 3570 3530 3614 Dl 3406 3528 3468 3570 Dl 3344 3487 3406 3527 Dl 3282 3447 3344 3486 Dl 3220 3410 3282 3447 Dl 3158 3373 3220 3409 Dl 3096 3338 3158 3373 Dl (\()3425 3980 w 10 I f (x)3466 3980 w 7 R f (0)3521 4000 w 10 R f (,)3572 3980 w 10 I f (y)3605 3980 w 7 R f (0)3660 4000 w 10 R f (\))3711 3980 w 4346 3880 4392 3880 Dl 4207 3880 4253 3880 Dl 4160 3880 4206 3880 Dl 4022 3880 4068 3880 Dl 3975 3880 4021 3880 Dl 3836 3880 3882 3880 Dl 3790 3880 3836 3880 Dl 3651 3880 3697 3880 Dl 3605 3880 3651 3880 Dl 3466 3880 3512 3880 Dl 3420 3880 3466 3880 Dl 3281 3880 3327 3880 Dl 3235 3880 3281 3880 Dl 3096 3880 3142 3880 Dl 4251 4330 4284 4360 Dl 4149 4240 4182 4270 Dl 4116 4210 4149 4240 Dl 4014 4120 4047 4150 Dl 3981 4089 4014 4119 Dl 3879 3999 3912 4029 Dl 3846 3970 3879 4000 Dl 3744 3880 3777 3910 Dl 3711 3850 3744 3880 Dl 3609 3760 3642 3790 Dl 3576 3729 3609 3759 Dl 3474 3639 3507 3669 Dl 3441 3610 3474 3640 Dl 3339 3520 3372 3550 Dl 3306 3490 3339 3520 Dl 3204 3400 3237 3430 Dl 10 S f (\347)4284 3260 w (\347)4284 3300 w (\347)4284 3400 w (\347)4284 3500 w (\347)4284 3600 w (\347)4284 3700 w (\347)4284 3800 w (\347)4284 3900 w (\347)4284 4000 w (\347)4284 4100 w (\347)4284 4200 w (\347)4284 4300 w (\347)4284 4400 w (\347)4284 4500 w (\347)4284 4600 w (\347)3204 3260 w (\347)3204 3300 w (\347)3204 3400 w (\347)3204 3500 w (\347)3204 3600 w (\347)3204 3700 w (\347)3204 3800 w (\347)3204 3900 w (\347)3204 4000 w (\347)3204 4100 w (\347)3204 4200 w (\347)3204 4300 w (\347)3204 4400 w (\347)3204 4500 w (\347)3204 4600 w 10 I f (B)4309 4360 w 10 S f (\242)4381 4355 w (=)4430 4360 w 10 R f (\()4501 4360 w 10 I f (x)4542 4360 w 7 R f (0)4597 4380 w 10 S f (+)4656 4360 w 10 S1 f ()4727 4360 w 4727 4360 m 75 build_12 4802 4360 m 10 R f (,)4810 4360 w 10 I f (y)4843 4360 w 7 R f (0)4898 4380 w 10 S f (-)4957 4360 w 10 S1 f ()5028 4360 w 5028 4360 m 75 build_12 5103 4360 m 10 R f (\))5111 4360 w (\()2388 3499 w 10 I f (x)2429 3499 w 7 R f (0)2484 3519 w 10 S f (-)2543 3499 w 10 S1 f ()2614 3499 w 2614 3499 m 75 build_12 2689 3499 m 10 R f (,)2697 3499 w 10 I f (y)2730 3499 w 7 R f (0)2785 3519 w 10 S f (+)2844 3499 w 10 S1 f ()2915 3499 w 2915 3499 m 75 build_12 2990 3499 m 10 R f (\))2998 3499 w 10 S f (=)3047 3499 w 10 I f (A)3118 3499 w 10 R f (Figure 6.)1 361 1 2699 4876 t 10 I f (a)2391 5326 w 10 S f (\242)2452 5321 w 7 R f (2)2482 5286 w 10 R f (\()2239 5166 w 10 I f (x)2280 5166 w 7 R f (0)2335 5186 w 10 S f (-)2418 5166 w 10 S1 f ()2513 5166 w 2513 5166 m 75 build_12 2588 5166 m 10 R f (\))2596 5166 w 7 R f (2)2634 5126 w 10 S1 f (_ _________)1 468 1 2224 5216 t 10 S f (+)2751 5246 w 10 I f (b)3032 5326 w 10 S f (\242)3093 5321 w 7 R f (2)3123 5286 w 10 R f (\()2880 5166 w 10 I f (y)2921 5166 w 7 R f (0)2976 5186 w 10 S f (+)3059 5166 w 10 S1 f ()3154 5166 w 3154 5166 m 75 build_12 3229 5166 m 10 R f (\))3237 5166 w 7 R f (2)3275 5126 w 10 S1 f (_ _________)1 468 1 2865 5216 t 10 S f (=)3392 5246 w 10 R f (1)3496 5246 w 10 I f (a)2391 5636 w 10 S f (\242)2452 5631 w 7 R f (2)2482 5596 w 10 R f (\()2239 5476 w 10 I f (x)2280 5476 w 7 R f (0)2335 5496 w 10 S f (+)2418 5476 w 10 S1 f ()2513 5476 w 2513 5476 m 75 build_12 2588 5476 m 10 R f (\))2596 5476 w 7 R f (2)2634 5436 w 10 S1 f (_ _________)1 468 1 2224 5526 t 10 S f (+)2751 5556 w 10 I f (b)3032 5636 w 10 S f (\242)3093 5631 w 7 R f (2)3123 5596 w 10 R f (\()2880 5476 w 10 I f (y)2921 5476 w 7 R f (0)2976 5496 w 10 S f (-)3059 5476 w 10 S1 f ()3154 5476 w 3154 5476 m 75 build_12 3229 5476 m 10 R f (\))3237 5476 w 7 R f (2)3275 5436 w 10 S1 f (_ _________)1 468 1 2865 5526 t 10 S f (=)3392 5556 w 10 R f (1)3496 5556 w (Solve simultaneously for)2 1000 1 720 5796 t 10 I f (a)1745 5796 w 10 S f (\242)1806 5791 w 7 R f (2)1836 5756 w 10 R f (and)1904 5796 w 10 I f (b)2073 5796 w 10 S f (\242)2134 5791 w 7 R f (2)2164 5756 w 10 R f (:)2207 5796 w 10 I f (a)2218 6036 w 10 S f (\242)2279 6031 w 7 R f (2)2309 5996 w 10 S f (=)2401 6036 w 10 I f (y)2974 6106 w 7 R f (0)3029 6126 w 10 R f (\()2530 5956 w 10 I f (x)2571 5956 w 7 R f (0)2626 5976 w 10 S f (+)2709 5956 w 10 I f (y)2804 5956 w 7 R f (0)2859 5976 w 10 R f (\) \()1 74 1 2910 5956 t 10 I f (x)2992 5956 w 7 R f (0)3047 5976 w 10 I f (y)3098 5956 w 7 R f (0)3153 5976 w 10 S f (+)3236 5956 w 10 R f (1)3331 5956 w 10 I f (/)3389 5956 w 10 R f (4 \))1 91 1 3425 5956 t 10 S1 f (_ ____________________)1 1016 1 2515 6006 t 10 I f (b)2218 6346 w 10 S f (\242)2279 6341 w 7 R f (2)2309 6306 w 10 S f (=)2401 6346 w 10 I f (x)2974 6416 w 7 R f (0)3029 6436 w 10 R f (\()2530 6266 w 10 I f (x)2571 6266 w 7 R f (0)2626 6286 w 10 S f (+)2709 6266 w 10 I f (y)2804 6266 w 7 R f (0)2859 6286 w 10 R f (\) \()1 74 1 2910 6266 t 10 I f (x)2992 6266 w 7 R f (0)3047 6286 w 10 I f (y)3098 6266 w 7 R f (0)3153 6286 w 10 S f (+)3236 6266 w 10 R f (1)3331 6266 w 10 I f (/)3389 6266 w 10 R f (4 \))1 91 1 3425 6266 t 10 S1 f (_ ____________________)1 1016 1 2515 6316 t 10 R f (Substituting in \(2\), we find)4 1081 1 720 6596 t 10 I f (Y)2483 6836 w 10 S f (\242)2550 6831 w 7 R f (2)2580 6796 w 10 S f (=)2672 6836 w 10 I f (y)2776 6836 w 7 R f (0)2825 6855 w (2)2825 6796 w 10 S f (+)2917 6836 w 10 R f (4)3046 6906 w (1)3046 6776 w 10 S1 f (_ _)1 80 1 3031 6806 t 10 I f (x)3154 6906 w 7 R f (0)3209 6926 w 10 I f (y)3154 6756 w 7 R f (0)3209 6776 w 10 S1 f (_ __)1 128 1 3139 6806 t 10 R f (\(5\))4924 6836 w 8 S1 f (__________________)720 7006 w 8 R f (* Somewhere between the ends of a chord of a smooth curve, the tangent to the curve must be parallel to the chord.)22 3671 1 720 7106 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 66 524 764 %%EndPage: 13 13 %%Page: 14 14 %%PageBoundingBox: (atend) /saveobj save def mark 14 pagesetup 10 R f (- 14 -)2 216 1 2772 480 t (Consider now the one-parameter family of ellipses that pass through)9 2741 1 970 840 t 10 I f (A)3737 840 w 10 R f ( member has arc)3 659(. One)1 242 2 3798 840 t 10 I f (AB)4725 840 w 10 S f (\242)4858 835 w 10 R f (. A)1 149 1 4891 840 t (second has arc)2 581 1 720 960 t 10 I f (AB)1326 960 w 10 R f (; let)1 153 1 1448 960 t 10 I f (a)1626 960 w 10 R f (and)1701 960 w 10 I f (b)1870 960 w 10 R f ( Then)1 255(be the lengths of its semiaxes.)5 1199 2 1945 960 t 10 I f (a)2406 1290 w 7 R f (2)2467 1250 w 10 R f (\()2239 1130 w 10 I f (x)2280 1130 w 7 R f (0)2335 1150 w 10 S f (-)2418 1130 w 10 S1 f ()2513 1130 w 2513 1130 m 75 build_12 2588 1130 m 10 R f (\))2596 1130 w 7 R f (2)2634 1090 w 10 S1 f (_ _________)1 468 1 2224 1180 t 10 S f (+)2751 1210 w 10 I f (b)3047 1290 w 7 R f (2)3108 1250 w 10 R f (\()2880 1130 w 10 I f (y)2921 1130 w 7 R f (0)2976 1150 w 10 S f (+)3059 1130 w 10 S1 f ()3154 1130 w 3154 1130 m 75 build_12 3229 1130 m 10 R f (\))3237 1130 w 7 R f (2)3275 1090 w 10 S1 f (_ _________)1 468 1 2865 1180 t 10 S f (=)3392 1210 w 10 R f (1)3496 1210 w (Since)720 1450 w 10 I f (x)974 1450 w 7 R f (0)1029 1470 w 10 R f (and)1104 1450 w 10 I f (y)1280 1450 w 7 R f (0)1335 1470 w 10 R f (are fixed,)1 384 1 1410 1450 t 10 I f (a)1827 1450 w 10 R f (and)1910 1450 w 10 I f (b)2087 1450 w 10 R f ( \()1 66( Let)1 191(must vary inversely with each other.)5 1493 3 2170 1450 t 10 I f (X)3928 1450 w 10 R f (,)3997 1450 w 10 I f (Y)4030 1450 w 10 R f (\) be the juncture of the)5 946 1 4094 1450 t ( that as)2 291( \(2\) we see)3 449( From)1 272(second ellipse.)1 593 4 720 1570 t 10 I f (a)2354 1570 w 10 R f (increases and)1 538 1 2433 1570 t 10 I f (b)3000 1570 w 10 R f (decreases,)3079 1570 w 10 I f (Y)3514 1570 w 10 R f ( Sup-)1 243(must decrease and vice versa.)4 1198 2 3599 1570 t (pose the curve in Figure 6a to be the curve)9 1798 1 720 1690 t 10 I f (AB)2554 1690 w 10 R f ( juncture must lie at least one half unit)8 1628( Its)1 162(in Figure 6b.)2 538 3 2712 1690 t (below)720 1810 w 10 I f (A)998 1810 w 10 R f (; thus)1 229 1 1059 1810 t 10 I f (Y)1322 1810 w 10 S f (<)1427 1810 w 10 I f (y)1531 1810 w 7 R f (0)1586 1830 w 10 R f ( \(5\),)1 175(. From)1 301 2 1629 1810 t 10 I f (y)2139 1810 w 7 R f (0)2194 1830 w 10 S f (<)2286 1810 w 10 I f (Y)2390 1810 w 10 S f (\242)2457 1805 w 10 R f (, so)1 148 1 2490 1810 t 10 I f (Y)2672 1810 w 10 S f (<)2777 1810 w 10 I f (Y)2881 1810 w 10 S f (\242)2948 1805 w 10 R f ( the inverse variation of)4 984(. From)1 301 2 2981 1810 t 10 I f (a)4300 1810 w 10 R f (and)4384 1810 w 10 I f (Y)4562 1810 w 10 R f (it follows)1 389 1 4651 1810 t (that)720 1930 w 10 I f (a)895 1930 w 10 S f (>)994 1930 w 10 I f (a)1098 1930 w 10 S f (\242)1159 1925 w 10 R f (and arc)1 290 1 1217 1930 t 10 I f (AB)1532 1930 w 10 R f (must lie outside arc)3 783 1 1680 1930 t 10 I f (AB)2489 1930 w 10 S f (\242)2622 1925 w 10 R f ( Figure 6b requires the ordinates of)6 1409( Therefore)1 443(, as shown.)2 446 3 2655 1930 t 10 I f (A)4979 1930 w 10 R f (and)720 2050 w 10 I f (B)895 2050 w 10 R f ( completes the)2 587( This)1 233( requires them to differ by more than 1.)8 1612(to differ by less than 1, while Figure 6a)8 1621 4 987 2050 t (proof of the impossibility of configuration)5 1691 1 720 2170 t 10 B f (3)2436 2170 w 10 R f ( does not depend on the assumption)6 1433( the proof)2 388(. Since)1 297 3 2486 2170 t 10 I f (a)4630 2170 w 10 S f (\263)4721 2170 w 10 I f (b)4817 2170 w 10 R f (, the)1 173 1 4867 2170 t (transposed configuration)1 990 1 720 2290 t 10 B f (9)1735 2290 w 10 R f (is also impossible.)2 737 1 1810 2290 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 547 514 764 %%EndPage: 14 14 %%Page: 15 15 %%PageBoundingBox: (atend) /saveobj save def mark 15 pagesetup 10 R f (- 15 -)2 216 1 2772 480 t 10 B f ( in C)2 206( Implementation)1 733(Appendix 3.)1 518 3 720 840 t 10 R f ( on the model in the text, and incorporates simplifications discussed there plus)12 3176(This program is based)3 894 2 970 996 t ( initializer for)2 549( The)1 205(other routine optimizations.)2 1108 3 720 1116 t 10 CW f (t)2607 1116 w 10 R f (has been specialized to take into account the known values)9 2348 1 2692 1116 t (of)720 1236 w 10 CW f (x)833 1236 w 10 R f (and)923 1236 w 10 CW f (y)1098 1236 w 10 R f ( strength of most multiplications)4 1325( The)1 211( calculations have been moved out of the loop.)8 1909(. Constant)1 437 4 1158 1236 t ( Variables)1 442( been reduced.)2 598(in the loop has)3 619 3 720 1356 t 10 CW f (xc)2414 1356 w 10 R f (and)2569 1356 w 10 CW f (yc)2748 1356 w 10 R f ( The)1 215(are the coordinates of the center of the ellipse.)8 1922 2 2903 1356 t (arithmetic fits in 32-bit)3 946 1 720 1476 t 10 CW f (long)1698 1476 w 10 R f (integers for values of)3 866 1 1970 1476 t 10 CW f (a)2868 1476 w 10 R f (and)2960 1476 w 10 CW f (b)3137 1476 w 10 R f (less than 896; the exact value has been con-)8 1810 1 3230 1476 t ( the comma operator denotes serial, not parallel, assignment.)8 2421( Beware,)1 379(firmed experimentally.)1 915 3 720 1596 t 10 CW f (extern void point\(int, int\);)3 1680 1 720 1752 t (#define incx\(\) x++, dxt += d2xt, t += dxt)8 2460 1 720 1908 t (#define incy\(\) y--, dyt += d2yt, t += dyt)8 2460 1 720 2028 t (void ellipse\(int xc, int yc, int a, int b\))8 2520 1 720 2184 t ( e\(x,y\) = b\3032*x\3032 + a\3032*y\3032 - a\3032*b\3032 */)8 2400({ /*)1 1560 2 720 2304 t (int x = 0, y = b;)6 1020 1 1200 2424 t (long a2 = \(long\)a*a, b2 = \(long\)b*b;)6 2160 1 1200 2544 t (long crit1 = -\(a2/4 + a%2 + b2\);)7 1920 1 1200 2664 t (long crit2 = -\(b2/4 + b%2 + a2\);)7 1920 1 1200 2784 t (long crit3 = -\(b2/4 + b%2\);)5 1620 1 1200 2904 t (long t = -a2*y; /* t = e\(x+1/2,y-1/2\) - \(a\3032+b\3032\)/4 */)10 3240 1 1200 3024 t (long dxt = 2*b2*x, dyt = -2*a2*y;)6 1980 1 1200 3144 t (long d2xt = 2*b2, d2yt = 2*a2;)6 1800 1 1200 3264 t (while\(y>=0 && x<=a\) {)3 1260 1 1200 3420 t (point\(xc+x, yc+y\);)1 1080 1 1680 3540 t (if\(x!=0 || y!=0\))2 960 1 1680 3660 t (point\(xc-x, yc-y\);)1 1080 1 2160 3780 t (if\(x!=0 && y!=0\) {)3 1080 1 1680 3900 t (point\(xc+x, yc-y\);)1 1080 1 2160 4020 t (point\(xc-x, yc+y\);)1 1080 1 2160 4140 t (})1680 4260 w ( e\(x+1,y-1/2\) <= 0 */)4 1260( /*)1 300(if\(t + b2*x <= crit1 ||)5 1380 3 1680 4380 t ( e\(x+1/2,y\) <= 0 */)4 1140( /*)1 420(t + a2*y <= crit3\))4 1080 3 1860 4500 t (incx\(\);)2160 4620 w (else if\(t - a2*y > crit2\) /* e\(x+1/2,y-1\) > 0 */)10 2880 1 1680 4740 t (incy\(\);)2160 4860 w (else {)1 360 1 1680 4980 t (incx\(\);)2160 5100 w (incy\(\);)2160 5220 w (})1680 5340 w (})1200 5460 w (})720 5580 w 10 B f (Optimization)720 5820 w 10 R f ( are justifiable unless point-drawing is)5 1639(Further modifications for efficiency are possible, but few)7 2431 2 970 5976 t ( subex-)1 296( Common)1 428( are unusually common in relation to other graphic primitives.)9 2526(unusually fast and ellipses)3 1070 4 720 6096 t ( multiplications can be reduced to additions.)6 1828( More)1 276( constants propagated.)2 910(pressions can be eliminated and)4 1306 4 720 6216 t (There is no need to test)5 934 1 720 6336 t 10 I f (x)1680 6336 w 10 S f (\243)1765 6336 w 10 I f (a)1861 6336 w 10 R f (unless)1937 6336 w 10 I f (y)2213 6336 w 10 S f (=)2306 6336 w 10 R f ( by splitting the single loop into)6 1274( tests can be eliminated)4 936(0. Further)1 420 3 2410 6336 t (four: vertical tail,)2 698 1 720 6456 t 8 R f (NNE)1444 6456 w 10 R f (arc,)1635 6456 w 8 R f (ENE)1802 6456 w 10 R f ( which)1 271( are only south steps in the vertical tail,)8 1576( There)1 283(arc, and horizontal tail.)3 926 4 1984 6456 t ( which continues while)3 927(continues while south steps are possible, only east steps in the horizontal tail,)12 3111 2 720 6576 t 10 I f (x)4784 6576 w 10 S f (\243)4869 6576 w 10 I f (a)4965 6576 w 10 R f (.)5015 6576 w (There are no east steps in the)6 1176 1 720 6696 t 8 R f (ENE)1924 6696 w 10 R f ( step in the)3 448(arc, which may begin at the first south)7 1552 2 2108 6696 t 8 R f (NNE)4137 6696 w 10 R f ( in the)2 258(arc. Points)1 451 2 4331 6696 t ( reflected vertically, in the arcs vertically and horizontally, and in the horizontal tail horizon-)14 3718(vertical tail are)2 602 2 720 6816 t (tally unless)1 453 1 720 6936 t 10 I f (a)1198 6936 w 10 S f (=)1297 6936 w 10 R f ( rarely effective)2 630(0. The)1 280 2 1401 6936 t 10 I f (EH)2336 6936 w 10 R f (test \()1 197 1 2494 6936 t 10 CW f (crit3)2691 6936 w 10 R f (\) may be placed last in the loop for the)9 1539 1 2991 6936 t 8 R f (NNE)4555 6936 w 10 R f (arc, the)1 294 1 4746 6936 t (only place where it remains necessary.)5 1540 1 720 7056 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 70 514 764 %%EndPage: 15 15 %%Page: 16 16 %%PageBoundingBox: (atend) /saveobj save def mark 16 pagesetup 10 R f (- 16 -)2 216 1 2772 480 t 10 B f (Testing)720 840 w 10 R f ( of theorems becomes more)4 1143(Mathematical proofs have lives of their own, and evolve as the context)11 2927 2 970 996 t (fully explored.)1 595 1 720 1116 t 7 R f (13)1320 1076 w 10 R f ( takes the form of)4 728(This phenomenon appears also in programming, where the exploration)8 2882 2 1430 1116 t ( bridging the gap between proof and imple-)7 1789( has the further complication of)5 1291( Programming)1 607(testing and use.)2 633 4 720 1236 t ( testing is never amiss.)4 908( Skeptical)1 422(mentation: does the code faithfully mirror what was proved?)8 2418 3 720 1356 t (Besides various sporadic checks, the C program has been tested)9 2545 1 720 1512 t (For all values of)3 652 1 970 1668 t 10 I f (a)1647 1668 w 10 R f (and)1722 1668 w 10 I f (b)1891 1668 w 10 R f (in the range 0 to 4.)5 749 1 1966 1668 t (For circles of integer radius up to 20 and of radius divisible by 100 up to 1000.)16 3150 1 970 1824 t ( 8 times the)3 467(Over small ranges around the first few critical points for tails, i.e. with one parameter near)15 3603 2 970 1980 t (square of the other, in both orientations.)6 1598 1 970 2100 t (For)970 2256 w 10 I f (a)1134 2256 w 10 S f (=)1233 2256 w 10 R f (1000 and)1 369 1 1337 2256 t 10 I f (b)1731 2256 w 10 S f (=)1830 2256 w 10 R f (1 and vice versa.)3 670 1 1934 2256 t (For)970 2412 w 10 I f (a)1134 2412 w 10 R f (and)1209 2412 w 10 I f (b)1378 2412 w 10 R f (in the range [890,900], which spans the onset of 32-bit overflow.)10 2593 1 1453 2412 t ( for only four radii less than 1000, namely 4,)9 1802(For square corners in approximate circles, which happen)7 2268 2 970 2568 t (11, 134, and 373.)3 694 1 970 2688 t 7 R f (1)1669 2648 w 10 R f (For the cases pictured in Appendix 1, the parameters for which were independently determined.)13 3822 1 970 2844 t ( each test case some of the)6 1094( For)1 195( checked mainly by mathematical, not merely visual, criteria.)8 2491(Outputs were)1 540 4 720 3000 t (following checks were made.)3 1169 1 720 3120 t ( 0 ,)2 91(Termination: a quadrant beginning at \()5 1545 2 970 3276 t 10 I f (b)2614 3276 w 10 R f (\) ends at \()3 396 1 2672 3276 t 10 I f (a)3076 3276 w 10 R f (, 0 \).)2 149 1 3134 3276 t (Symmetry: the approximation for \()4 1399 1 970 3432 t 10 I f (a)2377 3432 w 10 R f (,)2435 3432 w 10 I f (b)2468 3432 w 10 R f (\) mirrors that for \()4 726 1 2526 3432 t 10 I f (b)3260 3432 w 10 R f (,)3318 3432 w 10 I f (a)3351 3432 w 10 R f (\); circles have eight-fold symmetry.)4 1428 1 3409 3432 t (Continuity: no coordinate changes by more than one at any step.)10 2572 1 970 3588 t (Thinness: a quadrant has at most one square corner.)8 2065 1 970 3744 t (Square corners happen where predicted.)4 1597 1 970 3900 t (Spot checks against ellipse coordinates calculated in floating point.)8 2683 1 970 4056 t (The dimensions at onset of tails.)5 1294 1 970 4212 t ( from Wirth's algorithm, sometimes for agreement and sometimes for pre-)10 2990(Comparison against output)2 1080 2 970 4368 t (dicted disagreement.)1 826 1 970 4488 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 327 514 764 %%EndPage: 16 16 %%Page: 17 17 %%PageBoundingBox: (atend) /saveobj save def mark 17 pagesetup 10 R f (- 17 -)2 216 1 2772 480 t 12 B f (Math before Code:)2 972 1 2519 840 t (A Soundly Derived Ellipse-drawing Algorithm)4 2384 1 1813 960 t 10 I f (M. Douglas McIlroy)2 818 1 2596 1200 t 10 R f (AT&T Bell Laboratories)2 993 1 2508 1380 t (Murray Hill, New Jersey 07974)4 1267 1 2371 1500 t (ABSTRACT)2743 1740 w ( mathematically as the problem of construct-)6 1792(The problem of drawing an ellipse on a raster is posed)10 2168 2 720 2100 t ( to generate the)3 654( straightforward program is proved)4 1457( A)1 137(ing a Freeman approximation to a curve.)6 1712 4 720 2220 t ( logic having been)3 774( Its)1 163( into an efficient all-integer scheme.)5 1506(approximation, and then transformed)3 1517 4 720 2340 t ( free of anomalies that bedevil previously published programs,)8 2507(shaped by mathematics, the result is)5 1453 2 720 2460 t ( Mathematics)1 576( mathematical specification.)2 1143(the outlines of which were designed with only cursory)8 2241 3 720 2580 t ( for)1 148( outlines, borrowed from the most efficient algorithms)7 2226( The)1 213(was used mainly to fill in details.)6 1373 4 720 2700 t ( to be readily generalizable)4 1104(drawing circles, didn't work because they had been specialized too far)10 2856 2 720 2820 t (in a new direction.)3 743 1 720 2940 t ( serious pitfall in software evolution: advanced code is often)9 2428(This small example highlights a)4 1282 2 970 3096 t (an inappropriate platform from which to launch radical advances in code.)10 2929 1 720 3216 t 10 B f (Introduction)720 3456 w 10 R f ( beyond the relatively intuitive development in the accompanying paper, ``Getting)10 3337(This note attempts to go)4 983 2 720 3612 t (Raster Ellipses Right,'' to give a clear outline of a correctness proof of a simple ellipse-drawing algorithm.)16 4271 1 720 3732 t (The algorithm generates a Freeman approximation,)5 2071 1 720 3888 t 8 R f (1)2799 3856 w 10 R f ( by plotting on each grid)5 1010(wherein a curve is quantized)4 1161 2 2869 3888 t ( approximation enjoys)2 908( Freeman)1 407(line the nearest grid point to each intersection of that line with the curve.)13 3005 3 720 4008 t ( describable, readily computable, and respectful of the symmetries)8 2722(the distinction of being mathematically)4 1598 2 720 4128 t (of the grid, in the sense that approximation commutes with the symmetry operations.)12 3392 1 720 4248 t ( subtext is that formal analysis often has a place in the practical development even)14 3367(A not-so-deeply hidden)2 953 2 720 4404 t ( is not perfectly clear, it will help to spell it)10 1794( the objective of a program)5 1118( If)1 124(of quite ``obvious'' algorithms.)3 1284 4 720 4524 t ( The)1 208( for critical invariants.)3 896( Ditto)1 259( and to be clear about why the program meets the objective.)11 2417(out precisely,)1 540 5 720 4644 t ( redoubts of seat-of-the-)3 988(message is not new, but perhaps the application area is; graphics is one of many)14 3332 2 720 4764 t ( mathematical)1 570(pants programming, where mathematical understanding of the application is not matched by)11 3750 2 720 4884 t (analysis of the code.)3 815 1 720 5004 t ( the algorithm is contrasted with previous ones in the literature, all of which can)14 3333(At the end of the paper,)5 987 2 720 5160 t ( the method of development, from mathematics)6 1901( is argued that)3 569( It)1 113(produce mathematically anomalous results.)3 1737 4 720 5280 t (to program logic and not vice versa, is responsible for the better behavior of the present algorithm.)16 3932 1 720 5400 t 10 B f (Terminology)720 5640 w 10 R f (A)720 5796 w 10 I f (point P)1 292 1 817 5796 t 10 R f (is a coordinate pair \()4 820 1 1134 5796 t 10 I f (P. x)1 138 1 1962 5796 t 10 R f (,)2108 5796 w 10 I f (P. y)1 138 1 2141 5796 t 10 R f (\).)2287 5796 w (Point)720 5952 w 10 I f (P)960 5952 w 10 R f (is)1049 5952 w 10 I f (north)1144 5952 w 10 R f (of point)1 317 1 1389 5952 t 10 I f (Q)1734 5952 w 10 R f (if)1834 5952 w 10 I f (P. y)1 138 1 1923 5952 t 10 S f (>)2085 5952 w 10 I f (Q. y)1 149 1 2156 5952 t 10 R f (, and)1 197 1 2305 5952 t 10 I f (directly north)1 550 1 2530 5952 t 10 R f (of)3108 5952 w 10 I f (Q)3219 5952 w 10 R f (if)3319 5952 w 10 I f (P. x)1 138 1 3408 5952 t 10 S f (=)3570 5952 w 10 I f (Q. x)1 149 1 3641 5952 t 10 R f (and)3818 5952 w 10 I f (P. y)1 138 1 3990 5952 t 10 S f (>)4152 5952 w 10 I f (Q. y)1 149 1 4223 5952 t 10 R f ( defini-)1 295(. Similar)1 373 2 4372 5952 t (tions hold for east, south, and west.)6 1413 1 720 6072 t (Point)720 6228 w 10 I f (P)960 6228 w 10 R f (is)1049 6228 w 10 I f (northeast)1144 6228 w 10 R f (of point)1 317 1 1550 6228 t 10 I f (Q)1895 6228 w 10 R f (if)1995 6228 w 10 I f (P)2085 6228 w 10 R f (is both north and east of)5 983 1 2175 6228 t 10 I f (Q)3187 6228 w 10 R f (, and similarly for southeast, southwest, and)6 1781 1 3259 6228 t (northwest.)720 6348 w (A)720 6504 w 10 I f (grid point)1 401 1 820 6504 t 10 R f ( may be)2 324( it can be inferred from context, a grid point)9 1779( When)1 291(is a point with integer coordinates.)5 1397 4 1249 6504 t (referred to simply as a point.)5 1148 1 720 6624 t (The neighbor functions)2 946 1 720 6780 t 10 I f (north)1698 6780 w 10 R f (\()1923 6780 w 10 I f (P)1964 6780 w 10 R f (\) denotes the nearest to)4 948 1 2033 6780 t 10 I f (P)3013 6780 w 10 R f ( points directly north of point)5 1215(among all grid)2 597 2 3106 6780 t 10 I f (P)4951 6780 w 10 R f (;)5012 6780 w (and)720 6900 w 10 I f (northeast)891 6900 w 10 R f (\()1277 6900 w 10 I f (P)1318 6900 w 10 R f (\) denotes the nearest to grid point)6 1349 1 1387 6900 t 10 I f (P)2763 6900 w 10 R f ( grid points northeast of)4 959(among all)1 399 2 2851 6900 t 10 I f (P)4235 6900 w 10 R f ( in the)2 252(. Neighbors)1 492 2 4296 6900 t (remaining six principal compass directions are designated similarly.)7 2719 1 720 7020 t (Let function)1 503 1 720 7176 t 10 I f (e)1260 7176 w 10 R f (be defined by)2 567 1 1341 7176 t 10 I f (e)1945 7176 w 10 R f (\()1997 7176 w 10 I f (a)2038 7176 w 10 R f (,)2096 7176 w 10 I f (b)2129 7176 w 10 R f (,)2187 7176 w 10 I f (x)2220 7176 w 10 R f (,)2272 7176 w 10 I f (y)2305 7176 w 10 R f (\))2357 7176 w 10 S f (=)2447 7176 w 10 I f (b)2551 7176 w 7 R f (2)2612 7136 w 10 I f (x)2663 7176 w 7 R f (2)2718 7136 w 10 S f (+)2810 7176 w 10 I f (a)2914 7176 w 7 R f (2)2975 7136 w 10 I f (y)3026 7176 w 7 R f (2)3081 7136 w 10 S f (-)3173 7176 w 10 I f (a)3277 7176 w 7 R f (2)3338 7136 w 10 I f (b)3389 7176 w 7 R f (2)3450 7136 w 10 R f (. If)1 153 1 3493 7176 t 10 I f (a)3683 7176 w 10 S f (>)3757 7176 w 10 R f (0 and)1 231 1 3828 7176 t 10 I f (b)4096 7176 w 10 S f (>)4170 7176 w 10 R f (0,)4241 7176 w 10 I f (the ellipse)1 421 1 4354 7176 t 10 R f (is the)1 227 1 4813 7176 t (curve in the)2 471 1 720 7296 t 10 I f (x)1216 7296 w 10 R f (-)1260 7296 w 10 I f (y)1293 7296 w 10 R f (plane defined by)2 665 1 1362 7296 t 10 I f (e)2052 7296 w 10 R f (\()2104 7296 w 10 I f (a)2145 7296 w 10 R f (,)2203 7296 w 10 I f (b)2236 7296 w 10 R f (,)2294 7296 w 10 I f (x)2327 7296 w 10 R f (,)2379 7296 w 10 I f (y)2412 7296 w 10 R f (\))2464 7296 w 10 S f (=)2513 7296 w 10 R f (0, with the traditional canonical form)5 1487 1 2584 7296 t 10 I f (x)4096 7296 w 7 R f (2)4151 7256 w 10 I f (/ a)1 86 1 4202 7296 t 7 R f (2)4299 7256 w 10 S f (+)4391 7296 w 10 I f (y)4495 7296 w 7 R f (2)4550 7256 w 10 I f (/ b)1 86 1 4601 7296 t 7 R f (2)4698 7256 w 10 S f (=)4790 7296 w 10 R f (1.)4894 7296 w cleartomark showpage saveobj restore %%PageBoundingBox: 61 46 514 764 %%EndPage: 17 17 %%Page: 18 18 %%PageBoundingBox: (atend) /saveobj save def mark 18 pagesetup 10 R f (- 18 -)2 216 1 2772 480 t (In the first quadrant the ellipse may be equivalently specified by)10 2568 1 720 840 t 10 I f (y)3313 840 w 10 S f (=)3381 840 w 10 I f (f)3460 840 w 10 R f (\()3504 840 w 10 I f (a)3545 840 w 10 R f (,)3603 840 w 10 I f (b)3636 840 w 10 R f (,)3694 840 w 10 I f (x)3727 840 w 10 R f (\) or by)2 266 1 3779 840 t 10 I f (x)4070 840 w 10 S f (=)4138 840 w 10 I f (f)4217 840 w 10 R f (\()4261 840 w 10 I f (b)4302 840 w 10 R f (,)4360 840 w 10 I f (a)4393 840 w 10 R f (,)4451 840 w 10 I f (y)4484 840 w 10 R f (\), where)1 326 1 4536 840 t 10 I f (f)1220 1038 w 10 R f (\()1264 1038 w 10 I f (a)1305 1038 w 10 R f (,)1363 1038 w 10 I f (b)1396 1038 w 10 R f (,)1454 1038 w 10 I f (x)1487 1038 w 10 R f (\))1539 1038 w 10 S f (=)1629 1038 w 10 I f (b)1733 1038 w 12 S f (\326` ```````)1 513 1 1791 1038 t 10 R f (1)1865 1038 w 10 S f (-)1955 1038 w 10 I f (x)2050 1038 w 7 R f (2)2105 998 w 10 I f (/ a)1 86 1 2156 1038 t 7 R f (2)2253 998 w 10 R f (, 0)1 182 1 2320 1038 t 10 S f (\243)2518 1038 w 10 I f (x)2589 1038 w 10 S f (\243)2657 1038 w 10 I f (a)2728 1038 w 10 R f (, 0)1 116 1 2786 1038 t 10 S f (<)2918 1038 w 10 I f (a)2989 1038 w 10 R f (, 0)1 116 1 3047 1038 t 10 S f (<)3179 1038 w 10 I f (b.)3250 1038 w 10 R f (If)720 1254 w 10 I f (a)811 1254 w 10 S f (=)885 1254 w 10 R f (0 the ellipse degenerates to a north-south segment; if)8 2108 1 956 1254 t 10 I f (b)3089 1254 w 10 S f (=)3163 1254 w 10 R f (0 it degenerates to an east-west segment.)6 1627 1 3234 1254 t (Associated with each grid point in the first quadrant is a)10 2306 1 720 1410 t 10 I f (vertical bar)1 476 1 3058 1410 t 10 R f (denoted)3566 1410 w 10 I f (P. V)1 155 1 3914 1410 t 10 R f ( a north-south)2 571(, which is)2 400 2 4069 1410 t (segment of length 1 centered on)5 1323 1 720 1530 t 10 I f (P)2077 1530 w 10 R f (, and a similarly centered)4 1042 1 2138 1530 t 10 I f ( H)1 80(horizontal bar, P.)2 728 2 3214 1530 t 10 R f ( vertical bar is half)4 780(. The)1 238 2 4022 1530 t (open to the north; the horizontal bar is half open to the east.)12 2383 1 720 1650 t (The predicate)1 554 1 720 1806 t 10 I f (V)1308 1806 w 10 R f (\()1377 1806 w 10 I f (P)1418 1806 w 10 R f (\) means that the ellipse intersects)5 1368 1 1487 1806 t 10 I f (P. V)1 155 1 2889 1806 t 10 R f (and)3078 1806 w 10 I f (H)3256 1806 w 10 R f (\()3336 1806 w 10 I f (P)3377 1806 w 10 R f (\) means that the ellipse intersects)5 1368 1 3446 1806 t 10 I f (P. H)1 166 1 4849 1806 t 10 R f (.)5015 1806 w (Formally,)720 1926 w 10 I f (V)1220 2082 w 10 R f (\()1289 2082 w 10 I f (P)1330 2082 w 10 R f (\))1399 2082 w 10 S f (\272)1481 2082 w 10 I f (P. y)1 138 1 1577 2082 t 10 S f (-)1739 2082 w 10 S1 f ()1810 2082 w 1810 2082 m 75 build_12 1885 2082 m 10 S f (\243)1901 2082 w 10 I f (f)1980 2082 w 10 R f (\()2024 2082 w 10 I f (a)2065 2082 w 10 R f (,)2123 2082 w 10 I f (b)2156 2082 w 10 R f (,)2214 2082 w 10 I f (P. x)1 138 1 2247 2082 t 10 R f (\))2393 2082 w 10 S f (<)2442 2082 w 10 I f (P. y)1 138 1 2513 2082 t 10 S f (+)2675 2082 w 10 S1 f ()2746 2082 w 2746 2082 m 75 build_12 2821 2082 m 10 R f (,)2821 2082 w 10 I f (H)1220 2202 w 10 R f (\()1300 2202 w 10 I f (P)1341 2202 w 10 R f (\))1410 2202 w 10 S f (\272)1492 2202 w 10 I f (P. x)1 138 1 1588 2202 t 10 S f (-)1750 2202 w 10 S1 f ()1821 2202 w 1821 2202 m 75 build_12 1896 2202 m 10 S f (\243)1912 2202 w 10 I f (f)1991 2202 w 10 R f (\()2035 2202 w 10 I f (b)2076 2202 w 10 R f (,)2134 2202 w 10 I f (a)2167 2202 w 10 R f (,)2225 2202 w 10 I f (P. y)1 138 1 2258 2202 t 10 R f (\))2404 2202 w 10 S f (<)2453 2202 w 10 I f (P. x)1 138 1 2524 2202 t 10 S f (+)2686 2202 w 10 S1 f ()2757 2202 w 2757 2202 m 75 build_12 2832 2202 m 10 R f (.)2832 2202 w (Grid point)1 414 1 720 2358 t 10 I f (P)1159 2358 w 10 R f ( be)1 120(is said to)2 356 2 1245 2358 t 10 I f (lighted)1747 2358 w 10 R f (if the ellipse intersects either bar, i.e. if the ellipse passes near enough to)13 2902 1 2051 2358 t 10 I f (P)4979 2358 w 10 R f (to make)1 319 1 720 2478 t 10 I f (V)1064 2478 w 10 R f (\()1133 2478 w 10 I f (P)1174 2478 w 10 R f (\) or)1 141 1 1243 2478 t 10 I f (H)1409 2478 w 10 R f (\()1489 2478 w 10 I f (P)1530 2478 w 10 R f (\) true.)1 238 1 1599 2478 t 10 B f (Basic observations)1 792 1 720 2718 t 10 R f (1. Function)1 481 1 720 2874 t 10 I f (f)1226 2874 w 10 R f (is continuous and one-to-one.)3 1182 1 1279 2874 t ( value of)2 349(2. The)1 280 2 720 3030 t 10 I f (f)1374 3030 w 10 R f (\()1418 3030 w 10 I f (a)1459 3030 w 10 R f (,)1517 3030 w 10 I f (b)1550 3030 w 10 R f (,)1608 3030 w 10 I f (x)1641 3030 w 10 R f (\) decreases monotonically as)3 1150 1 1693 3030 t 10 I f (x)2868 3030 w 10 R f (increases from)1 584 1 2937 3030 t 10 I f (x)3546 3030 w 10 S f (=)3614 3030 w 10 R f (0 to)1 153 1 3685 3030 t 10 I f (x)3863 3030 w 10 S f (=)3931 3030 w 10 I f (a)4002 3030 w 10 R f (.)4052 3030 w ( slope of the curve)4 737(3. The)1 280 2 720 3186 t 10 I f (y)1762 3186 w 10 S f (=)1830 3186 w 10 I f (f)1909 3186 w 10 R f (\()1953 3186 w 10 I f (a)1994 3186 w 10 R f (,)2052 3186 w 10 I f (b)2085 3186 w 10 R f (,)2143 3186 w 10 I f (x)2176 3186 w 10 R f (\) decreases monotonically as)3 1150 1 2228 3186 t 10 I f (x)3403 3186 w 10 R f (increases from)1 584 1 3472 3186 t 10 I f (x)4081 3186 w 10 S f (=)4149 3186 w 10 R f (0 to)1 153 1 4220 3186 t 10 I f (x)4398 3186 w 10 S f (=)4466 3186 w 10 I f (a)4537 3186 w 10 R f (.)4587 3186 w (The following three lemmas are straightforward consequences of continuity and monotonicity.)10 3780 1 720 3342 t ( a first-quadrant point)3 919( If)1 133(Lemma 1.)1 422 3 720 3498 t 10 I f (P)2236 3498 w 10 R f (is lighted, no first-quadrant point northeast or southwest of)8 2487 1 2339 3498 t 10 I f (P)4869 3498 w 10 R f (is)4973 3498 w (lighted.)720 3618 w ( If)1 116(Lemma 2.)1 405 2 720 3774 t 10 I f (P)1266 3774 w 10 R f (is lighted, and)2 564 1 1352 3774 t 10 I f (P)1941 3774 w 10 R f ( \(at least\) one of)4 652(is interior to the first quadrant, then)6 1418 2 2027 3774 t 10 I f (east)4123 3774 w 10 R f (\()4292 3774 w 10 I f (P)4333 3774 w 10 R f (\),)4402 3774 w 10 I f (southeast)4486 3774 w 10 R f (\()4872 3774 w 10 I f (P)4913 3774 w 10 R f (\),)4982 3774 w (or)720 3894 w 10 I f (south)828 3894 w 10 R f (\()1053 3894 w 10 I f (P)1094 3894 w 10 R f (\) is lighted.)2 453 1 1163 3894 t ( points)1 272( grid)1 187( If)1 117(Lemma 3.)1 406 4 720 4050 t 10 I f (P)1729 4050 w 10 R f (and)1817 4050 w 10 I f (R)1988 4050 w 10 R f (, where)1 295 1 2049 4050 t 10 I f (R)2371 4050 w 10 R f (is directly east \(or south\) of)5 1111 1 2459 4050 t 10 I f (P)3597 4050 w 10 R f (, are lighted, then the ellipse inter-)6 1382 1 3658 4050 t (sects)720 4170 w 10 I f (Q. V)1 166 1 939 4170 t 10 R f (\(or)1130 4170 w 10 I f (Q. H)1 177 1 1271 4170 t 10 R f (\) for every grid point)4 837 1 1448 4170 t 10 I f (Q)2310 4170 w 10 R f (in the interior of the line segment from)7 1551 1 2407 4170 t 10 I f (P)3983 4170 w 10 R f (to)4069 4170 w 10 I f (R)4172 4170 w 10 R f (.)4233 4170 w 10 B f (Problem statement)1 806 1 720 4410 t 10 R f (Given nonnegative integers)2 1114 1 720 4566 t 10 I f (a)1867 4566 w 10 R f (and)1950 4566 w 10 I f (b)2127 4566 w 10 R f ( the ellipse in the first quad-)6 1175(, construct the Freeman approximation to)5 1688 2 2177 4566 t (rant, i.e. construct the set)4 1001 1 720 4686 t 10 I f (S)1746 4686 w 10 R f (of)1821 4686 w (lighted grid points in {)4 910 1 970 4806 t 10 I f (P)1880 4806 w 10 S f (\357)1941 4823 w 10 I f (P. x)1 138 1 2013 4806 t 10 S f (\263)2175 4806 w 10 R f (0 &)1 160 1 2246 4806 t 10 I f (P. y)1 138 1 2438 4806 t 10 S f (\263)2600 4806 w 10 R f (0} if)1 184 1 2671 4806 t 10 I f (a)2880 4806 w 10 S f (>)2954 4806 w 10 R f (0 and)1 219 1 3025 4806 t 10 I f (b)3269 4806 w 10 S f (>)3343 4806 w 10 R f (0;)3414 4806 w ( 0 ,)2 91(grid points in {\()3 640 2 970 4926 t 10 I f (y)1709 4926 w 10 R f (\))1761 4926 w 10 S f (\357)1794 4943 w 10 R f (0)1866 4926 w 10 S f (\243)1932 4926 w 10 I f (y)2003 4926 w 10 S f (\243)2071 4926 w 10 I f (b)2142 4926 w 10 R f (} if)1 134 1 2192 4926 t 10 I f (a)2351 4926 w 10 S f (=)2425 4926 w 10 R f (0;)2496 4926 w (grid points in {\()3 640 1 970 5046 t 10 I f (x)1618 5046 w 10 R f (, 0 \))2 124 1 1670 5046 t 10 S f (\357)1794 5063 w 10 R f (0)1866 5046 w 10 S f (\243)1932 5046 w 10 I f (x)2003 5046 w 10 S f (\243)2071 5046 w 10 I f (a)2142 5046 w 10 R f (} if)1 134 1 2192 5046 t 10 I f (b)2351 5046 w 10 S f (=)2425 5046 w 10 R f (0.)2496 5046 w (The algorithm should use only integer arithmetic.)6 1979 1 720 5166 t 10 B f (General plan)1 559 1 720 5406 t 10 R f ( the grid visiting lighted points and accumulating a set)9 2187(We shall walk)2 573 2 720 5562 t 10 I f (T)3507 5562 w 10 R f ( walk starts at)3 558( The)1 207(of lighted points.)2 685 3 3590 5562 t ( south whenever possible, otherwise southeast, and leaves the quadrant only when)11 3275(the north and steps east or)5 1045 2 720 5682 t ( quadrants may be filled in by symmetry.)7 1643( Other)1 277(all lighted points have been visited.)5 1416 3 720 5802 t ( the possibility of)3 704(Program 0 handles a narrowed problem, which omits)7 2126 2 720 5958 t 10 I f (a)3577 5958 w 10 S f (=)3651 5958 w 10 R f (0 or)1 160 1 3722 5958 t 10 I f (b)3909 5958 w 10 S f (=)3983 5958 w 10 R f (0 and uses real, not inte-)5 986 1 4054 5958 t ( to the)2 260( and generalizations finally lead to Program 4, an efficient solution)10 2731( Transformations)1 716(ger, arithmetic.)1 613 4 720 6078 t (full problem.)1 522 1 720 6198 t 10 B f (Invariant)720 6438 w 10 R f (When grid point)2 655 1 720 6594 t 10 I f (P)1400 6594 w 10 R f (is visited,)1 384 1 1486 6594 t 10 I f (P)1895 6594 w 10 R f (is lighted and)2 539 1 1981 6594 t 10 I f (T)2545 6594 w 10 R f (comprises all lighted points north or west of)7 1763 1 2626 6594 t 10 I f (P)4414 6594 w 10 R f (.)4475 6594 w cleartomark showpage saveobj restore %%PageBoundingBox: 61 117 514 764 %%EndPage: 18 18 %%Page: 19 19 %%PageBoundingBox: (atend) /saveobj save def mark 19 pagesetup 10 R f (- 19 -)2 216 1 2772 480 t 10 B f (Program 0.)1 482 1 2639 960 t 10 R f (Precondition:)970 1116 w 10 I f (a)1534 1116 w 10 S f (>)1608 1116 w 10 R f (0 &)1 169 1 1679 1116 t 10 I f (b)1889 1116 w 10 S f (>)1963 1116 w 10 R f (0)2034 1116 w 10 I f (T)970 1272 w 10 R f (:)1067 1272 w 10 S f (= \306)1 186 1 1111 1272 t 10 I f (P)970 1392 w 10 R f (:)1072 1392 w 10 S f (=)1116 1392 w 10 R f (\( 0 ,)2 124 1 1220 1392 t 10 I f (b)1352 1392 w 10 R f (\))1410 1392 w 10 CW f (while)970 1512 w 10 I f (P. y)1 138 1 1311 1512 t 10 S f (\263)1473 1512 w 10 R f (0 &)1 169 1 1544 1512 t 10 I f (P. x)1 138 1 1754 1512 t 10 S f (\243)1916 1512 w 10 I f (a)1987 1512 w (T)1330 1632 w 10 S f (\310 =)1 148 1 1427 1632 t 10 I f (P)1624 1632 w 10 CW f (if)1330 1752 w 10 I f (P. x)1 138 1 1491 1752 t 10 S f (<)1653 1752 w 10 I f (a)1724 1752 w 10 R f (& &)1 164 1 1815 1752 t 10 I f (V)2020 1752 w 10 R f (\()2089 1752 w 10 I f (east)2130 1752 w 10 R f (\()2299 1752 w 10 I f (P)2340 1752 w 10 R f (\) \))1 74 1 2409 1752 t 10 S f (\256)2532 1752 w 10 I f (P. x)1 138 1 2672 1752 t 10 S f (+ =)1 126 1 2850 1752 t 10 R f (1 {1})1 980 1 3016 1752 t ([ ])1 82 1 1330 1872 t 10 I f (P. x)1 138 1 1461 1872 t 10 S f (<)1623 1872 w 10 I f (a)1694 1872 w 10 R f (&)1785 1872 w 10 I f (H)1904 1872 w 10 R f (\()1984 1872 w 10 I f (east)2025 1872 w 10 R f (\()2194 1872 w 10 I f (P)2235 1872 w 10 R f (\) \))1 74 1 2304 1872 t 10 S f (\256)2427 1872 w 10 I f (P. x)1 138 1 2567 1872 t 10 S f (+ =)1 126 1 2745 1872 t 10 R f (1 {2})1 1085 1 2911 1872 t ([ ])1 82 1 1330 1992 t 10 I f (P. y)1 138 1 1461 1992 t 10 S f (>)1623 1992 w 10 R f ( &)1 86(0 &)1 169 2 1694 1992 t 10 I f (H)1990 1992 w 10 R f (\()2070 1992 w 10 I f (south)2111 1992 w 10 R f (\()2336 1992 w 10 I f (P)2377 1992 w 10 R f (\) \))1 74 1 2446 1992 t 10 S f (\256)2569 1992 w 10 I f (P. y)1 138 1 2709 1992 t 10 S f (- =)1 126 1 2887 1992 t 10 R f (1 {3})1 943 1 3053 1992 t ([ ])1 82 1 1330 2112 t 10 I f (P. y)1 138 1 1461 2112 t 10 S f (>)1623 2112 w 10 R f (0 &)1 169 1 1694 2112 t 10 I f (V)1904 2112 w 10 R f (\()1973 2112 w 10 I f (south)2014 2112 w 10 R f (\()2239 2112 w 10 I f (P)2280 2112 w 10 R f (\) \))1 74 1 2349 2112 t 10 S f (\256)2472 2112 w 10 I f (P. y)1 138 1 2612 2112 t 10 S f (- =)1 126 1 2790 2112 t 10 R f (1 {4})1 1040 1 2956 2112 t ([ ])1 82 1 1330 2232 t 10 I f (P. x)1 138 1 1461 2232 t 10 S f (=)1623 2232 w 10 I f (a)1694 2232 w 10 S f (\256)1785 2232 w 10 I f (P. y)1 138 1 1925 2232 t 10 S f (- =)1 126 1 2103 2232 t 10 R f (1 {5})1 1727 1 2269 2232 t ([ ])1 82 1 1330 2352 t 10 I f (P. y)1 138 1 1461 2352 t 10 S f (=)1623 2352 w 10 R f (0)1694 2352 w 10 S f (\256)1785 2352 w 10 I f (P. x)1 138 1 1925 2352 t 10 S f (+ =)1 126 1 2103 2352 t 10 R f (1 {6})1 1727 1 2269 2352 t 10 CW f (else)1330 2472 w 10 S f (\256)1611 2472 w 10 I f (P. x)1 138 1 1751 2472 t 10 S f (+ =)1 126 1 1929 2472 t 10 R f (1 ,)1 83 1 2095 2472 t 10 I f (P. y)1 138 1 2219 2472 t 10 S f (- =)1 126 1 2397 2472 t 10 R f (1 {7})1 1433 1 2563 2472 t (Postcondition:)970 2628 w 10 I f (T)1574 2628 w 10 S f (=)1679 2628 w 10 I f (S)1783 2628 w 10 R f ( from C, to forestall evaluating)5 1242( the conditional and operator)4 1152( &,)1 111(In program 0 some of the guards use &)8 1572 4 720 2784 t 10 I f (f)4824 2784 w 10 R f (out-)4879 2784 w ( set-updating operator)2 876( The)1 205(side its range.)2 552 3 720 2904 t 10 S f (\310 =)1 148 1 2378 2904 t 10 R f (is formed analogously to)3 991 1 2551 2904 t 10 S f (+ =)1 126 1 3567 2904 t 10 R f (in C.)1 195 1 3718 2904 t (The pseudoguard)1 690 1 720 3060 t 10 CW f (else)1435 3060 w 10 R f ( of all other guards,)4 783(, with the obvious meaning of the the negation of the disjunction)11 2582 2 1675 3060 t ( tests of)2 323( Direct)1 305(is used to control the execution of a southeast step.)9 2085 3 720 3180 t 10 I f (H)3464 3180 w 10 R f (\()3544 3180 w 10 I f (southeast)3585 3180 w 10 R f (\()3971 3180 w 10 I f (P)4012 3180 w 10 R f ( and)1 175(\) \))1 74 2 4081 3180 t 10 I f (V)4360 3180 w 10 R f (\()4429 3180 w 10 I f (southeast)4470 3180 w 10 R f (\()4856 3180 w 10 I f (P)4897 3180 w 10 R f (\) \))1 74 1 4966 3180 t ( because the event of the southeast neighbor being lighted is not necessarily incompati-)13 3494(would be inadequate)2 826 2 720 3300 t ( \(or south\) neighbor being lighted. In such a case, the other neighbor must be visited first)16 3647(ble with the east)3 673 2 720 3420 t (lest it be skipped.)3 700 1 720 3540 t 10 B f (Proof of Program 0)3 828 1 720 3780 t 10 I f (Initialization.)720 3936 w 10 R f ( 0 ,)2 91(The initial point \()3 727 2 1321 3936 t 10 I f (b)2147 3936 w 10 R f ( 0 ,)2 91( points to the north of \()6 976( are no lighted)3 598( There)1 290(\) is lighted.)2 469 5 2205 3936 t 10 I f (b)4637 3936 w 10 R f (\) and no)2 345 1 4695 3936 t ( the invariant is established.)4 1113( Thus)1 250(first quadrant points to the west.)5 1288 3 720 4056 t 10 I f (Termination.)720 4212 w 10 R f (At every iteration)2 710 1 1293 4212 t 10 I f (P. x)1 138 1 2031 4212 t 10 R f (increases or)1 477 1 2197 4212 t 10 I f (P. y)1 138 1 2703 4212 t 10 R f ( loop can)2 374( The)1 209(decreases towards its respective bound.)4 1587 3 2870 4212 t (\(and must\) terminate only after the extreme point \()8 2072 1 720 4332 t 10 I f (a)2800 4332 w 10 R f ( the postcondition is estab-)4 1092( truth of)2 334( The)1 211( is visited.)2 421(, 0 \))2 124 5 2858 4332 t (lished at case 5 below.)4 901 1 720 4452 t (The invariant will be proved by analyzing the numbered cases in the loop.)12 2959 1 720 4608 t ( guards assure that)3 752( The)1 208(Cases 1 and 2.)3 586 3 720 4764 t 10 I f (east)2295 4764 w 10 R f (\()2464 4764 w 10 I f (P)2505 4764 w 10 R f ( the invariant, all lighted points north or west)8 1834( By)1 171(\) is lighted.)2 461 3 2574 4764 t (of)720 4884 w 10 I f (P)838 4884 w 10 R f (are in)1 234 1 934 4884 t 10 I f (T)1203 4884 w 10 R f ( Lemma 1, no point southwest of)6 1373( By)1 176( step.)1 220(before the)1 411 4 1294 4884 t 10 I f (east)3508 4884 w 10 R f (\()3677 4884 w 10 I f (P)3718 4884 w 10 R f (\) nor northeast of)3 717 1 3787 4884 t 10 I f (P)4538 4884 w 10 R f (is lighted;)1 407 1 4633 4884 t (adding)720 5004 w 10 I f (P)1017 5004 w 10 R f (to)1103 5004 w 10 I f (T)1206 5004 w 10 R f (preserves the invariant.)2 928 1 1287 5004 t ( cases 1 and 2 with south exchanged for east, north for west, and)13 2575( Like)1 233(Cases 3 and 4.)3 577 3 720 5160 t 10 I f (V)4130 5160 w 10 R f (for)4216 5160 w 10 I f (H)4357 5160 w 10 R f (.)4429 5160 w (Case 5. The current point,)4 1041 1 720 5316 t 10 I f (P)1788 5316 w 10 S f (=)1873 5316 w 10 R f (\()1944 5316 w 10 I f (a)1985 5316 w 10 R f (,)2043 5316 w 10 I f (y)2076 5316 w 10 R f (\), is lighted, as is the extremum of the ellipse, \()10 1899 1 2128 5316 t 10 I f (a)4035 5316 w 10 R f ( \(by Lemma)2 492( Hence)1 306(, 0 \).)2 149 3 4093 5316 t (3, if)1 167 1 720 5436 t 10 I f (y)918 5436 w 10 S f (>)986 5436 w 10 R f (1\))1057 5436 w 10 I f (south)1171 5436 w 10 R f (\()1396 5436 w 10 I f (P)1437 5436 w 10 R f ( invariant is thus preserved when)5 1347( The)1 211(\) is lighted.)2 465 3 1506 5436 t 10 I f (P. y)1 138 1 3560 5436 t 10 S f (>)3722 5436 w 10 R f (0. If)1 197 1 3793 5436 t 10 I f (P. y)1 138 1 4021 5436 t 10 S f (=)4183 5436 w 10 R f (0, the invariant and)3 786 1 4254 5436 t (the fact that there are no lighted points east of \()10 1895 1 720 5556 t 10 I f (a)2623 5556 w 10 R f ( imply that the updating of)5 1069(, 0 \),)2 149 2 2681 5556 t 10 I f (T)3926 5556 w 10 R f (establishes the postcondi-)2 1031 1 4009 5556 t ( loop terminates by stepping to an unlighted point.)8 2014(tion. The)1 386 2 720 5676 t ( case 5, with)3 499( Like)1 233(Case 6.)1 294 3 720 5832 t 10 I f (P)1771 5832 w 10 S f (=)1856 5832 w 10 R f (\()1927 5832 w 10 I f (x)1968 5832 w 10 R f (, 0 \).)2 149 1 2020 5832 t ( other guards assure that neither)5 1287( The)1 208(Case 7.)1 297 3 720 5988 t 10 I f (east)2540 5988 w 10 R f (\()2709 5988 w 10 I f (P)2750 5988 w 10 R f (\) nor)1 194 1 2819 5988 t 10 I f (south)3041 5988 w 10 R f (\()3266 5988 w 10 I f (P)3307 5988 w 10 R f ( that)1 179(\) is lighted,)2 459 2 3376 5988 t 10 I f (P. y)1 138 1 4043 5988 t 10 S f (>)4205 5988 w 10 R f (0, and that)2 427 1 4276 5988 t 10 I f (P. x)1 138 1 4732 5988 t 10 S f (<)4894 5988 w 10 I f (a)4965 5988 w 10 R f (.)5015 5988 w (Thus, by Lemma 2,)3 786 1 720 6108 t 10 I f (southeast)1533 6108 w 10 R f (\()1919 6108 w 10 I f (P)1960 6108 w 10 R f ( Let)1 185(\) is lighted.)2 457 2 2029 6108 t 10 I f (Q)2697 6108 w 10 S f (=)2793 6108 w 10 I f (southeast)2864 6108 w 10 R f (\()3250 6108 w 10 I f (P)3291 6108 w 10 R f ( points northeast or directly north of)6 1449(\). No)1 231 2 3360 6108 t 10 I f (Q)720 6228 w 10 R f ( or directly west)3 667( no points southwest)3 838( Similarly)1 429(are lighted \(by the guards, Lemma 1, and the invariant\).)9 2283 4 823 6228 t (of)720 6348 w 10 I f (Q)828 6348 w 10 R f ( the step preserves the invariant.)5 1286( Thus)1 250(are lighted.)1 449 3 925 6348 t 10 B f (Transformation to simpler code)3 1357 1 720 6588 t 10 R f ( 1 follows from weakening some guards)6 1609( Program)1 395(Having proved Program 0, we proceed by transformation.)7 2316 3 720 6744 t (and noting that case 4 is superfluous.)6 1479 1 720 6864 t ( guarding term)2 592( The)1 207(Case 1.)1 296 3 720 7020 t 10 I f (V)1842 7020 w 10 R f (\()1911 7020 w 10 I f (east)1952 7020 w 10 R f (\()2121 7020 w 10 I f (P)2162 7020 w 10 R f ( means)1 282(\) \))1 74 2 2231 7020 t 10 I f (P. y)1 138 1 2614 7020 t 10 S f (-)2776 7020 w 10 S1 f ()2847 7020 w 2847 7020 m 75 build_12 2922 7020 m 10 S f (\243)2938 7020 w 10 I f (f)3017 7020 w 10 R f (\()3061 7020 w 10 I f (a)3102 7020 w 10 R f (,)3160 7020 w 10 I f (b)3193 7020 w 10 R f (,)3251 7020 w 10 I f (P. x)1 138 1 3284 7020 t 10 S f (+)3446 7020 w 10 R f (1 \))1 91 1 3517 7020 t 10 S f (<)3624 7020 w 10 I f (P. y)1 138 1 3695 7020 t 10 S f (+)3857 7020 w 10 S1 f ()3928 7020 w 3928 7020 m 75 build_12 4003 7020 m 10 R f (. If)1 143 1 4003 7020 t 10 I f (f)4174 7020 w 10 R f (\()4218 7020 w 10 I f (a)4259 7020 w 10 R f (,)4317 7020 w 10 I f (b)4350 7020 w 10 R f (,)4408 7020 w 10 I f (P. x)1 138 1 4441 7020 t 10 R f (\))4587 7020 w 10 S f (<)4636 7020 w 10 I f (P. y)1 138 1 4707 7020 t 10 S f (+)4869 7020 w 10 S1 f ()4940 7020 w 4940 7020 m 75 build_12 5015 7020 m 10 R f (,)5015 7020 w (monotonicity implies)1 891 1 720 7140 t 10 I f (f)1673 7140 w 10 R f (\()1717 7140 w 10 I f (a)1758 7140 w 10 R f (,)1816 7140 w 10 I f (b)1849 7140 w 10 R f (,)1907 7140 w 10 I f (east)1940 7140 w 10 R f (\()2109 7140 w 10 I f (P)2150 7140 w 10 R f (\).)2219 7140 w 10 I f (x)2285 7140 w 10 R f (\))2337 7140 w 10 S f (<)2386 7140 w 10 I f (east)2457 7140 w 10 R f (\()2626 7140 w 10 I f (P)2667 7140 w 10 R f (\).)2736 7140 w 10 I f (y)2802 7140 w 10 S f (+)2870 7140 w 10 S1 f ()2941 7140 w 2941 7140 m 75 build_12 3016 7140 m 10 R f ( in)1 139(, which is the second test)5 1184 2 3016 7140 t 10 I f (V)4400 7140 w 10 R f (\()4469 7140 w 10 I f (east)4510 7140 w 10 R f (\()4679 7140 w 10 I f (P)4720 7140 w 10 R f ( If)1 152(\) \).)1 99 2 4789 7140 t 10 I f (f)720 7260 w 10 R f (\()764 7260 w 10 I f (a)805 7260 w 10 R f (,)863 7260 w 10 I f (b)896 7260 w 10 R f (,)954 7260 w 10 I f (P. x)1 138 1 987 7260 t 10 R f (\))1133 7260 w 10 S f (\263)1182 7260 w 10 I f (P. y)1 138 1 1253 7260 t 10 S f (+)1415 7260 w 10 S1 f ()1486 7260 w 1486 7260 m 75 build_12 1561 7260 m 10 R f (, then for)2 377 1 1561 7260 t 10 I f (P)1970 7260 w 10 R f ( invariant requires,)2 767(to be lighted as the)4 783 2 2063 7260 t 10 I f (H)3646 7260 w 10 R f (\()3726 7260 w 10 I f (P)3767 7260 w 10 R f ( Monotonicity)1 603(\) must be true.)3 601 2 3836 7260 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 50 514 764 %%EndPage: 19 19 %%Page: 20 20 %%PageBoundingBox: (atend) /saveobj save def mark 20 pagesetup 10 R f (- 20 -)2 216 1 2772 480 t (precludes the curve from intersecting)4 1509 1 720 840 t 10 I f (P. H)1 166 1 2260 840 t 10 R f (and passing above)2 744 1 2457 840 t 10 I f (east)3232 840 w 10 R f (\()3401 840 w 10 I f (P)3442 840 w 10 R f (\).)3511 840 w 10 I f (V)3577 840 w 10 R f ( either event the second test in)6 1239(. In)1 163 2 3638 840 t 10 I f (V)720 960 w 10 R f (\()789 960 w 10 I f (east)830 960 w 10 R f (\()999 960 w 10 I f (P)1040 960 w 10 R f ( is satisfied; it can be dropped from the guard.)9 1836(\) \))1 74 2 1109 960 t ( guarding term)2 620( The)1 221(Case 2.)1 310 3 720 1116 t 10 I f (H)1912 1116 w 10 R f (\()1992 1116 w 10 I f (east)2033 1116 w 10 R f (\()2202 1116 w 10 I f (P)2243 1116 w 10 R f ( means)1 296(\) \))1 74 2 2312 1116 t 10 I f (P. x)1 138 1 2723 1116 t 10 S f (+)2885 1116 w 10 S1 f ()2956 1116 w 2956 1116 m 75 build_12 3031 1116 m 10 S f (\243)3047 1116 w 10 I f (f)3126 1116 w 10 R f (\()3170 1116 w 10 I f (b)3211 1116 w 10 R f (,)3269 1116 w 10 I f (a)3302 1116 w 10 R f (,)3360 1116 w 10 I f (P. y)1 138 1 3393 1116 t 10 R f (\))3539 1116 w 10 S f (<)3588 1116 w 10 I f (P. x)1 138 1 3659 1116 t 10 S f (+)3821 1116 w 10 S1 f ()3892 1116 w 3892 1116 m 75 build_12 3967 1116 m 10 R f ( inequality of)2 567(, the second)2 506 2 3967 1116 t ( ellipse not pass east of)5 934(which requires that the)3 921 2 720 1236 t 10 I f (east)2602 1236 w 10 R f (\()2771 1236 w 10 I f (P)2812 1236 w 10 R f (\).)2881 1236 w 10 I f (H)2947 1236 w 10 R f ( however, the ellipse does so, then some point)8 1853(. If,)1 168 2 3019 1236 t (directly east of)2 629 1 720 1356 t 10 I f (P)1392 1356 w 10 R f ( the ellipse must intersect)4 1092(must be lighted, so by Lemma 3)6 1394 2 1496 1356 t 10 I f (east)4026 1356 w 10 R f (\()4195 1356 w 10 I f (P)4236 1356 w 10 R f (\).)4305 1356 w 10 I f (V)4371 1356 w 10 R f (and case 1 is)3 564 1 4476 1356 t ( may be dropped from)4 899( case 1 and case 2 have the same outcome, the second inequality)12 2617(selectable. Because)1 804 3 720 1476 t 10 I f (H)720 1596 w 10 R f (\()800 1596 w 10 I f (east)841 1596 w 10 R f (\()1010 1596 w 10 I f (P)1051 1596 w 10 R f ( without changing the effect of the program.)7 1763(\) \))1 74 2 1120 1596 t ( term)1 215( guarding)1 386( The)1 211(Case 3.)1 300 4 720 1752 t 10 I f (H)1864 1752 w 10 R f (\()1944 1752 w 10 I f (south)1985 1752 w 10 R f (\()2210 1752 w 10 I f (P)2251 1752 w 10 R f ( means)1 287(\) \))1 74 2 2320 1752 t 10 I f (P. x)1 138 1 2713 1752 t 10 S f (-)2875 1752 w 10 S1 f ()2946 1752 w 2946 1752 m 75 build_12 3021 1752 m 10 S f (\243)3037 1752 w 10 I f (f)3116 1752 w 10 R f (\()3160 1752 w 10 I f (b)3201 1752 w 10 R f (,)3259 1752 w 10 I f (a)3292 1752 w 10 R f (,)3350 1752 w 10 I f (y)3383 1752 w 10 S f (-)3451 1752 w 10 R f (1 \))1 91 1 3522 1752 t 10 S f (<)3629 1752 w 10 I f (P. x)1 138 1 3700 1752 t 10 S f (+)3862 1752 w 10 S1 f ()3933 1752 w 3933 1752 m 75 build_12 4008 1752 m 10 R f ( first inequality can)3 795(. The)1 237 2 4008 1752 t (be dropped from the guard for reasons similar to those given for simplifying the guard in case 1.)17 3844 1 720 1872 t ( ellipse intersects)2 690( The)1 206(Case 4.)1 295 3 720 2028 t 10 I f (south)1937 2028 w 10 R f (\()2162 2028 w 10 I f (P)2203 2028 w 10 R f (\).)2272 2028 w 10 I f (V)2338 2028 w 10 R f ( it must also intersect)4 858( shall show that)3 628(. We)1 214 3 2399 2028 t 10 I f (south)4126 2028 w 10 R f (\()4351 2028 w 10 I f (P)4392 2028 w 10 R f (\).)4461 2028 w 10 I f (H)4527 2028 w 10 R f (, so case 4,)3 441 1 4599 2028 t (having the same outcome as case 3, is subsumed by case 3.)11 2360 1 720 2148 t (Since the ellipse intersects)3 1066 1 720 2304 t 10 I f (south)1814 2304 w 10 R f (\()2039 2304 w 10 I f (P)2080 2304 w 10 R f (\).)2149 2304 w 10 I f (V)2215 2304 w 10 R f (, it cannot intersect)3 769 1 2276 2304 t 10 I f (P. V)1 155 1 3073 2304 t 10 R f ( to be lighted, the ellipse must)6 1227( for P)2 228(. Hence,)1 357 3 3228 2304 t (intersect)720 2424 w 10 I f (P. H)1 166 1 1094 2424 t 10 R f ( of)1 118( monotonicity the intersection must lie west)6 1816(. By)1 203 3 1260 2424 t 10 I f (P)3432 2424 w 10 R f ( continuity, the ellipse must also)5 1345(. By)1 202 2 3493 2424 t (intersect the open segment)3 1068 1 720 2544 t 10 I f (K)1815 2544 w 10 R f ( \()1 61(that joins)1 372 2 1909 2544 t 10 I f (P. x)1 138 1 2350 2544 t 10 S f (-)2512 2544 w 10 S1 f ()2583 2544 w 2583 2544 m 75 build_12 2658 2544 m 10 R f (,)2666 2544 w 10 I f (P. y)1 138 1 2699 2544 t 10 S f (-)2861 2544 w 10 S1 f ()2932 2544 w 2932 2544 m 75 build_12 3007 2544 m 10 R f (\) to \()2 200 1 3015 2544 t 10 I f (P. x)1 138 1 3223 2544 t 10 R f (,)3369 2544 w 10 I f (P. y)1 138 1 3402 2544 t 10 S f (-)3564 2544 w 10 S1 f ()3635 2544 w 3635 2544 m 75 build_12 3710 2544 m 10 R f ( its intersections with)3 862(\). Between)1 460 2 3718 2544 t 10 I f (H)720 2664 w 10 R f (and)822 2664 w 10 I f (K)996 2664 w 10 R f (the ellipse has mean slope less than)6 1445 1 1093 2664 t 10 S f (-)2568 2664 w 10 R f ( monotonicity of the slope, the ellipse has slope)8 1934(1, hence by)2 467 2 2639 2664 t (less than)1 355 1 720 2784 t 10 S f (-)1108 2784 w 10 R f (1 at all points south of)5 932 1 1179 2784 t 10 I f (K)2144 2784 w 10 R f ( of slope less than)4 752( curve)1 254(. A)1 155 3 2211 2784 t 10 S f (-)3406 2784 w 10 R f (1 and continuous over the north-south)5 1563 1 3477 2784 t (range of)1 329 1 720 2904 t 10 I f (south)1074 2904 w 10 R f (\()1299 2904 w 10 I f (P)1340 2904 w 10 R f (\).)1409 2904 w 10 I f (V)1475 2904 w 10 R f (, which meets)2 552 1 1536 2904 t 10 I f (south)2113 2904 w 10 R f (\()2338 2904 w 10 I f (P)2379 2904 w 10 R f (\).)2448 2904 w 10 I f (V)2514 2904 w 10 R f (, must also meet)3 650 1 2575 2904 t 10 I f (south)3250 2904 w 10 R f (\()3475 2904 w 10 I f (P)3516 2904 w 10 R f (\).)3585 2904 w 10 I f (H)3651 2904 w 10 R f (.)3723 2904 w 10 B f (Program 1.)1 482 1 2639 3144 t 10 R f (Precondition:)970 3300 w 10 I f (a)1534 3300 w 10 S f (>)1608 3300 w 10 R f (0 &)1 169 1 1679 3300 t 10 I f (b)1889 3300 w 10 S f (>)1963 3300 w 10 R f (0)2034 3300 w 10 I f (T)970 3456 w 10 R f (:)1067 3456 w 10 S f (= \306)1 186 1 1111 3456 t 10 I f (P)970 3576 w 10 R f (:)1072 3576 w 10 S f (=)1116 3576 w 10 R f (\( 0 ,)2 124 1 1220 3576 t 10 I f (b)1352 3576 w 10 R f (\))1410 3576 w 10 CW f (while)970 3696 w 10 I f (P. y)1 138 1 1311 3696 t 10 S f (\263)1473 3696 w 10 R f (0 &)1 169 1 1544 3696 t 10 I f (P. x)1 138 1 1754 3696 t 10 S f (\243)1916 3696 w 10 I f (a)1987 3696 w (T)1330 3816 w 10 S f (\310 =)1 148 1 1427 3816 t 10 I f (P)1624 3816 w 10 CW f (if)1330 3936 w 10 I f (P. x)1 138 1 1491 3936 t 10 S f (<)1653 3936 w 10 I f (a)1724 3936 w 10 R f (& &)1 164 1 1815 3936 t 10 I f (P. y)1 138 1 2020 3936 t 10 S f (-)2182 3936 w 10 S1 f ()2253 3936 w 2253 3936 m 75 build_12 2328 3936 m 10 S f (\243)2344 3936 w 10 I f (f)2423 3936 w 10 R f (\()2467 3936 w 10 I f (a)2508 3936 w 10 R f (,)2566 3936 w 10 I f (b)2599 3936 w 10 R f (,)2657 3936 w 10 I f (P. x)1 138 1 2690 3936 t 10 S f (+)2852 3936 w 10 R f (1 \))1 91 1 2923 3936 t 10 S f (\256)3063 3936 w 10 I f (P. x)1 138 1 3203 3936 t 10 S f (+ =)1 126 1 3381 3936 t 10 R f (1 {1})1 449 1 3547 3936 t ([ ])1 82 1 1330 4056 t 10 I f (P. x)1 138 1 1461 4056 t 10 S f (<)1623 4056 w 10 I f (a)1694 4056 w 10 R f (&)1785 4056 w 10 I f (P. x)1 138 1 1904 4056 t 10 S f (+)2066 4056 w 10 S1 f ()2137 4056 w 2137 4056 m 75 build_12 2212 4056 m 10 S f (\243)2228 4056 w 10 I f (f)2307 4056 w 10 R f (\()2351 4056 w 10 I f (b)2392 4056 w 10 R f (,)2450 4056 w 10 I f (a)2483 4056 w 10 R f (,)2541 4056 w 10 I f (P. y)1 138 1 2574 4056 t 10 R f (\))2720 4056 w 10 S f (\256)2802 4056 w 10 I f (P. x)1 138 1 2942 4056 t 10 S f (+ =)1 126 1 3120 4056 t 10 R f (1 {2})1 710 1 3286 4056 t ([ ])1 82 1 1330 4176 t 10 I f (P. y)1 138 1 1461 4176 t 10 S f (>)1623 4176 w 10 R f ( &)1 86(0 &)1 169 2 1694 4176 t 10 I f (P. x)1 138 1 1990 4176 t 10 S f (+)2152 4176 w 10 S1 f ()2223 4176 w 2223 4176 m 75 build_12 2298 4176 m 10 S f (>)2314 4176 w 10 I f (f)2393 4176 w 10 R f (\()2437 4176 w 10 I f (b)2478 4176 w 10 R f (,)2536 4176 w 10 I f (a)2569 4176 w 10 R f (,)2627 4176 w 10 I f (P. y)1 138 1 2660 4176 t 10 S f (-)2822 4176 w 10 R f (1 \))1 91 1 2893 4176 t 10 S f (\256)3033 4176 w 10 I f (P. y)1 138 1 3173 4176 t 10 S f (- =)1 126 1 3351 4176 t 10 R f (1 {3})1 479 1 3517 4176 t ([ ])1 82 1 1330 4296 t 10 I f (P. x)1 138 1 1461 4296 t 10 S f (=)1623 4296 w 10 I f (a)1694 4296 w 10 S f (\256)1785 4296 w 10 I f (P. y)1 138 1 1925 4296 t 10 S f (- =)1 126 1 2103 4296 t 10 R f (1 {5})1 1727 1 2269 4296 t ([ ])1 82 1 1330 4416 t 10 I f (P. y)1 138 1 1461 4416 t 10 S f (=)1623 4416 w 10 R f (0)1694 4416 w 10 S f (\256)1785 4416 w 10 I f (P. x)1 138 1 1925 4416 t 10 S f (+ =)1 126 1 2103 4416 t 10 R f (1 {6})1 1727 1 2269 4416 t ([ ])1 82 1 1330 4536 t 10 CW f (else)1461 4536 w 10 S f (\256)1742 4536 w 10 I f (P. x)1 138 1 1882 4536 t 10 S f (+ =)1 126 1 2060 4536 t 10 R f (1 ,)1 83 1 2226 4536 t 10 I f (P. y)1 138 1 2350 4536 t 10 S f (- =)1 126 1 2528 4536 t 10 R f (1 {7})1 1302 1 2694 4536 t (Postcondition:)970 4692 w 10 I f (T)1574 4692 w 10 S f (=)1679 4692 w 10 I f (S)1783 4692 w 10 R f (To get rid of the square root we replace formulas in)10 2056 1 720 4848 t 10 I f (f)2801 4848 w 10 R f (by equivalent formulas in)3 1024 1 2854 4848 t 10 I f (e)3904 4848 w 10 R f (. If)1 142 1 3948 4848 t 10 I f (y)4116 4848 w 10 R f (can be less than zero,)4 854 1 4186 4848 t (an inequality of the form)4 1061 1 720 4968 t 10 I f (y)1823 4968 w 10 S f (\243)1891 4968 w 10 I f (f)1970 4968 w 10 R f (\()2014 4968 w 10 I f (a)2055 4968 w 10 R f (,)2113 4968 w 10 I f (b)2146 4968 w 10 R f (,)2204 4968 w 10 I f (x)2237 4968 w 10 R f (\) is replaced by)3 663 1 2289 4968 t 10 I f (y)2994 4968 w 10 S f (<)3062 4968 w 10 R f (0)3133 4968 w 10 S f (\357)3207 4985 w 10 I f (e)3279 4968 w 10 R f (\()3331 4968 w 10 I f (a)3372 4968 w 10 R f (,)3430 4968 w 10 I f (b)3463 4968 w 10 R f (,)3521 4968 w 10 I f (x)3554 4968 w 10 R f (,)3606 4968 w 10 I f (y)3639 4968 w 10 R f (\))3691 4968 w 10 S f (\243)3740 4968 w 10 R f ( is)1 108(0; the resulting disjunction)3 1121 2 3811 4968 t ( identity)1 332( The)1 206( 1a\).)1 178(interpreted as a case split \(between cases 1 and)8 1872 4 720 5088 t 10 I f (e)3334 5088 w 10 R f (\()3386 5088 w 10 I f (b)3427 5088 w 10 R f (,)3485 5088 w 10 I f (a)3518 5088 w 10 R f (,)3576 5088 w 10 I f (y)3609 5088 w 10 R f (,)3661 5088 w 10 I f (x)3694 5088 w 10 R f (\))3746 5088 w 10 S f (=)3795 5088 w 10 I f (e)3866 5088 w 10 R f (\()3918 5088 w 10 I f (a)3959 5088 w 10 R f (,)4017 5088 w 10 I f (b)4050 5088 w 10 R f (,)4108 5088 w 10 I f (x)4141 5088 w 10 R f (,)4193 5088 w 10 I f (y)4226 5088 w 10 R f (\) allows arguments)2 762 1 4278 5088 t (to be rearranged.)2 672 1 720 5208 t ( operators are no longer necessary because)6 1699( &)1 86(The conditional &)2 733 3 720 5364 t 10 I f (e)3263 5364 w 10 R f (, unlike)1 300 1 3307 5364 t 10 I f (f)3632 5364 w 10 R f (, has no range restrictions.)4 1048 1 3660 5364 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 240 514 764 %%EndPage: 20 20 %%Page: 21 21 %%PageBoundingBox: (atend) /saveobj save def mark 21 pagesetup 10 R f (- 21 -)2 216 1 2772 480 t 10 B f (Program 2.)1 482 1 2639 960 t 10 R f (Precondition:)970 1116 w 10 I f (a)1534 1116 w 10 S f (>)1608 1116 w 10 R f (0 &)1 169 1 1679 1116 t 10 I f (b)1889 1116 w 10 S f (>)1963 1116 w 10 R f (0)2034 1116 w 10 I f (T)970 1272 w 10 R f (:)1067 1272 w 10 S f (= \306)1 186 1 1111 1272 t 10 I f (P)970 1392 w 10 R f (:)1072 1392 w 10 S f (=)1116 1392 w 10 R f (\( 0 ,)2 124 1 1220 1392 t 10 I f (b)1352 1392 w 10 R f (\))1410 1392 w 10 CW f (while)970 1512 w 10 I f (P. y)1 138 1 1311 1512 t 10 S f (\263)1473 1512 w 10 R f (0 &)1 169 1 1544 1512 t 10 I f (P. x)1 138 1 1754 1512 t 10 S f (\243)1916 1512 w 10 I f (a)1987 1512 w (T)1330 1632 w 10 S f (\310 =)1 148 1 1427 1632 t 10 I f (P)1624 1632 w 10 CW f (if)1330 1752 w 10 I f (P. x)1 138 1 1491 1752 t 10 S f (<)1653 1752 w 10 I f (a)1724 1752 w 10 R f (&)1815 1752 w 10 I f (e)1934 1752 w 10 R f (\()1986 1752 w 10 I f (a)2027 1752 w 10 R f (,)2085 1752 w 10 I f (b)2118 1752 w 10 R f (,)2176 1752 w 10 I f (P. x)1 138 1 2209 1752 t 10 S f (+)2371 1752 w 10 R f (1 ,)1 83 1 2442 1752 t 10 I f (P. y)1 138 1 2533 1752 t 10 S f (-)2695 1752 w 10 S1 f ()2766 1752 w 2766 1752 m 75 build_12 2841 1752 m 10 R f (\))2849 1752 w 10 S f (\243)2898 1752 w 10 R f (0)2969 1752 w 10 S f (\256)3060 1752 w 10 I f (P. x)1 138 1 3200 1752 t 10 S f (+ =)1 126 1 3378 1752 t 10 R f (1 {1})1 452 1 3544 1752 t ([ ])1 82 1 1330 1872 t 10 I f (P. x)1 138 1 1461 1872 t 10 S f (<)1623 1872 w 10 I f (a)1694 1872 w 10 R f (&)1785 1872 w 10 I f (P. y)1 138 1 1904 1872 t 10 S f (-)2066 1872 w 10 S1 f ()2137 1872 w 2137 1872 m 75 build_12 2212 1872 m 10 S f (<)2228 1872 w 10 R f (0)2299 1872 w 10 S f (\256)2390 1872 w 10 I f (P. x)1 138 1 2530 1872 t 10 S f (+ =)1 126 1 2708 1872 t 10 R f (1 {1a})1 1166 1 2874 1872 t ([ ])1 82 1 1330 1992 t 10 I f (P. x)1 138 1 1461 1992 t 10 S f (<)1623 1992 w 10 I f (a)1694 1992 w 10 R f (&)1785 1992 w 10 I f (e)1904 1992 w 10 R f (\()1956 1992 w 10 I f (a)1997 1992 w 10 R f (,)2055 1992 w 10 I f (b)2088 1992 w 10 R f (,)2146 1992 w 10 I f (P. x)1 138 1 2179 1992 t 10 S f (+)2341 1992 w 10 S1 f ()2412 1992 w 2412 1992 m 75 build_12 2487 1992 m 10 R f (,)2495 1992 w 10 I f (P. y)1 138 1 2528 1992 t 10 R f (\))2674 1992 w 10 S f (\243)2723 1992 w 10 R f (0)2794 1992 w 10 S f (\256)2885 1992 w 10 I f (P. x)1 138 1 3025 1992 t 10 S f (+ =)1 126 1 3203 1992 t 10 R f (1 {2})1 627 1 3369 1992 t ([ ])1 82 1 1330 2112 t 10 I f (P. y)1 138 1 1461 2112 t 10 S f (>)1623 2112 w 10 R f (0 &)1 169 1 1694 2112 t 10 I f (e)1904 2112 w 10 R f (\()1956 2112 w 10 I f (a)1997 2112 w 10 R f (,)2055 2112 w 10 I f (b)2088 2112 w 10 R f (,)2146 2112 w 10 I f (P. x)1 138 1 2179 2112 t 10 S f (+)2341 2112 w 10 S1 f ()2412 2112 w 2412 2112 m 75 build_12 2487 2112 m 10 R f (,)2495 2112 w 10 I f (P. y)1 138 1 2528 2112 t 10 S f (-)2690 2112 w 10 R f (1 \))1 91 1 2761 2112 t 10 S f (>)2868 2112 w 10 R f (0)2939 2112 w 10 S f (\256)3030 2112 w 10 I f (P. y)1 138 1 3170 2112 t 10 S f (- =)1 126 1 3348 2112 t 10 R f (1 {3})1 482 1 3514 2112 t ([ ])1 82 1 1330 2232 t 10 I f (P. x)1 138 1 1461 2232 t 10 S f (=)1623 2232 w 10 I f (a)1694 2232 w 10 S f (\256)1785 2232 w 10 I f (P. y)1 138 1 1925 2232 t 10 S f (- =)1 126 1 2103 2232 t 10 R f (1 {5})1 1727 1 2269 2232 t ([ ])1 82 1 1330 2352 t 10 I f (P. y)1 138 1 1461 2352 t 10 S f (=)1623 2352 w 10 R f (0)1694 2352 w 10 S f (\256)1785 2352 w 10 I f (P. x)1 138 1 1925 2352 t 10 S f (+ =)1 126 1 2103 2352 t 10 R f (1 {6})1 1727 1 2269 2352 t 10 CW f (else)1330 2472 w 10 S f (\256)1611 2472 w 10 I f (P. x)1 138 1 1751 2472 t 10 S f (+ =)1 126 1 1929 2472 t 10 R f (1 ,)1 83 1 2095 2472 t 10 I f (P. y)1 138 1 2219 2472 t 10 S f (- =)1 126 1 2397 2472 t 10 R f (1 {7})1 1433 1 2563 2472 t (Postcondition:)970 2628 w 10 I f (T)1574 2628 w 10 S f (=)1679 2628 w 10 I f (S)1783 2628 w 10 R f (For all)1 270 1 720 2784 t 10 I f (x)1021 2784 w 10 S f (\263)1089 2784 w 10 I f (a)1160 2784 w 10 R f (, we have)2 391 1 1210 2784 t 10 I f (e)1632 2784 w 10 R f (\()1684 2784 w 10 I f (a)1725 2784 w 10 R f (,)1783 2784 w 10 I f (b)1816 2784 w 10 R f (,)1874 2784 w 10 I f (x)1907 2784 w 10 R f (,)1959 2784 w 10 I f (y)1992 2784 w 10 R f (\))2044 2784 w 10 S f (>)2093 2784 w 10 R f (0 unless \()2 395 1 2164 2784 t 10 I f (x)2567 2784 w 10 R f (,)2619 2784 w 10 I f (y)2652 2784 w 10 R f (\))2704 2784 w 10 S f (=)2753 2784 w 10 R f (\()2824 2784 w 10 I f (a)2865 2784 w 10 R f ( If)1 122(, 0 \).)2 149 2 2923 2784 t 10 I f (P)3225 2784 w 10 S f (=)3310 2784 w 10 R f (\()3381 2784 w 10 I f (a)3422 2784 w 10 R f ( the outcomes of cases 1 and 2 are)8 1411(, 0 \),)2 149 2 3480 2784 t ( the test)2 321( Thus)1 255( the second tests in cases 1 and 2 will fail.)10 1723( Otherwise)1 465(the same as that of case 6.)6 1075 5 720 2904 t 10 I f (P. x)1 138 1 4589 2904 t 10 S f (<)4751 2904 w 10 I f (a)4822 2904 w 10 R f (can)4902 2904 w (be dropped from the guards in cases 1 and 2.)9 1785 1 720 3024 t (Case 1a is subsumed by case 6.)6 1251 1 720 3180 t (Because)720 3336 w 10 I f (e)1088 3336 w 10 R f (\()1140 3336 w 10 I f (a)1181 3336 w 10 R f (,)1239 3336 w 10 I f (b)1272 3336 w 10 R f (,)1330 3336 w 10 I f (x)1363 3336 w 10 S f (+)1431 3336 w 10 S1 f ()1502 3336 w 1502 3336 m 75 build_12 1577 3336 m 10 R f (, 0 \))2 124 1 1585 3336 t 10 S f (<)1725 3336 w 10 R f (0 for all integer)3 651 1 1796 3336 t 10 I f (x)2483 3336 w 10 S f (<)2551 3336 w 10 I f (a)2622 3336 w 10 R f (, case 2 will be selectable when)6 1321 1 2672 3336 t 10 I f (P. y)1 138 1 4029 3336 t 10 S f (=)4191 3336 w 10 R f (0 and)1 230 1 4262 3336 t 10 I f (P. x)1 138 1 4528 3336 t 10 S f (<)4690 3336 w 10 I f (a)4761 3336 w 10 R f (, thus)1 229 1 4811 3336 t ( 6 unless)2 356(subsuming case)1 634 2 720 3456 t 10 I f (P. x)1 138 1 1738 3456 t 10 S f (=)1900 3456 w 10 I f (a)1971 3456 w 10 R f ( the latter situation,)3 781(. In)1 161 2 2021 3456 t 10 I f (P)2991 3456 w 10 S f (=)3076 3456 w 10 R f (\()3147 3456 w 10 I f (a)3188 3456 w 10 R f ( and case 6 has the same outcome \(termi-)8 1670(, 0 \))2 124 2 3246 3456 t ( case 6 may be dropped.)5 964( Thus)1 250(nation\) as case 5.)3 687 3 720 3576 t ( weakened\) case 2 fails, we can prevent case 3 from being considered when)13 3038(By trying case 3 only when \(the)6 1282 2 720 3732 t 10 I f (P. y)1 138 1 720 3852 t 10 S f (=)882 3852 w 10 R f (0, unless)1 360 1 953 3852 t 10 I f (P. x)1 138 1 1348 3852 t 10 S f (=)1510 3852 w 10 I f (a)1581 3852 w 10 R f ( were executable in the latter situation, it would have the same outcome)12 2967( case 3)2 291(. If)1 151 3 1631 3852 t ( the test)2 311( Thus)1 250(\(termination\) as case 5.)3 931 3 720 3972 t 10 I f (P. y)1 138 1 2237 3972 t 10 S f (>)2399 3972 w 10 R f (0 may be dropped from guard 3.)6 1289 1 2470 3972 t (Since)720 4128 w 10 I f (e)970 4128 w 10 R f (\()1022 4128 w 10 I f (a)1063 4128 w 10 R f (,)1121 4128 w 10 I f (b)1154 4128 w 10 R f (,)1212 4128 w 10 I f (a)1245 4128 w 10 S f (+)1319 4128 w 10 S1 f ()1390 4128 w 1390 4128 m 75 build_12 1465 4128 m 10 R f (,)1473 4128 w 10 I f (y)1506 4128 w 10 R f (\))1558 4128 w 10 S f (>)1607 4128 w 10 R f (0 for all)2 322 1 1678 4128 t 10 I f (y)2028 4128 w 10 R f (, the weakened guard 3 will always be true when)9 1972 1 2072 4128 t 10 I f (P. x)1 138 1 4073 4128 t 10 S f (=)4235 4128 w 10 I f (a)4306 4128 w 10 R f (, thus subsuming)2 684 1 4356 4128 t (case 5.)1 271 1 720 4248 t 10 B f (Program 3.)1 482 1 2639 4488 t 10 R f (Precondition:)970 4644 w 10 I f (a)1534 4644 w 10 S f (>)1608 4644 w 10 R f (0 &)1 169 1 1679 4644 t 10 I f (b)1889 4644 w 10 S f (>)1963 4644 w 10 R f (0)2034 4644 w 10 I f (T)970 4800 w 10 R f (:)1067 4800 w 10 S f (= \306)1 186 1 1111 4800 t 10 I f (P)970 4920 w 10 R f (:)1072 4920 w 10 S f (=)1116 4920 w 10 R f (\( 0 ,)2 124 1 1220 4920 t 10 I f (b)1352 4920 w 10 R f (\))1410 4920 w 10 CW f (while)970 5040 w 10 I f (P. y)1 138 1 1311 5040 t 10 S f (\263)1473 5040 w 10 R f (0 &)1 169 1 1544 5040 t 10 I f (P. x)1 138 1 1754 5040 t 10 S f (\243)1916 5040 w 10 I f (a)1987 5040 w (T)1330 5160 w 10 S f (\310 =)1 148 1 1427 5160 t 10 I f (P)1624 5160 w 10 CW f (if)1330 5280 w 10 I f (e)1491 5280 w 10 R f (\()1543 5280 w 10 I f (a)1584 5280 w 10 R f (,)1642 5280 w 10 I f (b)1675 5280 w 10 R f (,)1733 5280 w 10 I f (P. x)1 138 1 1766 5280 t 10 S f (+)1928 5280 w 10 R f (1 ,)1 83 1 1999 5280 t 10 I f (P. y)1 138 1 2090 5280 t 10 S f (-)2252 5280 w 10 S1 f ()2323 5280 w 2323 5280 m 75 build_12 2398 5280 m 10 R f (\))2406 5280 w 10 S f (\243)2455 5280 w 10 R f (0)2526 5280 w 10 S f (\256)2617 5280 w 10 I f (P. x)1 138 1 2757 5280 t 10 S f (+ =)1 126 1 2935 5280 t 10 R f (1 {1})1 895 1 3101 5280 t ([ ])1 82 1 1330 5400 t 10 I f (e)1461 5400 w 10 R f (\()1513 5400 w 10 I f (a)1554 5400 w 10 R f (,)1612 5400 w 10 I f (b)1645 5400 w 10 R f (,)1703 5400 w 10 I f (P. x)1 138 1 1736 5400 t 10 S f (+)1898 5400 w 10 S1 f ()1969 5400 w 1969 5400 m 75 build_12 2044 5400 m 10 R f (,)2052 5400 w 10 I f (P. y)1 138 1 2085 5400 t 10 R f (\))2231 5400 w 10 S f (\243)2280 5400 w 10 R f (0)2351 5400 w 10 S f (\256)2442 5400 w 10 I f (P. x)1 138 1 2582 5400 t 10 S f (+ =)1 126 1 2760 5400 t 10 R f (1 {2})1 1070 1 2926 5400 t 10 CW f (else)1330 5520 w 10 S f (\256)1611 5520 w 10 CW f (if)1690 5640 w 10 I f (e)1851 5640 w 10 R f (\()1903 5640 w 10 I f (a)1944 5640 w 10 R f (,)2002 5640 w 10 I f (b)2035 5640 w 10 R f (,)2093 5640 w 10 I f (P. x)1 138 1 2126 5640 t 10 S f (+)2288 5640 w 10 S1 f ()2359 5640 w 2359 5640 m 75 build_12 2434 5640 m 10 R f (,)2442 5640 w 10 I f (P. y)1 138 1 2475 5640 t 10 S f (-)2637 5640 w 10 R f (1 \))1 91 1 2708 5640 t 10 S f (>)2815 5640 w 10 R f (0)2886 5640 w 10 S f (\256)2977 5640 w 10 I f (P. y)1 138 1 3117 5640 t 10 S f (- =)1 126 1 3295 5640 t 10 R f (1 {3})1 535 1 3461 5640 t 10 CW f (else)1690 5760 w 10 S f (\256)1971 5760 w 10 I f (P. x)1 138 1 2111 5760 t 10 S f (+ =)1 126 1 2289 5760 t 10 R f (1 ,)1 83 1 2455 5760 t 10 I f (P. y)1 138 1 2579 5760 t 10 S f (- =)1 126 1 2757 5760 t 10 R f (1 {7})1 1073 1 2923 5760 t (Postcondition:)970 5916 w 10 I f (T)1574 5916 w 10 S f (=)1679 5916 w 10 I f (S)1783 5916 w 10 B f (Meeting the full specification)3 1235 1 720 6156 t 10 R f ( to)1 104(The precondition can be weakened)4 1390 2 720 6312 t 10 I f (a)2240 6312 w 10 S f (\263)2314 6312 w 10 R f (0 &)1 169 1 2385 6312 t 10 I f (b)2595 6312 w 10 S f (>)2669 6312 w 10 R f (0 because)1 391 1 2740 6312 t 10 I f (e)3157 6312 w 10 R f (\( 0 ,)2 124 1 3209 6312 t 10 I f (b)3341 6312 w 10 R f (,)3399 6312 w (. . .)2 125 1 3457 6287 t (\) would be positive in the guards of)7 1425 1 3615 6312 t ( When)1 293(cases 1 and 2.)3 569 2 720 6432 t 10 I f (a)1612 6432 w 10 S f (=)1686 6432 w 10 R f (0, only cases 3 and 7 can happen; the program would generate a north-south line)14 3283 1 1757 6432 t (segment as required.)2 823 1 720 6552 t (The precondition can be further weakened to)6 1795 1 720 6708 t 10 I f (a)2541 6708 w 10 S f (\263)2615 6708 w 10 R f (0 &)1 169 1 2686 6708 t 10 I f (b)2896 6708 w 10 S f (\263)2970 6708 w 10 R f (0 because)1 391 1 3041 6708 t 10 I f (e)3458 6708 w 10 R f (\()3510 6708 w 10 I f (a)3551 6708 w 10 R f (, 0 ,)2 116 1 3609 6708 t 10 I f (x)3733 6708 w 10 R f (, 0 \))2 124 1 3785 6708 t 10 S f (=)3925 6708 w 10 R f ( of cases)2 347(0 and at least one)4 697 2 3996 6708 t ( selectable when)2 673(1 and 2 would always be)5 1030 2 720 6828 t 10 I f (b)2455 6828 w 10 S f (=)2529 6828 w 10 R f (0 and)1 226 1 2600 6828 t 10 I f (P. y)1 138 1 2858 6828 t 10 S f (=)3020 6828 w 10 R f (0. For)1 271 1 3091 6828 t 10 I f (b)3394 6828 w 10 S f (=)3468 6828 w 10 R f (0, the initial value of)4 858 1 3539 6828 t 10 I f (P. y)1 138 1 4429 6828 t 10 R f (is 0; hence)2 441 1 4599 6828 t (the program would generate an east-west line segment as required.)9 2660 1 720 6948 t ( values may appear in the terms)6 1360( Noninteger)1 515( 3 involve integer multiples of 1/4.)6 1482(The guards in Program)3 963 4 720 7104 t 10 I f (b)720 7224 w 7 R f (2)781 7184 w 10 R f (\()832 7224 w 10 I f (P. x)1 138 1 873 7224 t 10 S f (+)1035 7224 w 10 S1 f ()1106 7224 w 1106 7224 m 75 build_12 1181 7224 m 10 R f (\))1189 7224 w 7 R f (2)1227 7184 w 10 R f (and)1303 7224 w 10 I f (a)1479 7224 w 7 R f (2)1540 7184 w 10 R f (\()1591 7224 w 10 I f (P. y)1 138 1 1632 7224 t 10 S f (-)1794 7224 w 10 S1 f ()1865 7224 w 1865 7224 m 75 build_12 1940 7224 m 10 R f (\))1948 7224 w 7 R f (2)1986 7184 w 10 R f (, which result when the definition of)6 1493 1 2029 7224 t 10 I f (e)3554 7224 w 10 R f ( The)1 212(is substituted into the guards.)4 1198 2 3630 7224 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 54 514 764 %%EndPage: 21 21 %%Page: 22 22 %%PageBoundingBox: (atend) /saveobj save def mark 22 pagesetup 10 R f (- 22 -)2 216 1 2772 480 t ( however, hastens the onset of overflow;)6 1622( Scaling,)1 376(requirement for integer arithmetic can be meet by scaling.)8 2322 3 720 840 t (judicious rounding is better.)3 1122 1 720 960 t (The function)1 525 1 720 1116 t 10 I f (h)1282 1116 w 7 R f (1)1343 1136 w 10 R f (\()1394 1116 w 10 I f (a)1435 1116 w 10 R f (,)1493 1116 w 10 I f (b)1526 1116 w 10 R f (,)1584 1116 w 10 I f (x)1617 1116 w 10 R f (,)1669 1116 w 10 I f (y)1702 1116 w 10 R f (\))1754 1116 w 10 S f (=)1803 1116 w 10 I f (e)1874 1116 w 10 R f (\()1926 1116 w 10 I f (a)1967 1116 w 10 R f (,)2025 1116 w 10 I f (b)2058 1116 w 10 R f (,)2116 1116 w 10 I f (x)2149 1116 w 10 S f (+)2217 1116 w 10 R f (1 ,)1 83 1 2288 1116 t 10 I f (y)2379 1116 w 10 S f (-)2447 1116 w 10 S1 f ()2518 1116 w 2518 1116 m 75 build_12 2593 1116 m 10 R f (\))2601 1116 w 10 S f (-)2650 1116 w 10 I f (a)2721 1116 w 7 R f (2)2782 1076 w 10 I f (/)2833 1116 w 10 R f ( of)1 121( terms)1 259( In)1 145(4 is a polynomial over the integers.)6 1479 4 2869 1116 t 10 I f (h)4911 1116 w 7 R f (1)4972 1136 w 10 R f (,)5015 1116 w (the guard in case 1 becomes)5 1202 1 720 1236 t 10 I f (h)1963 1236 w 7 R f (1)2024 1256 w 10 R f (\()2075 1236 w 10 I f (a)2116 1236 w 10 R f (,)2174 1236 w 10 I f (b)2207 1236 w 10 R f (,)2265 1236 w 10 I f (x)2298 1236 w 10 R f (,)2350 1236 w 10 I f (y)2383 1236 w 10 R f (\))2435 1236 w 10 S f (\243 -)1 126 1 2484 1236 t 10 I f (a)2626 1236 w 7 R f (2)2687 1196 w 10 I f (/)2738 1236 w 10 R f ( to the all-)3 453(4, which at integer arguments is equivalent)6 1813 2 2774 1236 t (integer expression)1 742 1 720 1356 t 10 I f (h)1500 1356 w 7 R f (1)1561 1376 w 10 R f (\()1612 1356 w 10 I f (a)1653 1356 w 10 R f (,)1711 1356 w 10 I f (b)1744 1356 w 10 R f (,)1802 1356 w 10 I f (x)1835 1356 w 10 R f (,)1887 1356 w 10 I f (y)1920 1356 w 10 R f (\))1972 1356 w 10 S f (\243 - \353)2 192 1 2021 1356 t 10 I f (a)2221 1356 w 7 R f (2)2282 1316 w 10 I f (/)2333 1356 w 10 R f (4)2369 1356 w 10 S f (\373 -)1 109 1 2427 1356 t 10 I f (a)2552 1356 w 10 CW f (mod)2634 1356 w 10 R f ( an all-)2 305( each guard similarly, we obtain)5 1346(2. Replacing)1 543 3 2846 1356 t (integer program.)1 665 1 720 1476 t 10 B f (Program 4.)1 482 1 2639 1716 t 10 R f (Precondition:)970 1872 w 10 I f (a)1534 1872 w 10 S f (\263)1608 1872 w 10 R f (0 &)1 169 1 1679 1872 t 10 I f (b)1889 1872 w 10 S f (\263)1963 1872 w 10 R f (0)2034 1872 w 10 CW f (let)970 2028 w 10 I f (h)1330 2148 w 7 R f (1)1391 2168 w 10 R f (\()1442 2148 w 10 I f (a)1483 2148 w 10 R f (,)1541 2148 w 10 I f (b)1574 2148 w 10 R f (,)1632 2148 w 10 I f (x)1665 2148 w 10 R f (,)1717 2148 w 10 I f (y)1750 2148 w 10 R f (\))1802 2148 w 10 S f (=)1892 2148 w 10 I f (e)1996 2148 w 10 R f (\()2048 2148 w 10 I f (a)2089 2148 w 10 R f (,)2147 2148 w 10 I f (b)2180 2148 w 10 R f (,)2238 2148 w 10 I f (x)2271 2148 w 10 S f (+)2339 2148 w 10 R f (1 ,)1 83 1 2410 2148 t 10 I f (y)2501 2148 w 10 S f (-)2569 2148 w 10 S1 f ()2640 2148 w 2640 2148 m 75 build_12 2715 2148 m 10 R f (\))2723 2148 w 10 S f (-)2813 2148 w 10 I f (a)2917 2148 w 7 R f (2)2978 2108 w 10 I f (/)3029 2148 w 10 R f (4)3065 2148 w 10 I f (h)1330 2268 w 7 R f (2)1391 2288 w 10 R f (\()1442 2268 w 10 I f (a)1483 2268 w 10 R f (,)1541 2268 w 10 I f (b)1574 2268 w 10 R f (,)1632 2268 w 10 I f (x)1665 2268 w 10 R f (,)1717 2268 w 10 I f (y)1750 2268 w 10 R f (\))1802 2268 w 10 S f (=)1892 2268 w 10 I f (e)1996 2268 w 10 R f (\()2048 2268 w 10 I f (a)2089 2268 w 10 R f (,)2147 2268 w 10 I f (b)2180 2268 w 10 R f (,)2238 2268 w 10 I f (x)2271 2268 w 10 S f (+)2339 2268 w 10 S1 f ()2410 2268 w 2410 2268 m 75 build_12 2485 2268 m 10 R f (,)2493 2268 w 10 I f (y)2526 2268 w 10 R f (\))2578 2268 w 10 S f (-)2668 2268 w 10 I f (b)2772 2268 w 7 R f (2)2833 2228 w 10 I f (/)2884 2268 w 10 R f (4)2920 2268 w 10 I f (h)1330 2388 w 7 R f (3)1391 2408 w 10 R f (\()1442 2388 w 10 I f (a)1483 2388 w 10 R f (,)1541 2388 w 10 I f (b)1574 2388 w 10 R f (,)1632 2388 w 10 I f (x)1665 2388 w 10 R f (,)1717 2388 w 10 I f (y)1750 2388 w 10 R f (\))1802 2388 w 10 S f (=)1892 2388 w 10 I f (e)1996 2388 w 10 R f (\()2048 2388 w 10 I f (a)2089 2388 w 10 R f (,)2147 2388 w 10 I f (b)2180 2388 w 10 R f (,)2238 2388 w 10 I f (x)2271 2388 w 10 S f (+)2339 2388 w 10 S1 f ()2410 2388 w 2410 2388 m 75 build_12 2485 2388 m 10 R f (,)2493 2388 w 10 I f (y)2526 2388 w 10 S f (-)2594 2388 w 10 R f (1 \))1 91 1 2665 2388 t 10 S f (-)2813 2388 w 10 I f (b)2917 2388 w 7 R f (2)2978 2348 w 10 I f (/)3029 2388 w 10 R f (4)3065 2388 w 10 I f (T)970 2508 w 10 R f (:)1067 2508 w 10 S f (= \306)1 186 1 1111 2508 t 10 I f (P)970 2628 w 10 R f (:)1072 2628 w 10 S f (=)1116 2628 w 10 R f (\( 0 ,)2 124 1 1220 2628 t 10 I f (b)1352 2628 w 10 R f (\))1410 2628 w 10 CW f (while)970 2748 w 10 I f (P. y)1 138 1 1311 2748 t 10 S f (\263)1473 2748 w 10 R f (0 &)1 169 1 1544 2748 t 10 I f (P. x)1 138 1 1754 2748 t 10 S f (\243)1916 2748 w 10 I f (a)1987 2748 w (T)1330 2868 w 10 S f (\310 =)1 148 1 1427 2868 t 10 I f (P)1624 2868 w 10 CW f (if)1330 2988 w 10 I f (h)1491 2988 w 7 R f (1)1552 3008 w 10 R f (\()1603 2988 w 10 I f (a)1644 2988 w 10 R f (,)1702 2988 w 10 I f (b)1735 2988 w 10 R f (,)1793 2988 w 10 I f (x)1826 2988 w 10 R f (,)1878 2988 w 10 I f (y)1911 2988 w 10 R f (\))1963 2988 w 10 S f (\243 - \353)2 192 1 2012 2988 t 10 I f (a)2212 2988 w 7 R f (2)2273 2948 w 10 I f (/)2324 2988 w 10 R f (4)2360 2988 w 10 S f (\373 -)1 109 1 2418 2988 t 10 I f (a)2543 2988 w 10 CW f (mod)2625 2988 w 10 R f (2)2837 2988 w 10 S f (\256)2928 2988 w 10 I f (P. x)1 138 1 3068 2988 t 10 S f (+ =)1 126 1 3246 2988 t 10 R f (1 {1})1 584 1 3412 2988 t ([ ])1 82 1 1330 3108 t 10 I f (h)1461 3108 w 7 R f (2)1522 3128 w 10 R f (\()1573 3108 w 10 I f (a)1614 3108 w 10 R f (,)1672 3108 w 10 I f (b)1705 3108 w 10 R f (,)1763 3108 w 10 I f (x)1796 3108 w 10 R f (,)1848 3108 w 10 I f (y)1881 3108 w 10 R f (\))1933 3108 w 10 S f (\243 - \353)2 192 1 1982 3108 t 10 I f (b)2182 3108 w 7 R f (2)2243 3068 w 10 I f (/)2294 3108 w 10 R f (4)2330 3108 w 10 S f (\373 -)1 109 1 2388 3108 t 10 I f (b)2513 3108 w 10 CW f (mod)2595 3108 w 10 R f (2)2807 3108 w 10 S f (\256)2898 3108 w 10 I f (P. x)1 138 1 3038 3108 t 10 S f (+ =)1 126 1 3216 3108 t 10 R f (1 {2})1 614 1 3382 3108 t 10 CW f (else)1330 3228 w 10 S f (\256)1611 3228 w 10 CW f (if)1690 3348 w 10 I f (h)1851 3348 w 7 R f (3)1912 3368 w 10 R f (\()1963 3348 w 10 I f (a)2004 3348 w 10 R f (,)2062 3348 w 10 I f (b)2095 3348 w 10 R f (,)2153 3348 w 10 I f (x)2186 3348 w 10 R f (,)2238 3348 w 10 I f (y)2271 3348 w 10 R f (\))2323 3348 w 10 S f (> - \353)2 192 1 2372 3348 t 10 I f (b)2572 3348 w 7 R f (2)2633 3308 w 10 I f (/)2684 3348 w 10 R f (4)2720 3348 w 10 S f (\373 -)1 109 1 2778 3348 t 10 I f (b)2903 3348 w 10 CW f (mod)2985 3348 w 10 R f (2)3197 3348 w 10 S f (\256)3288 3348 w 10 I f (P. y)1 138 1 3428 3348 t 10 S f (- =)1 126 1 3606 3348 t 10 R f (1 {3})1 224 1 3772 3348 t 10 CW f (else)1690 3468 w 10 S f (\256)1971 3468 w 10 I f (P. x)1 138 1 2111 3468 t 10 S f (+ =)1 126 1 2289 3468 t 10 R f (1 ,)1 83 1 2455 3468 t 10 I f (P. y)1 138 1 2579 3468 t 10 S f (- =)1 126 1 2757 3468 t 10 R f (1 {7})1 1073 1 2923 3468 t (Postcondition:)970 3624 w 10 I f (T)1574 3624 w 10 S f (=)1679 3624 w 10 I f (S)1783 3624 w 10 R f ( reducing the strength of multiplications, elimi-)6 1951(Further arithmetic transformations \(propagating constants,)4 2369 2 720 3780 t ( expressions out of the loop\) can significantly improve)8 2277(nating common subexpressions, moving invariant)4 2043 2 720 3900 t ( incorporated in every published ellipse-tracing algorithm,)6 2501( generic optimizations are)3 1118(efficiency. Such)1 701 3 720 4020 t ( since they are independent of the details of the)9 1963( However,)1 449( this trilogy.)2 505(including that in the first paper of)6 1403 4 720 4140 t (ellipse problem, we shall not pursue them further here.)8 2185 1 720 4260 t 10 B f (Discussion)720 4500 w 10 R f (Although the topic has been studied for years,)7 1843 1 720 4656 t 8 R f (2-6)2571 4624 w 10 R f (previous algorithms yield ad hoc approximations that lack)7 2337 1 2703 4656 t ( published descrip-)2 784( The)1 217( specification independent of the details of the algorithm.)8 2379(a precise mathematical)2 940 4 720 4776 t ( program for efficiency, while ignoring the ques-)7 1986(tions tend to concentrate on algebraic manipulation of the)8 2334 2 720 4896 t (tion of just what locus the program traces.)7 1681 1 720 5016 t (In particular, all but one of the cited algorithms can yield different approximations for the same ellipse)16 4320 1 720 5172 t ( single exception)2 695( The)1 214( axis.)1 220(depending on the orientation of the major)6 1719 4 720 5292 t 8 R f (6)3576 5260 w 10 R f (is an artifact of treating the major)6 1390 1 3650 5292 t (and minor axes differently.)3 1081 1 720 5412 t ( optimizations as suggested above.)4 1389(All the cited algorithms bear a superficial resemblance to Program 4 with)11 2931 2 720 5568 t ( example, with-)2 626( For)1 190( lack leads to asymmetry under transposition.)6 1818( This)1 230(None, however, incorporates case 2.)4 1456 5 720 5688 t ( \()1 68( would be missed when)4 974( 1 , 3 \))4 190(out case 2 the lighted point \()6 1192 4 720 5808 t 10 I f (a)3152 5808 w 10 R f (,)3210 5808 w 10 I f (b)3243 5808 w 10 R f (\))3301 5808 w 10 S f (=)3350 5808 w 10 R f ( would be)2 414( 3 , 1 \))4 190( yet its transpose \()4 767(\( 2 , 3 \),)4 248 4 3421 5808 t (visited when \()2 566 1 720 5928 t 10 I f (a)1294 5928 w 10 R f (,)1352 5928 w 10 I f (b)1385 5928 w 10 R f (\))1443 5928 w 10 S f (=)1492 5928 w 10 R f (\( 3 , 2 \).)4 248 1 1563 5928 t ( came first; mathematics)3 1003(From the published descriptions, I infer that the logical scheme of the algorithms)12 3317 2 720 6084 t ( octant-plotting)1 617( basic intuition was to imitate as closely as possible the)10 2245( The)1 209(was used only to fill in details.)6 1249 4 720 6204 t ( location and treat-)3 775( Proper)1 324( had proved so successful with circles.)6 1581(technique originated by Pitteway, which)4 1640 4 720 6324 t ( octant join, which had been a trivial subproblem for circles, became the central, and finally)15 3839(ment of the)2 481 2 720 6444 t ( solutions were heuristic.)3 999( All)1 178(defeating, problem for ellipses.)3 1245 3 720 6564 t ( the mathematics to the logical structure of)7 1769(The present derivation proceeds in an opposite direction, from)8 2551 2 720 6720 t ( proving)1 345( In)1 142( plan to walk from north to east.)7 1345( is still an initial algorithmic intuition: the)7 1742( There)1 292(Program 4.)1 454 6 720 6840 t (Program 0 it was necessary to confirm the intuition that the walk would visit all lighted points.)16 3785 1 720 6960 t ( published algorithms represent attempts to generalize from highly spe-)9 2867(From a more abstract viewpoint, the)5 1453 2 720 7116 t ( circle codes assume 8-fold symmetry.)5 1671( The)1 232(cialized code for the circle.)4 1190 3 720 7236 t 8 R f (5, 7, 8)2 200 1 3821 7204 t 10 R f (This paper instead has)3 968 1 4072 7236 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 52 514 764 %%EndPage: 22 22 %%Page: 23 23 %%PageBoundingBox: (atend) /saveobj save def mark 23 pagesetup 10 R f (- 23 -)2 216 1 2772 480 t ( Bresenham's less refined algorithm for a quadrant,)7 2114(proceeded by generalizing)2 1068 2 720 840 t 8 R f (9)3910 808 w 10 R f (which only assumes four-)3 1056 1 3984 840 t ( latter generalization involves little more than)6 1856( The)1 211( all that can be counted on in an ellipse.)9 1635(fold symmetry,)1 618 4 720 960 t ( test,)1 200(the discovery of case 2, while generalization from octant algorithms requires at least a sound octant)15 4120 2 720 1080 t ( of the published)3 699( None)1 276(separate treatment for each octant, and a satisfactory treatment of the octant join.)12 3345 3 720 1200 t (algorithms meets this multiple challenge fully.)5 1861 1 720 1320 t ( and general-)2 524(In diagrammatic terms, the two approaches may be understood as proceeding by specialization)12 3796 2 720 1476 t ( stum-)1 257( The)1 209( the two paths do not commute.)6 1282( Unfortunately)1 614( in Figure 1.)3 501(ization in opposite orders, as shown)5 1457 6 720 1596 t ( is much)2 355( It)1 119( been specialized too far.)4 1026(bling block is generalizing from octant circle algorithms, which have)9 2820 4 720 1716 t (easier to generalize from the more malleable quadrant circle algorithm.)9 2837 1 720 1836 t (Quadrant circle)1 617 1 1762 2102 t (\(Bresenham\))1813 2222 w 3240 2142 2520 2142 Dl 3240 2142 3168 2160 Dl 3240 2142 3168 2124 Dl 10 I f (generalize)2672 2102 w 10 R f (Quadrant ellipse)1 657 1 3362 2102 t (\(Program 0\))1 485 1 3448 2222 t 3690 2574 3690 2286 Dl 3690 2574 3672 2502 Dl 3690 2574 3708 2502 Dl 10 I f (specialize)3263 2450 w 10 R f (Efficient ellipse)1 635 1 3373 2738 t 2070 2574 2070 2286 Dl 2070 2574 2052 2502 Dl 2070 2574 2088 2502 Dl 10 I f (specialize)1643 2450 w 10 R f (Octant circle)1 512 1 1814 2678 t (\(Horn\))1935 2798 w 3240 2718 2520 2718 Dl 3240 2718 3168 2736 Dl 3240 2718 3168 2700 Dl 10 I f (generalize)2672 2678 w 10 R f (Figure 1.)1 361 1 2699 3060 t ( strength'')1 425( ``Industrial)1 511( less sophisticated code.)3 996( easier to reengineer)3 838( It's)1 194(The lesson is broadly applicable.)4 1356 6 720 3216 t ( major steps)2 489( this reason,)2 488( For)1 193(programs are likely to be less amenable to major changes than are prototypes.)12 3150 4 720 3336 t ( necessitate stepping back in the phylogenetic tree in order to surge forward in an)14 3271(in software evolution may)3 1049 2 720 3456 t ( it would be well to preserve primitive versions of evolving programs as a kind of)15 3275( Perhaps)1 367(altered direction.)1 678 3 720 3576 t (health insurance against software sclerosis.)4 1721 1 720 3696 t (Moral: You can't add raisins after the bread is baked.)9 2128 1 720 3852 t 10 B f (References)720 4092 w 10 R f ( H., ``Computer processing of line-drawing images,'')6 2132([1] Freeman,)1 565 2 889 4308 t 10 I f (Computing Surveys)1 780 1 3611 4308 t 10 B f (6)4416 4308 w 10 R f (, p. 63 \(1974\).)3 566 1 4466 4308 t ( M. L. V., ``Algorithms for drawing ellipses or hyperbolae with a digital plotter,'')13 3333([2] Pitteway,)1 566 2 889 4488 t 10 I f (Com-)4818 4488 w (puter J.)1 305 1 1080 4608 t 10 B f (10)1410 4608 w 10 R f (, pp. 282-289 \(1967\).)3 849 1 1510 4608 t ( V., ``Techniques for conic splines,'')5 1504([3] Pratt,)1 405 2 889 4788 t 10 I f (Computer Graphics)1 803 1 2829 4788 t 10 B f (19)3663 4788 w 10 R f ( A.)1 129(\(3\), pp. 151-159, Barsky, B.)4 1148 2 3763 4788 t ( '85 Conference Proceedings.)3 1180( SIGGRAPH)1 550(\(Ed.\), ACM \(1985\).)2 796 3 1080 4908 t ( Aken, J. R., ``An efficient ellipse-drawing algorithm,'')7 2253([4] Van)1 357 2 889 5088 t 10 I f ( Graphics and Appli-)3 865(IEEE Computer)1 646 2 3529 5088 t (cations)1080 5208 w 10 B f (4)1394 5208 w 10 R f (\(9\), pp. 24-35 \(1984\).)3 865 1 1444 5208 t ( S. K., and Hughes, J. F.,)6 1057( J. D., Van Dam, A., Feiner,)6 1177([5] Foley,)1 444 3 889 5388 t 10 I f (Computer Graphics Principles and)3 1438 1 3602 5388 t (Practice)1080 5508 w 10 R f (, Addison-Wesley \(1990\).)2 1037 1 1418 5508 t ( N., ``Drawing lines, circles and ellipses in a raster,'' pp. 427-434 in)12 2737([6] Wirth,)1 449 2 889 5688 t 10 I f (Beauty is our Business)3 914 1 4101 5688 t 10 R f (,)5015 5688 w ( D., and Misra, J. \(Eds.\), Springer-Verlag, New)7 1974(Feijen, W. H. J., van Gasteren, A. J. M., Gries,)9 1986 2 1080 5808 t (York \(1990\).)1 521 1 1080 5928 t ( B. K. P., ``Circle generators for display devices,'')8 2055([7] Horn,)1 421 2 889 6108 t 10 I f (Computer Graphics and Image Process-)4 1644 1 3396 6108 t (ing)1080 6228 w 10 B f (5)1233 6228 w 10 R f (\(1976\).)1308 6228 w ( D., ``Best approximate circles on integer grids,'')7 2073( M.)1 153([8] McIlroy,)1 543 3 889 6408 t 10 I f (ACM Trans. on Graphics)3 1062 1 3698 6408 t 10 B f (2)4800 6408 w 10 R f (, pp.)1 190 1 4850 6408 t (237-264 \(Oct. 1983\).)2 843 1 1080 6528 t ( ``A linear algorithm for incremental digital display of circular arcs,'')10 2814( J.,)1 117([9] Bresenham,)1 665 3 889 6708 t 10 I f (Comm. ACM)1 526 1 4514 6708 t 10 B f (20)1080 6828 w 10 R f (, pp. 100-106 \(1977\).)3 849 1 1180 6828 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 93 514 764 %%EndPage: 23 23 %%Page: 24 24 %%PageBoundingBox: (atend) /saveobj save def mark 24 pagesetup 10 R f (- 24 -)2 216 1 2772 480 t 12 B f (Ellipses Not Yet Made Easy)4 1421 1 2169 840 t 10 I f (M. D. McIlroy)2 576 1 2592 1080 t 10 R f (AT&T Bell Laboratories)2 993 1 2383 1260 t (Murray Hill, New Jersey 07974)4 1267 1 2246 1380 t (ABSTRACT)2618 1620 w ( Circles and Ellipses in a Raster,'')6 1374(A paper by N. Wirth, ``Drawing Lines,)6 1566 2 1080 2016 t 8 R f (1)4028 1984 w 10 R f (excerpted here)1 585 1 4095 2016 t ( problem of draw-)3 729(by permission, illustrates methodological issues that arise in the simple)9 2871 2 1080 2136 t ( highly)1 291( paper, like others on the subject, attempts to generalize from a)11 2608( The)1 214(ing ellipses.)1 487 4 1080 2256 t ( result is foredoomed because the model has)7 1773( The)1 206(optimized algorithm for drawing circles.)4 1621 3 1080 2376 t ( engagingly and crisply written than)5 1450( More)1 268( return.)1 290(been specialized beyond the point of no)6 1592 4 1080 2496 t ( computer science, the paper affords an attractive and instruc-)9 2504(much practical literature in)3 1096 2 1080 2616 t ( the literature which Wirth himself has acknowledged as)8 2355(tive addition to that branch of)5 1245 2 1080 2736 t (having ``taught how not to do it,'')6 1363 1 1080 2856 t 8 R f (2)2451 2824 w 10 R f (in matters of both style and substance.)6 1529 1 2516 2856 t (Wirth's words appear in full-size type, my annotations in small size.)10 2730 1 1080 3012 t 10 B f (Abstract.)720 3168 w 10 R f ( lines and circles are developed)5 1289(In a tutorial style, Bresenham's algorithms for drawing straight)8 2578 2 1173 3168 t ( circle algorithm is then generalized for drawing ellipses.)8 2276( The)1 205(using Dijkstra's notation and discipline.)4 1597 3 720 3288 t 9 R f ( suited as a case study, for it reveals just how the design of the)14 2376(The ``tutorial'' exposition is admirably)4 1444 2 970 3444 t ( does not,)2 351( It)1 103(ellipse-drawing algorithm went astray, as had many similar algorithms published previously.)10 3366 3 970 3564 t ( later admits: ``We adopt his notation but deviate from his)10 2165(however, well illustrate Dijkstra's rigor, as it)6 1655 2 970 3684 t ( on)1 126( Concentrating)1 569( specifying the task algorithmically rather than by a result predicate.'')10 2648(discipline by)1 477 4 970 3804 t ( for purpose, the development fails to consider the problem and the pro-)12 2702(method without careful regard)3 1118 2 970 3924 t ( are analyzed in isola-)4 816( precise objective is stated, and single statements)7 1813( No)1 163(gram as a connected whole.)4 1028 4 970 4044 t ( result is a program with)5 1000( The)1 210( context.)1 335(tion, without reference to boundary conditions imposed by)7 2275 4 970 4164 t (unknown properties, which cannot be trusted for general use.)8 2205 1 970 4284 t 9 S f (\347)1948 4440 w 9 R f (Beware of programs with imprecise specifications.)5 1833 1 1963 4440 t 9 S f ( _________________________________________)1 1818( _)1 -1818( `````````````````````````````````````````)1 1818(\347 `)1 -1818 4 3811 4440 t 10 B f (Introduction.)720 4596 w 10 R f ( The)1 209( of my programs.)3 700(Recently, I needed to incorporate a raster drawing algorithm into one)10 2788 3 1343 4596 t ( quickly)1 326( Literature)1 444( was the target of my search.)6 1150(Bresenham algorithm is known to be efficient and therefore)8 2400 4 720 4716 t ( my favourite)2 562(revealed descriptions in several sources [1,3]; all I needed to do was to translate them into)15 3758 2 720 4836 t ( but to)2 322( I wished\320in contrast to the computer\320not to interpret the algorithms)10 3171(notation. However,)1 827 3 720 4956 t 10 I f (understand)720 5076 w 10 R f ( had to discover that the sources picked were, albeit typical, quite inadequate for this)14 3517(them. I)1 318 2 1205 5076 t ( courses are to teach the use of a \(specific\))9 1727( reflected the widespread view that programming)6 1992(purpose. They)1 601 3 720 5196 t (programming language, whereas the algorithms are simply given.)7 2621 1 720 5316 t 9 R f ( first sentence to suggest labor at the sci-)8 1501(How frequently technical papers utter the word ``recent'' in the)9 2319 2 970 5472 t ( the frontier, has more than passing interest.)7 1617( present topic, though admittedly not at)6 1452( The)1 192(entific frontier!)1 559 4 970 5592 t ( type seems)2 423(Everybody \(including me\) who conscientiously studies algorithms of the Pitteway-Bresenham)9 3397 2 970 5712 t ( analysis.)1 345(impelled to improve the never quite complete)6 1703 2 970 5832 t 7 R f (3-5)3025 5804 w 9 R f (The analysis is delicate\320more delicate than)5 1640 1 3150 5832 t (the paper recognizes.)2 764 1 970 5952 t ( texts intend to help teach programming languages, then so does the present)12 2753(If, as Wirth charges, graphics)4 1067 2 970 6108 t ( languages like Pascal than at critical under-)7 1640( analysis is aimed at getting efficient code for)8 1708(paper. More)1 472 3 970 6228 t ( avowed concern for efficiency upstages considerations of purpose.)8 2427( The)1 186(standing of the problem.)3 882 3 970 6348 t ( diction of the last phrase, ``whereas the algorithms are simply given,'' inspires a comple-)14 3369(The unusual)1 451 2 970 6504 t ( tutorial will strive to give algorithms simply, but it will also strive to jus-)14 2711(mentary interpretation: a good)3 1109 2 970 6624 t ( presentation, but not justification; one can't justify the unjustifi-)9 2344( succeeds on)2 456( Wirth)1 256(tify them adequately.)2 764 4 970 6744 t ( more thorough attempt might have uncovered the impasse.)8 2142(able. A)1 284 2 970 6864 t 10 R f ( early and outspoken critic of this view, and he correctly pointed out that the difficul-)15 3445(Dijkstra was an)2 625 2 970 7020 t ( order to)2 362( In)1 145( are primarily inherent in the subject, namely in constructive reasoning.)10 2971(ties of programming)2 842 4 720 7140 t ( notational issue to a bare minimum by postulating his own)10 2377(emphasize this central theme, he compressed the)6 1943 2 720 7260 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 50 514 764 %%EndPage: 24 24 %%Page: 25 25 %%PageBoundingBox: (atend) /saveobj save def mark 25 pagesetup 10 R f (- 25 -)2 216 1 2772 480 t (notation that is concisely defined within a few formulas [2].)9 2391 1 720 840 t 9 R f ( notation: its suppression of spurious detail about)7 1806(The capsule description misses the genius of Dijkstra's)7 2014 2 970 996 t ( been the central purpose, the notation)6 1466( mere ``compression of the notational issue'')6 1714(sequencing. Had)1 640 3 970 1116 t (would not stand out among others.)5 1248 1 970 1236 t ( com-)1 211( Guarded)1 359(The present treatment illustrates Dijkstra's notation little more than it does his discipline.)12 3250 3 970 1392 t (mands appear only as trivial equivalents for everyday)7 2036 1 970 1512 t 9 CW f (while)3043 1512 w 9 R f (and)3350 1512 w 9 CW f (if-then-else)3517 1512 w 9 R f (constructs. The)1 588 1 4202 1512 t ( that is)2 291( notation)1 342( The)1 210(deployment of synonyms is window-dressing, not a methodological advance.)8 2977 4 970 1632 t ( is elementary algebra; no computer scientist should be)8 2104(mainly\320and productively\320used in the paper)4 1716 2 970 1752 t (without it.)1 371 1 970 1872 t 10 R f ([Further introduction, a section on lines, and a section on circles are omitted.])12 3098 1 720 2028 t 10 B f (Ellipses.)720 2184 w 10 R f ( the circle algorithm, we wish to design an algorithm for plotting ellipses by proceed-)14 3438(Similarly to)1 477 2 1125 2184 t (ing in steps to find raster points to be marked.)9 1832 1 720 2304 t 9 R f ( gives, ``We wish simi-)4 905(Grammatically the adverb ``similarly'' has to modify the main verb, but that)11 2915 2 970 2460 t ( it is as computers do; see Wirth's)7 1240( Perhaps)1 333( it is probably not as we do.)7 1013(larly.'' However algorithms wish,)3 1234 4 970 2580 t ( slapdash English, even though well above threshold for most computing)10 2758( The)1 199(introductory paragraph.)1 863 3 970 2700 t ( writing inexactly one hides)4 1042( In)1 132(journals, symptomizes a less than careful approach to the whole work.)10 2646 3 970 2820 t (inexact reasoning, even from oneself.)4 1353 1 970 2940 t 9 S f (\347)2107 3096 w 9 R f (What's worth telling is worth telling well.)6 1516 1 2122 3096 t 9 S f ( __________________________________)1 1501( _)1 -1501( ``````````````````````````````````)1 1501(\347 `)1 -1501 4 3653 3096 t 10 R f ( other three quadrants can be covered by symmetry arguments and)10 2679(We concentrate on the first quadrant; the)6 1641 2 720 3252 t (require no additional computation.)3 1382 1 720 3372 t 9 R f (``No additional computation'' really means ``no more code to be displayed in this paper.'')13 3262 1 970 3528 t 10 R f ( without loss of generality, we assume)6 1595( Again)1 305( equation.)1 405(Let the ellipse be defined by the following)7 1765 4 970 3684 t (0)720 3804 w 10 S f (<)819 3804 w 10 I f (a)923 3804 w 10 S f (\243)1014 3804 w 10 I f (b)1110 3804 w 10 R f (.)1160 3804 w 10 I f (E)1220 3984 w 10 R f (: \()1 102 1 1289 3984 t 10 I f (x / a)2 138 1 1399 3984 t 10 R f (\))1545 3984 w 7 R f (2)1583 3944 w 10 S f (+)1675 3984 w 10 R f (\()1779 3984 w 10 I f (y / b)2 138 1 1820 3984 t 10 R f (\))1966 3984 w 7 R f (2)2004 3944 w 10 S f (=)2096 3984 w 10 R f (1)2200 3984 w 9 R f ( problem)1 325(The customary meaning of ``without loss of generality'' is that in some obvious way the general)15 3495 2 970 4200 t ( possibility of)2 504( The)1 190( however, generality has certainly been lost.)6 1613( Here,)1 248(can be mapped into a special case.)6 1265 5 970 4320 t 9 I f (a)970 4440 w 9 S f (=)1060 4440 w 9 R f ( limiting case, should be restored in any real)8 1653(0, an ellipse of zero width, and a perfectly reasonable)9 1982 2 1155 4440 t ( dying at the moment the rings appear edge)8 1631( a time-lapse animation of Saturn)5 1250(implementation. Imagine)1 939 3 970 4560 t (on.)970 4680 w 9 S f (\347)2463 4836 w 9 R f (Handle limiting cases.)2 804 1 2478 4836 t 9 S f ( __________________)1 789( _)1 -789( ``````````````````)1 789(\347 `)1 -789 4 3297 4836 t 9 R f (More seriously, the highly technical restriction)5 1753 1 970 4992 t 9 I f (a)2759 4992 w 9 S f (\243)2841 4992 w 9 I f (b)2928 4992 w 9 R f (is ``simply given''\320never explained and never)5 1780 1 3010 4992 t ( the program can fail without it.)6 1146( Yet)1 176(appealed to in the development.)4 1150 3 970 5112 t 10 R f (We start with the point)4 944 1 720 5268 t 10 I f (P)1697 5268 w 10 R f (\( 0 ,)2 124 1 1766 5268 t 10 I f (b)1898 5268 w 10 R f (\) and proceed by incrementing)4 1251 1 1956 5268 t 10 I f (x)3240 5268 w 10 R f (in each step, and decrementing)4 1265 1 3317 5268 t 10 I f (y)4615 5268 w 10 R f (if neces-)1 348 1 4692 5268 t (sary.)720 5388 w 9 R f (Here, as throughout the paper, the reader is left to infer that)11 2166 1 970 5544 t 9 I f (a)3161 5544 w 9 R f (and)3231 5544 w 9 I f (b)3386 5544 w 9 R f ( assumption is cen-)3 703( The)1 188(are integers.)1 443 3 3456 5544 t (tral to the correctness of the algorithm.)6 1401 1 970 5664 t (The extra identifier)2 696 1 970 5820 t 9 I f (P)1689 5820 w 9 R f (, like)1 181 1 1744 5820 t 9 I f (E)1948 5820 w 9 R f (in the previous equation, serves no purpose whatever.)7 1937 1 2026 5820 t 10 R f (The exact ordinate of the next point follows from the defining equation:)11 2871 1 720 5976 t 10 I f (Y)1220 6156 w 10 S f (=)1325 6156 w 10 I f (b)1429 6156 w 10 S f (\326)1511 6174 w 10 R f (\( 1)1 91 1 1574 6156 t 10 S f (-)1705 6156 w 10 R f (\( \()1 74 1 1800 6156 t 10 I f (x)1882 6156 w 10 S f (+)1966 6156 w 10 R f (1 \))1 91 1 2061 6156 t 10 I f (/ a)1 86 1 2160 6156 t 10 R f (\))2254 6156 w 7 R f (2)2292 6116 w 10 R f (\))2343 6156 w 9 R f ( the omitted part of the paper.)6 1077(The notation here depends on)4 1067 2 970 6372 t 9 I f (Y)3161 6372 w 9 R f (is an ordinate on the true ellipse;)6 1184 1 3235 6372 t 9 I f (y)4443 6372 w 9 R f (is a ras-)2 283 1 4507 6372 t ( by usage to mean the point on the true ellipse)10 1710( point'' was defined informally)4 1151( ``Next)1 287(ter approximation.)1 672 4 970 6492 t (at the next integer abscissa.)4 990 1 970 6612 t 10 R f (The raster point coordinate must satisfy)5 1584 1 720 6768 t 10 I f (y)1220 6948 w 10 S f (-)1313 6948 w 10 R f (1)1417 6948 w 10 I f (/)1475 6948 w 10 R f (2)1511 6948 w 10 S f (<)1610 6948 w 10 I f (b)1714 6948 w 10 S f (\326)1796 6966 w 10 R f (\( 1)1 91 1 1859 6948 t 10 S f (-)1990 6948 w 10 R f (\( \()1 74 1 2085 6948 t 10 I f (x)2167 6948 w 10 S f (+)2251 6948 w 10 R f (1 \))1 91 1 2346 6948 t 10 I f (/ a)1 86 1 2445 6948 t 10 R f (\))2539 6948 w 7 R f (2)2577 6908 w 10 R f (\))2628 6948 w 10 I f (y)1220 7128 w 7 R f (2)1275 7088 w 10 S f (-)1367 7128 w 10 I f (y)1471 7128 w 10 S f (+)1564 7128 w 10 R f (1)1668 7128 w 10 I f (/)1726 7128 w 10 R f (4)1762 7128 w 10 S f (<)1861 7128 w 10 I f (b)1965 7128 w 7 R f (2)2026 7088 w 10 S f (-)2085 7128 w 10 I f (b)2156 7128 w 7 R f (2)2217 7088 w 10 R f (\()2268 7128 w 10 I f (x)2309 7128 w 10 S f (+)2393 7128 w 10 R f (1 \))1 91 1 2488 7128 t 7 R f (2)2584 7088 w 10 I f (/ a)1 86 1 2635 7128 t 7 R f (2)2732 7088 w 10 I f (a)1220 7308 w 7 R f (2)1281 7268 w 10 I f (y)1332 7308 w 7 R f (2)1387 7268 w 10 S f (-)1479 7308 w 10 I f (a)1583 7308 w 7 R f (2)1644 7268 w 10 I f (y)1695 7308 w 10 S f (+)1788 7308 w 10 I f (a)1892 7308 w 7 R f (2)1953 7268 w 10 I f (/)2004 7308 w 10 R f (4)2040 7308 w 10 S f (<)2139 7308 w 10 I f (a)2243 7308 w 7 R f (2)2304 7268 w 10 I f (b)2355 7308 w 7 R f (2)2416 7268 w 10 S f (-)2508 7308 w 10 I f (b)2612 7308 w 7 R f (2)2673 7268 w 10 I f (x)2724 7308 w 7 R f (2)2779 7268 w 10 S f (-)2871 7308 w 10 R f (2)2975 7308 w 10 I f (b)3033 7308 w 7 R f (2)3094 7268 w 10 I f (x)3145 7308 w 10 S f (-)3238 7308 w 10 I f (b)3342 7308 w 7 R f (2)3403 7268 w cleartomark showpage saveobj restore %%PageBoundingBox: 61 45 514 764 %%EndPage: 25 25 %%Page: 26 26 %%PageBoundingBox: (atend) /saveobj save def mark 26 pagesetup 10 R f (- 26 -)2 216 1 2772 480 t 10 I f (b)1220 840 w 7 R f (2)1281 800 w 10 I f (x)1332 840 w 7 R f (2)1387 800 w 10 S f (+)1479 840 w 10 R f (2)1583 840 w 10 I f (b)1641 840 w 7 R f (2)1702 800 w 10 I f (x)1753 840 w 10 S f (+)1846 840 w 10 I f (a)1950 840 w 7 R f (2)2011 800 w 10 I f (y)2062 840 w 7 R f (2)2117 800 w 10 S f (-)2209 840 w 10 I f (a)2313 840 w 7 R f (2)2374 800 w 10 I f (y)2425 840 w 10 S f (+)2518 840 w 10 I f (a)2622 840 w 7 R f (2)2683 800 w 10 I f (/)2734 840 w 10 R f (4)2770 840 w 10 S f (-)2869 840 w 10 I f (a)2973 840 w 7 R f (2)3034 800 w 10 I f (b)3085 840 w 7 R f (2)3146 800 w 10 S f (+)3238 840 w 10 I f (b)3342 840 w 7 R f (2)3403 800 w 10 S f (<)3495 840 w 10 R f (0)3599 840 w 9 R f ( if)1 79(The second line of the derivation is unjustified)7 1681 2 970 1056 t 9 I f (y)2754 1056 w 9 S f (<)2839 1056 w 9 R f (1)2934 1056 w 9 I f (/)2986 1056 w 9 R f (2 or if)2 223 1 3018 1056 t 9 I f (x)3265 1056 w 9 S f (+)3342 1056 w 9 R f (1)3429 1056 w 9 S f (>)3519 1056 w 9 I f (a)3614 1056 w 9 R f ( former event can happen)4 921(. The)1 210 2 3659 1056 t ( latter cannot, but that fact is not foreseeable at this stage of)12 2185( The)1 190(and cause trouble, as we shall see later.)7 1445 3 970 1176 t (the derivation.)1 521 1 970 1296 t 9 S f (\347)2304 1452 w 9 R f (Attend to boundary conditions.)3 1122 1 2319 1452 t 9 S f ( _________________________)1 1107( _)1 -1107( `````````````````````````)1 1107(\347 `)1 -1107 4 3456 1452 t 10 R f (The necessary and sufficient condition for decrementing)6 2292 1 720 1608 t 10 I f (y)3044 1608 w 10 R f (is therefore)1 459 1 3120 1608 t 10 I f (h)3612 1608 w 10 S f (\263)3703 1608 w 10 R f (0 with the auxiliary variable)4 1158 1 3799 1608 t 10 I f (h)4990 1608 w 10 R f (being defined as)2 654 1 720 1728 t 10 I f (h)1220 1908 w 10 S f (=)1319 1908 w 10 I f (b)1423 1908 w 7 R f (2)1484 1868 w 10 I f (x)1535 1908 w 7 R f (2)1590 1868 w 10 S f (+)1682 1908 w 10 R f (2)1786 1908 w 10 I f (b)1844 1908 w 7 R f (2)1905 1868 w 10 I f (x)1956 1908 w 10 S f (+)2049 1908 w 10 I f (a)2153 1908 w 7 R f (2)2214 1868 w 10 I f (y)2265 1908 w 7 R f (2)2320 1868 w 10 S f (-)2412 1908 w 10 I f (a)2516 1908 w 7 R f (2)2577 1868 w 10 I f (y)2628 1908 w 10 S f (+)2721 1908 w 10 I f (a)2825 1908 w 7 R f (2)2886 1868 w 10 I f (/)2937 1908 w 10 R f (4)2973 1908 w 10 S f (-)3072 1908 w 10 I f (a)3176 1908 w 7 R f (2)3237 1868 w 10 I f (b)3288 1908 w 7 R f (2)3349 1868 w 10 S f (+)3441 1908 w 10 I f (b)3545 1908 w 7 R f (2)3606 1868 w 9 R f ( discussion about decrementing)3 1149(Although it appears suddenly and unexplainedly here, the)7 2107 2 970 2124 t 9 I f (y)4254 2124 w 9 R f (parallels that)1 468 1 4322 2124 t (in the omitted discussion of circles.)5 1278 1 970 2244 t 10 R f ( as soon as)3 460(As in the case of the circle, the termination condition is met)11 2485 2 720 2400 t 10 I f (y)3700 2400 w 10 R f (might have to be decreased by)5 1261 1 3779 2400 t (more than 1 after an increase of)6 1280 1 720 2520 t 10 I f (x)2028 2520 w 10 R f ( is greater than 45)4 723(by 1, i.e. when the tangent to the curve)8 1574 2 2100 2520 t 10 S f (\260)4397 2520 w 10 R f ( in the)2 254(. Unlike)1 349 2 4437 2520 t ( obviously given by)3 804(case of the circle, however, this condition is not)8 1932 2 720 2640 t 10 I f (x)3485 2640 w 10 S f (=)3578 2640 w 10 R f ( reject the obvious solution)4 1099(y. We)1 267 2 3674 2640 t ( is -1, [sic] because this computation alone would)8 1978(of computing the ordinate for which the curve's derivative)8 2342 2 720 2760 t (involve at least the square root function.)6 1606 1 720 2880 t 9 R f ( paragraph, and especially of the sentence beginning, ``Unlike in the case of,'' won't)13 3136(The English of the)3 684 2 970 3036 t (stand up to scrutiny.)3 732 1 970 3156 t ( of decreasing)2 520(The notion)1 399 2 970 3312 t 9 I f (y)1919 3312 w 9 R f (after increasing)1 565 1 1989 3312 t 9 I f (x)2584 3312 w 9 R f ( simplify the maintenance of)4 1060( To)1 153(is excessively sequential.)2 923 3 2654 3312 t ( would prefer to say that at each)7 1184(the loop invariant, and to avoid needlessly overspecifying the code, one)10 2636 2 970 3432 t (step either)1 383 1 970 3552 t 9 I f (x)1386 3552 w 9 R f (alone is modified, or)3 777 1 1459 3552 t 9 I f (x)2270 3552 w 9 R f (and)2344 3552 w 9 I f (y)2508 3552 w 9 R f ( presentation here, which)3 940( The)1 197(are modified simultaneously.)2 1071 3 2582 3552 t (decides which to do first, bears on Pascal more than on the problem.)12 2467 1 970 3672 t ( ordinate in question is)4 878( The)1 200(The last sentence betrays a lack of analysis.)7 1672 3 970 3828 t 9 I f (y)3758 3828 w 9 S f (=)3843 3828 w 9 I f (b)3938 3828 w 6 R f (2)3993 3792 w 9 R f (\()4037 3828 w 9 I f (a)4074 3828 w 6 R f (2)4129 3792 w 9 S f (+)4203 3828 w 9 I f (b)4290 3828 w 6 R f (2)4345 3792 w 9 R f (\))4389 3828 w 6 S f (-)4429 3792 w 6 R f (1)4472 3792 w 6 I f (/)4507 3792 w 6 R f (2)4529 3792 w 9 R f (. The)1 224 1 4566 3828 t ( removed by squaring to get a polynomial discriminator function, as Wirth has just)13 3125(square root can be)3 695 2 970 3948 t (done to obtain)2 540 1 970 4068 t 9 I f (h)1545 4068 w 9 R f ( Unless)1 304( registers.)1 364( resulting fourth powers, however, threaten to overflow small)8 2311(. The)1 221 4 1590 4068 t ( which the paper)3 614(unusually wide arithmetic is at hand, it is well to seek a discriminator of lower degree,)15 3206 2 970 4188 t (proceeds to do in a novel way.)6 1101 1 970 4308 t 10 R f (Instead we compute a function)4 1225 1 720 4464 t 10 I f (g)1970 4464 w 10 R f (, similar to)2 431 1 2020 4464 t 10 I f (h)2476 4464 w 10 R f ( origin stems [sic] from the inequality)6 1510( Its)1 150(, incrementally.)1 624 3 2526 4464 t 10 I f (y)1220 4644 w 10 S f (-)1313 4644 w 10 R f (3)1417 4644 w 10 I f (/)1475 4644 w 10 R f (2)1511 4644 w 10 S f (<)1610 4644 w 10 I f (b)1714 4644 w 10 S f (\326)1796 4662 w 10 R f (\( 1)1 91 1 1859 4644 t 10 S f (-)1990 4644 w 10 R f (\( \()1 74 1 2085 4644 t 10 I f (x)2167 4644 w 10 S f (+)2251 4644 w 10 R f (1 \))1 91 1 2346 4644 t 10 I f (/ a)1 86 1 2445 4644 t 10 R f (\))2539 4644 w 7 R f (2)2577 4604 w 10 R f (\))2628 4644 w ( a)1 72( Therefore,)1 470( current raster point.)3 818(implying that the ordinate of the next point be at least 3/2 units below the)14 2960 4 720 4824 t (decrease of)1 452 1 720 4944 t 10 I f (y)1199 4944 w 10 R f (by 2 would be necessary for an increase of)8 1708 1 1270 4944 t 10 I f (x)3004 4944 w 10 R f ( similar development as for)4 1097( A)1 123(by 1 only.)2 405 3 3074 4944 t 10 I f (h)4725 4944 w 10 R f (yields)4801 4944 w (the function)1 480 1 720 5064 t 10 I f (g)1225 5064 w 10 R f (as)1300 5064 w 10 I f (g)1220 5244 w 10 S f (=)1319 5244 w 10 I f (b)1423 5244 w 7 R f (2)1484 5204 w 10 I f (x)1535 5244 w 7 R f (2)1590 5204 w 10 S f (+)1682 5244 w 10 R f (2)1786 5244 w 10 I f (b)1844 5244 w 7 R f (2)1905 5204 w 10 I f (x)1956 5244 w 10 S f (+)2049 5244 w 10 I f (a)2153 5244 w 7 R f (2)2214 5204 w 10 I f (y)2265 5244 w 7 R f (2)2320 5204 w 10 S f (-)2412 5244 w 10 R f (3)2516 5244 w 10 I f (a)2574 5244 w 7 R f (2)2635 5204 w 10 I f (y)2686 5244 w 10 S f (+)2779 5244 w 10 R f (9)2883 5244 w 10 I f (a)2941 5244 w 7 R f (2)3002 5204 w 10 I f (/)3053 5244 w 10 R f (4)3089 5244 w 10 S f (-)3188 5244 w 10 I f (a)3292 5244 w 7 R f (2)3353 5204 w 10 I f (b)3404 5244 w 7 R f (2)3465 5204 w 10 S f (+)3557 5244 w 10 I f (b)3661 5244 w 7 R f (2)3722 5204 w 10 R f (and)720 5424 w 10 I f (x)889 5424 w 10 R f (can be incremented as long as)5 1194 1 958 5424 t 10 I f (g)2177 5424 w 10 S f (<)2276 5424 w 10 R f (0.)2380 5424 w 9 R f ( satisfaction, of the inequality would imply the unde-)8 1920( not)1 138( Violation,)1 409(Beware, the explanation is backward.)4 1353 4 970 5580 t ( if the ellipse is sufficiently narrow,)6 1322( Furthermore,)1 525(sired outcome.)1 537 3 970 5700 t 9 I f (y)3383 5700 w 9 R f (can decrease by any integer amount,)5 1338 1 3452 5700 t ( what if)2 282( significantly,)1 498( More)1 243(not just 1 or 2.)4 533 4 970 5820 t 9 I f (y)2552 5820 w 9 R f ( happens when)2 542( This)1 209(should never decrease by more than 1?)6 1421 3 2618 5820 t 9 I f (a)970 5940 w 9 S f (=)1060 5940 w 9 I f (b)1155 5940 w 9 S f (=)1245 5940 w 9 R f ( this case the)3 464(1. In)1 190 2 1310 5940 t 9 I f (g)1987 5940 w 9 R f (test turns out to work by luck of a compensating error: the derivation of)13 2584 1 2055 5940 t 9 I f (g)4662 5940 w 9 R f (is)4730 5940 w (flawed by the same inattention to range restrictions as was the derivation of)12 2726 1 970 6060 t 9 I f (h)3719 6060 w 9 R f (.)3764 6060 w 10 R f ( starting at the point)4 804(The first quadrant of the ellipse is then completed by the same process,)12 2837 2 720 6216 t 10 I f (P)4387 6216 w 10 R f (\()4456 6216 w 10 I f (a)4497 6216 w 10 R f ( of [sic])2 312( \),)1 66(, 0)1 107 3 4555 6216 t (incrementing)720 6336 w 10 I f (y)1273 6336 w 10 R f (and conditionally decrementing)2 1267 1 1343 6336 t 10 I f (x)2636 6336 w 10 R f ( auxiliary function here is obtained from the previous)8 2130(. The)1 230 2 2680 6336 t (case of)1 279 1 720 6456 t 10 I f (h)1024 6456 w 10 R f (by systematically substituting)2 1190 1 1099 6456 t 10 I f (x)2314 6456 w 10 R f (,)2366 6456 w 10 I f (y)2399 6456 w 10 R f (,)2451 6456 w 10 I f (a)2484 6456 w 10 R f (,)2542 6456 w 10 I f (b)2575 6456 w 10 R f (for)2650 6456 w 10 I f (y)2791 6456 w 10 R f (,)2843 6456 w 10 I f (x)2876 6456 w 10 R f (,)2928 6456 w 10 I f (b)2961 6456 w 10 R f (,)3019 6456 w 10 I f (a)3052 6456 w 10 R f (.)3102 6456 w 9 R f ( termination the same way,)4 995( one understands ``the same process'' to test for)8 1761( If)1 110(The wording is imprecise.)3 954 4 970 6612 t ( the asymmetry)2 572( Here)1 229( skinny ellipse.)2 560(then it will not necessarily work for drawing the long branch of a)12 2459 4 970 6732 t (imposed by the unexplained precondition)4 1492 1 970 6852 t 9 I f (a)2485 6852 w 9 S f (\243)2567 6852 w 9 I f (b)2654 6852 w 9 R f (comes into play.)2 594 1 2722 6852 t 10 R f (The derivation of the incrementing values for)6 1885 1 720 7008 t 10 I f (h)2642 7008 w 10 R f (and)2730 7008 w 10 I f (g)2912 7008 w 10 R f (follow [sic] from the application of the axiom of)8 2040 1 3000 7008 t (assignment: on incrementing)2 1155 1 720 7128 t 10 I f (x)1900 7128 w 10 R f (the incrementation of)2 854 1 1969 7128 t 10 I f (h)2848 7128 w 10 R f (is obtained from)2 655 1 2923 7128 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 63 514 764 %%EndPage: 26 26 %%Page: 27 27 %%PageBoundingBox: (atend) /saveobj save def mark 27 pagesetup 10 R f (- 27 -)2 216 1 2772 480 t ({)1080 900 w 10 I f (h)1128 900 w 10 S f (=)1227 900 w 10 I f (b)1331 900 w 7 R f (2)1392 860 w 10 I f (x)1443 900 w 7 R f (2)1498 860 w 10 S f (+)1590 900 w 10 R f (2)1694 900 w 10 I f (b)1752 900 w 7 R f (2)1813 860 w 10 I f (x)1864 900 w 10 S f (+)1957 900 w 10 I f (k)2061 900 w 10 R f (})2105 900 w 10 I f (h)1080 1020 w 10 R f (:)1171 1020 w 10 S f (=)1215 1020 w 10 I f (h)1319 1020 w 10 S f (+)1418 1020 w 10 I f (b)1522 1020 w 7 R f (2)1583 980 w 10 R f (\( 2)1 91 1 1634 1020 t 10 I f (x)1733 1020 w 10 S f (+)1826 1020 w 10 R f (3 \))1 91 1 1930 1020 t ({)1080 1140 w 10 I f (h)1128 1140 w 10 S f (=)1227 1140 w 10 I f (b)1331 1140 w 7 R f (2)1392 1100 w 10 I f (x)1443 1140 w 7 R f (2)1498 1100 w 10 S f (+)1590 1140 w 10 R f (2)1694 1140 w 10 I f (b)1752 1140 w 7 R f (2)1813 1100 w 10 I f (x)1864 1140 w 10 S f (+)1957 1140 w 10 I f (b)2061 1140 w 7 R f (2)2122 1100 w 10 S f (+)2214 1140 w 10 R f (2)2318 1140 w 10 I f (b)2376 1140 w 7 R f (2)2437 1100 w 10 I f (x)2488 1140 w 10 S f (+)2581 1140 w 10 R f (2)2685 1140 w 10 I f (b)2743 1140 w 7 R f (2)2804 1100 w 10 S f (+)2896 1140 w 10 I f (k)3000 1140 w 10 R f (})3044 1140 w 10 I f (x)1080 1260 w 10 R f (:)1165 1260 w 10 S f (=)1209 1260 w 10 I f (x)1313 1260 w 10 S f (+)1406 1260 w 10 R f (1)1510 1260 w ({)1080 1380 w 10 I f (h)1128 1380 w 10 S f (=)1227 1380 w 10 I f (b)1331 1380 w 7 R f (2)1392 1340 w 10 I f (x)1443 1380 w 7 R f (2)1498 1340 w 10 S f (+)1590 1380 w 10 R f (2)1694 1380 w 10 I f (b)1752 1380 w 7 R f (2)1813 1340 w 10 I f (x)1864 1380 w 10 S f (+)1957 1380 w 10 I f (k)2061 1380 w 10 R f (})2105 1380 w (on incrementing)1 652 1 720 1560 t 10 I f (y)1397 1560 w 10 R f (, the incrementation of)3 904 1 1441 1560 t 10 I f (h)2370 1560 w 10 R f (is obtained from)2 655 1 2445 1560 t ({)1080 1740 w 10 I f (h)1128 1740 w 10 S f (=)1227 1740 w 10 I f (a)1331 1740 w 7 R f (2)1392 1700 w 10 I f (y)1443 1740 w 7 R f (2)1498 1700 w 10 S f (-)1590 1740 w 10 I f (a)1694 1740 w 7 R f (2)1755 1700 w 10 I f (y)1806 1740 w 10 S f (+)1899 1740 w 10 I f (k)2003 1740 w 10 R f (})2047 1740 w 10 I f (h)1080 1860 w 10 R f (:)1171 1860 w 10 S f (=)1215 1860 w 10 I f (h)1319 1860 w 10 S f (-)1418 1860 w 10 R f (2)1489 1860 w 10 I f (a)1547 1860 w 7 R f (2)1608 1820 w 10 R f (\()1659 1860 w 10 I f (y)1700 1860 w 10 S f (-)1784 1860 w 10 R f (1 \))1 91 1 1879 1860 t ({)1080 1980 w 10 I f (h)1128 1980 w 10 S f (=)1227 1980 w 10 I f (a)1331 1980 w 7 R f (2)1392 1940 w 10 I f (y)1443 1980 w 7 R f (2)1498 1940 w 10 S f (-)1590 1980 w 10 R f (2)1694 1980 w 10 I f (a)1752 1980 w 7 R f (2)1813 1940 w 10 I f (y)1864 1980 w 10 S f (+)1957 1980 w 10 I f (a)2061 1980 w 7 R f (2)2122 1940 w 10 S f (-)2214 1980 w 10 R f (\()2318 1980 w 10 I f (a)2359 1980 w 7 R f (2)2420 1940 w 10 I f (y)2471 1980 w 10 S f (-)2564 1980 w 10 I f (a)2668 1980 w 7 R f (2)2729 1940 w 10 R f (\))2780 1980 w 10 S f (+)2870 1980 w 10 I f (k)2974 1980 w 10 R f (})3018 1980 w 10 I f (y)1080 2100 w 10 R f (:)1165 2100 w 10 S f (=)1209 2100 w 10 I f (y)1313 2100 w 10 S f (-)1406 2100 w 10 R f (1)1510 2100 w ({)1080 2220 w 10 I f (h)1128 2220 w 10 S f (=)1227 2220 w 10 I f (a)1331 2220 w 7 R f (2)1392 2180 w 10 I f (y)1443 2220 w 7 R f (2)1498 2180 w 10 S f (+)1590 2220 w 10 I f (a)1694 2220 w 7 R f (2)1755 2180 w 10 I f (y)1806 2220 w 10 S f (+)1899 2220 w 10 I f (k)2003 2220 w 10 R f (})2047 2220 w (and the incrementation of)3 1023 1 720 2400 t 10 I f (g)1768 2400 w 10 R f (is obtained from)2 655 1 1843 2400 t ({)1080 2580 w 10 I f (g)1128 2580 w 10 S f (=)1227 2580 w 10 I f (a)1331 2580 w 7 R f (2)1392 2540 w 10 I f (y)1443 2580 w 7 R f (2)1498 2540 w 10 S f (-)1590 2580 w 10 R f (3)1694 2580 w 10 I f (a)1752 2580 w 7 R f (2)1813 2540 w 10 I f (y)1864 2580 w 10 S f (+)1957 2580 w 10 I f (k)2061 2580 w 10 R f (})2105 2580 w 10 I f (g)1080 2700 w 10 R f (:)1171 2700 w 10 S f (=)1215 2700 w 10 I f (g)1319 2700 w 10 S f (-)1418 2700 w 10 R f (2)1522 2700 w 10 I f (a)1580 2700 w 7 R f (2)1641 2660 w 10 R f (\()1692 2700 w 10 I f (y)1733 2700 w 10 S f (-)1817 2700 w 10 R f (2 \))1 91 1 1912 2700 t ({)1080 2820 w 10 I f (g)1128 2820 w 10 S f (=)1227 2820 w 10 I f (a)1331 2820 w 7 R f (2)1392 2780 w 10 I f (y)1443 2820 w 7 R f (2)1498 2780 w 10 S f (-)1590 2820 w 10 R f (2)1694 2820 w 10 I f (a)1752 2820 w 7 R f (2)1813 2780 w 10 I f (y)1864 2820 w 10 S f (+)1957 2820 w 10 I f (a)2061 2820 w 7 R f (2)2122 2780 w 10 S f (-)2214 2820 w 10 R f (3 \()1 91 1 2318 2820 t 10 I f (a)2417 2820 w 7 R f (2)2478 2780 w 10 I f (y)2529 2820 w 10 S f (-)2622 2820 w 10 I f (a)2726 2820 w 7 R f (2)2787 2780 w 10 R f (\))2838 2820 w 10 S f (+)2928 2820 w 10 I f (k)3032 2820 w 10 R f (})3076 2820 w 10 I f (y)1080 2940 w 10 R f (:)1165 2940 w 10 S f (=)1209 2940 w 10 I f (y)1313 2940 w 10 S f (-)1406 2940 w 10 R f (1)1510 2940 w ({)1080 3060 w 10 I f (g)1128 3060 w 10 S f (=)1227 3060 w 10 I f (a)1331 3060 w 7 R f (2)1392 3020 w 10 I f (y)1443 3060 w 7 R f (2)1498 3020 w 10 S f (-)1590 3060 w 10 R f (3)1694 3060 w 10 I f (a)1752 3060 w 7 R f (2)1813 3020 w 10 I f (y)1864 3060 w 10 S f (+)1957 3060 w 10 I f (k)2061 3060 w 10 R f (})2105 3060 w 9 R f ( derivation,)1 412(In each stretch of the preceding)5 1135 2 970 3276 t 9 I f (k)2541 3276 w 9 R f (represents nonchanging terms, as was explained in the omit-)8 2185 1 2605 3276 t ( simply says that the)4 773( It)1 110( methodology.)1 530( this formalism, however, is misplaced)5 1446( All)1 171(ted part of the paper.)4 790 6 970 3396 t (update step is)2 491 1 970 3516 t 9 I f (x)1170 3696 w 9 R f (,)1217 3696 w 9 I f (y)1277 3696 w 9 R f (,)1324 3696 w 9 I f (h)1384 3696 w 9 R f (,)1436 3696 w 9 I f (g)1496 3696 w 9 R f (:)1578 3696 w 9 S f (=)1618 3696 w 9 I f (x)1713 3696 w 9 S f (+)1790 3696 w 9 R f (1 ,)1 75 1 1877 3696 t 9 I f (y)1989 3696 w 9 S f (+ D)1 142 1 2096 3696 t 9 I f (y)2245 3696 w 9 R f (,)2292 3696 w 9 I f (h)2352 3696 w 9 S f (+ D)1 142 1 2434 3696 t 9 I f (h)2583 3696 w 9 R f (,)2635 3696 w 9 I f (g)2695 3696 w 9 S f (+ D)1 142 1 2777 3696 t 9 I f (g)2926 3696 w 9 R f (where)970 3876 w 9 S f (D)1222 3876 w 9 I f (h)1284 3876 w 9 S f (=)1374 3876 w 9 I f (h)1469 3876 w 9 R f (\()1521 3876 w 9 I f (x)1558 3876 w 9 S f (+)1620 3876 w 9 R f (1 ,)1 75 1 1685 3876 t 9 I f (y)1789 3876 w 9 S f (+ D)1 120 1 1851 3876 t 9 I f (y)1978 3876 w 9 R f (\))2025 3876 w 9 S f (-)2108 3876 w 9 I f (h)2203 3876 w 9 R f (\()2255 3876 w 9 I f (x)2292 3876 w 9 R f (,)2339 3876 w 9 I f (y)2369 3876 w 9 R f (\) and)1 192 1 2416 3876 t 9 S f (D)2640 3876 w 9 I f (g)2702 3876 w 9 R f ( the development is con-)4 927( of)1 107( Most)1 240(is defined similarly.)2 737 4 2779 3876 t ( inflict them on an)4 678( necessity in Pascal is no reason to)7 1281( Their)1 246(cerned with inconsistent intermediate states.)4 1615 4 970 3996 t (exposition of an algorithmic idea.)4 1215 1 970 4116 t 9 S f (\347)2137 4272 w 9 R f (Use formalism for function, not fashion.)5 1456 1 2152 4272 t 9 S f ( _________________________________)1 1441( _)1 -1441( `````````````````````````````````)1 1441(\347 `)1 -1441 4 3623 4272 t 10 R f (This completes the design considerations for the following algorithm.)8 2784 1 720 4428 t 10 I f (x)1080 4608 w 10 R f (:)1165 4608 w 10 S f (=)1209 4608 w 10 R f (0 ;)1 86 1 1313 4608 t 10 I f (y)1440 4608 w 10 R f (:)1525 4608 w 10 S f (=)1569 4608 w 10 R f (0 ;)1 86 1 1673 4608 t 10 I f (h)1080 4728 w 10 R f (:)1171 4728 w 10 S f (=)1215 4728 w 10 R f (\()1319 4728 w 10 I f (a)1360 4728 w 7 R f (2)1421 4688 w 10 I f (DIV)1496 4728 w 10 R f (4 \))1 91 1 1694 4728 t 10 S f (-)1842 4728 w 10 I f (ba)1946 4728 w 7 R f (2)2057 4688 w 10 S f (+)2149 4728 w 10 I f (b)2253 4728 w 7 R f (2)2314 4688 w 10 R f (;)2365 4728 w 10 I f (g)2434 4728 w 10 R f (:)2525 4728 w 10 S f (=)2569 4728 w 10 R f (\( 9)1 91 1 2673 4728 t 10 I f (/)2772 4728 w 10 R f (4 \))1 91 1 2808 4728 t 10 I f (a)2915 4728 w 7 R f (2)2976 4688 w 10 S f (-)3068 4728 w 10 R f (3)3172 4728 w 10 I f (ba)3230 4728 w 7 R f (2)3341 4688 w 10 S f (+)3433 4728 w 10 I f (b)3537 4728 w 7 R f (2 ;)1 60 1 3598 4688 t 10 B f (do)1080 4848 w 10 I f (g)1227 4848 w 10 S f (<)1326 4848 w 10 R f (0)1430 4848 w 10 S f (\256)1554 4848 w 10 I f (Mark)1727 4848 w 10 R f (\()1951 4848 w 10 I f (x)1992 4848 w 10 R f (,)2044 4848 w 10 I f (y)2077 4848 w 10 R f (\) ;)1 77 1 2129 4848 t 10 B f (if)1330 4968 w 10 I f (h)1432 4968 w 10 S f (<)1531 4968 w 10 R f (0)1635 4968 w 10 S f (\256)1759 4968 w 10 I f (d)1932 4968 w 10 R f (:)2023 4968 w 10 S f (=)2067 4968 w 10 R f (\( 2)1 91 1 2171 4968 t 10 I f (x)2270 4968 w 10 S f (+)2354 4968 w 10 R f (3 \))1 91 1 2425 4968 t 10 I f (b)2532 4968 w 7 R f (2)2593 4928 w 10 R f (;)2644 4968 w 10 I f (g)2713 4968 w 10 R f (:)2804 4968 w 10 S f (=)2848 4968 w 10 I f (g)2952 4968 w 10 S f (+)3051 4968 w 10 I f (d)3155 4968 w 10 R f ([ ])1 82 1 1330 5088 t 10 I f (h)1461 5088 w 10 S f (\263)1552 5088 w 10 R f (0)1648 5088 w 10 S f (\256)1772 5088 w 10 I f (d)1945 5088 w 10 R f (:)2036 5088 w 10 S f (=)2080 5088 w 10 R f (\( 2)1 91 1 2184 5088 t 10 I f (x)2283 5088 w 10 S f (+)2367 5088 w 10 R f (3 \))1 91 1 2462 5088 t 10 I f (b)2569 5088 w 7 R f (2)2630 5048 w 10 S f (-)2722 5088 w 10 R f (2 \()1 91 1 2826 5088 t 10 I f (y)2925 5088 w 10 S f (-)3009 5088 w 10 R f (1 \))1 91 1 3104 5088 t 10 I f (a)3211 5088 w 7 R f (2)3272 5048 w 10 R f (;)3323 5088 w 10 I f (g)1920 5208 w 10 R f (:)2011 5208 w 10 S f (=)2055 5208 w 10 I f (g)2159 5208 w 10 S f (+)2258 5208 w 10 I f (d)2362 5208 w 10 S f (+)2461 5208 w 10 R f (2)2565 5208 w 10 I f (a)2623 5208 w 7 R f (2)2684 5168 w 10 R f (;)2735 5208 w 10 I f (y)1920 5328 w 10 R f (:)2005 5328 w 10 S f (=)2049 5328 w 10 I f (y)2153 5328 w 10 S f (-)2246 5328 w 10 R f (1)2350 5328 w 10 I f (fi)1330 5448 w 10 R f (;)1394 5448 w 10 I f (h)1330 5568 w 10 R f (:)1421 5568 w 10 S f (=)1465 5568 w 10 I f (h)1569 5568 w 10 S f (+)1668 5568 w 10 I f (d)1772 5568 w 10 R f (;)1830 5568 w 10 I f (x)1899 5568 w 10 R f (:)1984 5568 w 10 S f (=)2028 5568 w 10 I f (x)2132 5568 w 10 S f (+)2225 5568 w 10 R f (1)2329 5568 w 10 B f (od)1080 5688 w 10 R f (;)1194 5688 w 10 I f (x)1080 5808 w 10 R f (:)1165 5808 w 10 S f (=)1209 5808 w 10 I f (a)1313 5808 w 10 R f (;)1371 5808 w 10 I f (y)1440 5808 w 10 R f (1 :)1 119 1 1492 5808 t 10 S f (=)1627 5808 w 10 I f (y)1731 5808 w 10 R f (;)1783 5808 w 10 I f (y)1852 5808 w 10 R f (:)1937 5808 w 10 S f (=)1981 5808 w 10 R f (0 ;)1 86 1 2085 5808 t 10 I f (h)1080 5928 w 10 R f (:)1171 5928 w 10 S f (=)1215 5928 w 10 R f (\()1319 5928 w 10 I f (b)1360 5928 w 7 R f (2)1421 5888 w 10 I f (DIV)1496 5928 w 10 R f (4 \))1 91 1 1694 5928 t 10 S f (-)1842 5928 w 10 I f (ab)1946 5928 w 7 R f (2)2057 5888 w 10 S f (+)2149 5928 w 10 R f (2)2253 5928 w 10 I f (a)2311 5928 w 7 R f (2)2372 5888 w 10 R f (;)2423 5928 w 10 B f (do)1080 6048 w 10 I f (y)1227 6048 w 10 S f (\243)1312 6048 w 10 I f (y)1408 6048 w 10 R f (1)1460 6048 w 10 S f (\256)1584 6048 w 10 I f (Mark)1724 6048 w 10 R f (\()1948 6048 w 10 I f (x)1989 6048 w 10 R f (,)2041 6048 w 10 I f (y)2074 6048 w 10 R f (\) ;)1 77 1 2126 6048 t 10 B f (if)1330 6168 w 10 I f (h)1432 6168 w 10 S f (<)1531 6168 w 10 R f (0)1635 6168 w 10 S f (\256)1759 6168 w 10 I f (h)1932 6168 w 10 R f (:)2023 6168 w 10 S f (=)2067 6168 w 10 I f (h)2171 6168 w 10 S f (+)2270 6168 w 10 R f (\( 2)1 91 1 2374 6168 t 10 I f (y)2473 6168 w 10 S f (+)2557 6168 w 10 R f (3 \))1 91 1 2652 6168 t 10 I f (a)2759 6168 w 7 R f (2)2820 6128 w 10 R f ([ ])1 82 1 1330 6288 t 10 I f (h)1461 6288 w 10 S f (\263)1552 6288 w 10 R f (0)1648 6288 w 10 S f (\256)1772 6288 w 10 I f (h)1945 6288 w 10 R f (:)2036 6288 w 10 S f (=)2080 6288 w 10 I f (h)2184 6288 w 10 S f (+)2283 6288 w 10 R f (\( 2)1 91 1 2387 6288 t 10 I f (y)2486 6288 w 10 S f (+)2570 6288 w 10 R f (3 \))1 91 1 2665 6288 t 10 I f (a)2772 6288 w 7 R f (2)2833 6248 w 10 S f (-)2925 6288 w 10 R f (2 \()1 91 1 3029 6288 t 10 I f (x)3128 6288 w 10 S f (-)3212 6288 w 10 R f (1 \))1 91 1 3307 6288 t 10 I f (b)3414 6288 w 7 R f (2)3475 6248 w 10 R f (;)3526 6288 w 10 I f (x)3595 6288 w 10 R f (:)3680 6288 w 10 S f (=)3724 6288 w 10 I f (x)3828 6288 w 10 S f (-)3921 6288 w 10 R f (1)4025 6288 w 10 I f (fi)1330 6408 w 10 R f (;)1394 6408 w 10 I f (y)1330 6528 w 10 R f (:)1415 6528 w 10 S f (=)1459 6528 w 10 I f (y)1563 6528 w 10 S f (+)1656 6528 w 10 R f (1)1760 6528 w 10 B f (od)1080 6648 w 9 R f ( The)1 193( the inconsistent division operators in the second line.)8 1993(The reader is left to puzzle out)6 1139 3 970 6864 t 9 I f (h)4325 6864 w 9 R f (in the pro-)2 390 1 4400 6864 t (gram is not the same as the)6 984 1 970 6984 t 9 I f (h)1978 6984 w 9 R f ( the omit-)2 351( As)1 146( the nearest integer.)3 707( is rounded down to)4 721( It)1 102(in the development.)2 716 6 2047 6984 t ( Simi-)1 247( does not change the outcome of any test in the algorithm.)11 2102(ted part of the paper explained, rounding)6 1471 3 970 7104 t (larly,)970 7224 w 9 I f (g)1181 7224 w 9 R f ( 9)1 52(may be rounded down and the first term of its initializer may be replaced by \()15 2800 2 1249 7224 t 9 I f (a)4108 7224 w 6 R f (2)4163 7188 w 9 R f (\))4207 7224 w 9 I f (DIV)4282 7224 w 9 R f (4.)4469 7224 w cleartomark showpage saveobj restore %%PageBoundingBox: 61 54 489 764 %%EndPage: 27 27 %%Page: 28 28 %%PageBoundingBox: (atend) /saveobj save def mark 28 pagesetup 10 R f (- 28 -)2 216 1 2772 480 t 9 R f (The second initialization of)3 989 1 970 840 t 9 I f (h)1982 840 w 9 R f (should be the same as the first with)7 1271 1 2050 840 t 9 I f (a)3344 840 w 9 R f (and)3412 840 w 9 I f (b)3565 840 w 9 R f (interchanged.)3633 840 w (The second loop is fatally flawed, because the unstated side condition for the validity of the)15 3338 1 970 996 t 9 I f (h)4333 996 w 9 R f (test can be)2 387 1 4403 996 t ( condition,)1 396(violated. That)1 534 2 970 1116 t 9 I f (x)1933 1116 w 9 S f (\263)2010 1116 w 9 R f (1)2097 1116 w 9 I f (/)2149 1116 w 9 R f ( is so narrow as to be rendered with)8 1356(2, is violated whenever an ellipse)5 1253 2 2181 1116 t ( the program)2 475( Apparently)1 458( the accompanying figure for the result.)6 1468( See)1 183( wide at either end.)4 718(tails one pixel)2 518 6 970 1236 t (was never tested against such obviously stressful cases.)7 1999 1 970 1356 t ( not necessarily coincide with the last point)7 1589(A subtler trouble is that the endpoint of the second loop does)11 2231 2 970 1512 t ( with)1 188( example,)1 357( For)1 177(calculated \(but not plotted\) in the first loop.)7 1616 4 970 1632 t 9 I f (a)3336 1632 w 9 S f (=)3426 1632 w 9 R f (2 and)1 203 1 3521 1632 t 9 I f (b)3752 1632 w 9 S f (=)3842 1632 w 9 R f (3, the first loop ends at)5 853 1 3937 1632 t ( infer that it was simply assumed that the two endpoints)10 2110( I)1 86( 0 , 3 \).)4 194( at \()2 161( while the second loop ends)5 1045(\( 1 , 3 \),)4 224 6 970 1752 t ( had been recognized, there should have been some analy-)9 2118( the possibility of mismatch)4 1014( If)1 109(would coincide.)1 579 4 970 1872 t (sis of how bad it can be.)6 876 1 970 1992 t 10 R f 1584 2812 50 50 De 1592 2812 33 33 De 1600 2812 16 16 De 1584 3151 50 50 De 1592 3151 33 33 De 1600 3151 16 16 De 4125 2812 50 50 De 4133 2812 33 33 De 4142 2812 16 16 De 4125 3151 50 50 De 4133 3151 33 33 De 4142 3151 16 16 De 1668 2897 50 50 De 1677 2897 33 33 De 1685 2897 16 16 De 1668 3067 50 50 De 1677 3067 33 33 De 1685 3067 16 16 De 4040 2897 50 50 De 4048 2897 33 33 De 4057 2897 16 16 De 4040 3067 50 50 De 4048 3067 33 33 De 4057 3067 16 16 De 1753 2982 50 50 De 1761 2982 33 33 De 1770 2982 16 16 De 1753 2982 50 50 De 1761 2982 33 33 De 1770 2982 16 16 De 3955 2982 50 50 De 3964 2982 33 33 De 3972 2982 16 16 De 3955 2982 50 50 De 3964 2982 33 33 De 3972 2982 16 16 De 1838 3067 50 50 De 1846 3067 33 33 De 1855 3067 16 16 De 1838 2897 50 50 De 1846 2897 33 33 De 1855 2897 16 16 De 3871 3067 50 50 De 3879 3067 33 33 De 3888 3067 16 16 De 3871 2897 50 50 De 3879 2897 33 33 De 3888 2897 16 16 De 1922 3067 50 50 De 1931 3067 33 33 De 1939 3067 16 16 De 1922 2897 50 50 De 1931 2897 33 33 De 1939 2897 16 16 De 3786 3067 50 50 De 3794 3067 33 33 De 3803 3067 16 16 De 3786 2897 50 50 De 3794 2897 33 33 De 3803 2897 16 16 De 2007 3067 50 50 De 2016 3067 33 33 De 2024 3067 16 16 De 2007 2897 50 50 De 2016 2897 33 33 De 2024 2897 16 16 De 3701 3067 50 50 De 3710 3067 33 33 De 3718 3067 16 16 De 3701 2897 50 50 De 3710 2897 33 33 De 3718 2897 16 16 De 2092 3067 50 50 De 2100 3067 33 33 De 2109 3067 16 16 De 2092 2897 50 50 De 2100 2897 33 33 De 2109 2897 16 16 De 3616 3067 50 50 De 3625 3067 33 33 De 3633 3067 16 16 De 3616 2897 50 50 De 3625 2897 33 33 De 3633 2897 16 16 De 2176 3067 50 50 De 2185 3067 33 33 De 2193 3067 16 16 De 2176 2897 50 50 De 2185 2897 33 33 De 2193 2897 16 16 De 3532 3067 50 50 De 3540 3067 33 33 De 3549 3067 16 16 De 3532 2897 50 50 De 3540 2897 33 33 De 3549 2897 16 16 De 2261 3067 50 50 De 2270 3067 33 33 De 2278 3067 16 16 De 2261 2897 50 50 De 2270 2897 33 33 De 2278 2897 16 16 De 3447 3067 50 50 De 3456 3067 33 33 De 3464 3067 16 16 De 3447 2897 50 50 De 3456 2897 33 33 De 3464 2897 16 16 De 2346 3067 50 50 De 2354 3067 33 33 De 2363 3067 16 16 De 2346 2897 50 50 De 2354 2897 33 33 De 2363 2897 16 16 De 3362 3067 50 50 De 3371 3067 33 33 De 3379 3067 16 16 De 3362 2897 50 50 De 3371 2897 33 33 De 3379 2897 16 16 De 2431 3067 50 50 De 2439 3067 33 33 De 2448 3067 16 16 De 2431 2897 50 50 De 2439 2897 33 33 De 2448 2897 16 16 De 3278 3067 50 50 De 3286 3067 33 33 De 3295 3067 16 16 De 3278 2897 50 50 De 3286 2897 33 33 De 3295 2897 16 16 De 2515 3067 50 50 De 2524 3067 33 33 De 2532 3067 16 16 De 2515 2897 50 50 De 2524 2897 33 33 De 2532 2897 16 16 De 3193 3067 50 50 De 3201 3067 33 33 De 3210 3067 16 16 De 3193 2897 50 50 De 3201 2897 33 33 De 3210 2897 16 16 De 2600 3067 50 50 De 2608 3067 33 33 De 2617 3067 16 16 De 2600 2897 50 50 De 2608 2897 33 33 De 2617 2897 16 16 De 3108 3067 50 50 De 3117 3067 33 33 De 3125 3067 16 16 De 3108 2897 50 50 De 3117 2897 33 33 De 3125 2897 16 16 De 2685 3067 50 50 De 2693 3067 33 33 De 2702 3067 16 16 De 2685 2897 50 50 De 2693 2897 33 33 De 2702 2897 16 16 De 3024 3067 50 50 De 3032 3067 33 33 De 3040 3067 16 16 De 3024 2897 50 50 De 3032 2897 33 33 De 3040 2897 16 16 De 2769 3067 50 50 De 2778 3067 33 33 De 2786 3067 16 16 De 2769 2897 50 50 De 2778 2897 33 33 De 2786 2897 16 16 De 2939 3067 50 50 De 2947 3067 33 33 De 2956 3067 16 16 De 2939 2897 50 50 De 2947 2897 33 33 De 2956 2897 16 16 De 2854 3067 50 50 De 2863 3067 33 33 De 2871 3067 16 16 De 2854 2897 50 50 De 2863 2897 33 33 De 2871 2897 16 16 De 2854 3067 50 50 De 2863 3067 33 33 De 2871 3067 16 16 De 2854 2897 50 50 De 2863 2897 33 33 De 2871 2897 16 16 De 1668 2474 50 50 De 1677 2474 33 33 De 1685 2474 16 16 De 4040 2474 50 50 De 4048 2474 33 33 De 4057 2474 16 16 De 1753 2474 50 50 De 1761 2474 33 33 De 1770 2474 16 16 De 3955 2474 50 50 De 3964 2474 33 33 De 3972 2474 16 16 De 1838 2389 50 50 De 1846 2389 33 33 De 1855 2389 16 16 De 3871 2558 50 50 De 3879 2558 33 33 De 3888 2558 16 16 De 1838 2558 50 50 De 1846 2558 33 33 De 1855 2558 16 16 De 3871 2389 50 50 De 3879 2389 33 33 De 3888 2389 16 16 De 1922 2389 50 50 De 1931 2389 33 33 De 1939 2389 16 16 De 3786 2558 50 50 De 3794 2558 33 33 De 3803 2558 16 16 De 1922 2558 50 50 De 1931 2558 33 33 De 1939 2558 16 16 De 3786 2389 50 50 De 3794 2389 33 33 De 3803 2389 16 16 De 2007 2389 50 50 De 2016 2389 33 33 De 2024 2389 16 16 De 3701 2558 50 50 De 3710 2558 33 33 De 3718 2558 16 16 De 2007 2558 50 50 De 2016 2558 33 33 De 2024 2558 16 16 De 3701 2389 50 50 De 3710 2389 33 33 De 3718 2389 16 16 De 2092 2389 50 50 De 2100 2389 33 33 De 2109 2389 16 16 De 3616 2558 50 50 De 3625 2558 33 33 De 3633 2558 16 16 De 2092 2558 50 50 De 2100 2558 33 33 De 2109 2558 16 16 De 3616 2389 50 50 De 3625 2389 33 33 De 3633 2389 16 16 De 2176 2389 50 50 De 2185 2389 33 33 De 2193 2389 16 16 De 3532 2558 50 50 De 3540 2558 33 33 De 3549 2558 16 16 De 2176 2558 50 50 De 2185 2558 33 33 De 2193 2558 16 16 De 3532 2389 50 50 De 3540 2389 33 33 De 3549 2389 16 16 De 2261 2389 50 50 De 2270 2389 33 33 De 2278 2389 16 16 De 3447 2558 50 50 De 3456 2558 33 33 De 3464 2558 16 16 De 2261 2558 50 50 De 2270 2558 33 33 De 2278 2558 16 16 De 3447 2389 50 50 De 3456 2389 33 33 De 3464 2389 16 16 De 2346 2389 50 50 De 2354 2389 33 33 De 2363 2389 16 16 De 3362 2558 50 50 De 3371 2558 33 33 De 3379 2558 16 16 De 2346 2558 50 50 De 2354 2558 33 33 De 2363 2558 16 16 De 3362 2389 50 50 De 3371 2389 33 33 De 3379 2389 16 16 De 2431 2389 50 50 De 2439 2389 33 33 De 2448 2389 16 16 De 3278 2558 50 50 De 3286 2558 33 33 De 3295 2558 16 16 De 2431 2558 50 50 De 2439 2558 33 33 De 2448 2558 16 16 De 3278 2389 50 50 De 3286 2389 33 33 De 3295 2389 16 16 De 2515 2389 50 50 De 2524 2389 33 33 De 2532 2389 16 16 De 3193 2558 50 50 De 3201 2558 33 33 De 3210 2558 16 16 De 2515 2558 50 50 De 2524 2558 33 33 De 2532 2558 16 16 De 3193 2389 50 50 De 3201 2389 33 33 De 3210 2389 16 16 De 2600 2389 50 50 De 2608 2389 33 33 De 2617 2389 16 16 De 3108 2558 50 50 De 3117 2558 33 33 De 3125 2558 16 16 De 2600 2558 50 50 De 2608 2558 33 33 De 2617 2558 16 16 De 3108 2389 50 50 De 3117 2389 33 33 De 3125 2389 16 16 De 2685 2389 50 50 De 2693 2389 33 33 De 2702 2389 16 16 De 3024 2558 50 50 De 3032 2558 33 33 De 3040 2558 16 16 De 2685 2558 50 50 De 2693 2558 33 33 De 2702 2558 16 16 De 3024 2389 50 50 De 3032 2389 33 33 De 3040 2389 16 16 De 2769 2389 50 50 De 2778 2389 33 33 De 2786 2389 16 16 De 2939 2558 50 50 De 2947 2558 33 33 De 2956 2558 16 16 De 2769 2558 50 50 De 2778 2558 33 33 De 2786 2558 16 16 De 2939 2389 50 50 De 2947 2389 33 33 De 2956 2389 16 16 De 2854 2558 50 50 De 2863 2558 33 33 De 2871 2558 16 16 De 2854 2389 50 50 De 2863 2389 33 33 De 2871 2389 16 16 De 1584 2474 50 50 De 1592 2474 33 33 De 1600 2474 16 16 De 4125 2474 50 50 De 4133 2474 33 33 De 4142 2474 16 16 De 10 S1 f (_ _________________________________________________________)1 2880 1 1440 2220 t (_ _________________________________________________________)1 2880 1 1440 2304 t (_ _________________________________________________________)1 2880 1 1440 2389 t (_ _________________________________________________________)1 2880 1 1440 2474 t (_ _________________________________________________________)1 2880 1 1440 2558 t (_ _________________________________________________________)1 2880 1 1440 2643 t (_ _________________________________________________________)1 2880 1 1440 2728 t (_ _________________________________________________________)1 2880 1 1440 2812 t (_ _________________________________________________________)1 2880 1 1440 2897 t (_ _________________________________________________________)1 2880 1 1440 2982 t (_ _________________________________________________________)1 2880 1 1440 3067 t (_ _________________________________________________________)1 2880 1 1440 3151 t (_ _________________________________________________________)1 2880 1 1440 3236 t 10 S f (\347)4320 2320 w (\347)4320 2336 w (\347)4320 2436 w (\347)4320 2536 w (\347)4320 2636 w (\347)4320 2736 w (\347)4320 2836 w (\347)4320 2936 w (\347)4320 3036 w (\347)4320 3136 w (\347)4320 3236 w (\347)4235 2320 w (\347)4235 2336 w (\347)4235 2436 w (\347)4235 2536 w (\347)4235 2636 w (\347)4235 2736 w (\347)4235 2836 w (\347)4235 2936 w (\347)4235 3036 w (\347)4235 3136 w (\347)4235 3236 w (\347)4150 2320 w (\347)4150 2336 w (\347)4150 2436 w (\347)4150 2536 w (\347)4150 2636 w (\347)4150 2736 w (\347)4150 2836 w (\347)4150 2936 w (\347)4150 3036 w (\347)4150 3136 w (\347)4150 3236 w (\347)4065 2320 w (\347)4065 2336 w (\347)4065 2436 w (\347)4065 2536 w (\347)4065 2636 w (\347)4065 2736 w (\347)4065 2836 w (\347)4065 2936 w (\347)4065 3036 w (\347)4065 3136 w (\347)4065 3236 w (\347)3981 2320 w (\347)3981 2336 w (\347)3981 2436 w (\347)3981 2536 w (\347)3981 2636 w (\347)3981 2736 w (\347)3981 2836 w (\347)3981 2936 w (\347)3981 3036 w (\347)3981 3136 w (\347)3981 3236 w (\347)3896 2320 w (\347)3896 2336 w (\347)3896 2436 w (\347)3896 2536 w (\347)3896 2636 w (\347)3896 2736 w (\347)3896 2836 w (\347)3896 2936 w (\347)3896 3036 w (\347)3896 3136 w (\347)3896 3236 w (\347)3811 2320 w (\347)3811 2336 w (\347)3811 2436 w (\347)3811 2536 w (\347)3811 2636 w (\347)3811 2736 w (\347)3811 2836 w (\347)3811 2936 w (\347)3811 3036 w (\347)3811 3136 w (\347)3811 3236 w (\347)3727 2320 w (\347)3727 2336 w (\347)3727 2436 w (\347)3727 2536 w (\347)3727 2636 w (\347)3727 2736 w (\347)3727 2836 w (\347)3727 2936 w (\347)3727 3036 w (\347)3727 3136 w (\347)3727 3236 w (\347)3642 2320 w (\347)3642 2336 w (\347)3642 2436 w (\347)3642 2536 w (\347)3642 2636 w (\347)3642 2736 w (\347)3642 2836 w (\347)3642 2936 w (\347)3642 3036 w (\347)3642 3136 w (\347)3642 3236 w (\347)3557 2320 w (\347)3557 2336 w (\347)3557 2436 w (\347)3557 2536 w (\347)3557 2636 w (\347)3557 2736 w (\347)3557 2836 w (\347)3557 2936 w (\347)3557 3036 w (\347)3557 3136 w (\347)3557 3236 w (\347)3472 2320 w (\347)3472 2336 w (\347)3472 2436 w (\347)3472 2536 w (\347)3472 2636 w (\347)3472 2736 w (\347)3472 2836 w (\347)3472 2936 w (\347)3472 3036 w (\347)3472 3136 w (\347)3472 3236 w (\347)3388 2320 w (\347)3388 2336 w (\347)3388 2436 w (\347)3388 2536 w (\347)3388 2636 w (\347)3388 2736 w (\347)3388 2836 w (\347)3388 2936 w (\347)3388 3036 w (\347)3388 3136 w (\347)3388 3236 w (\347)3303 2320 w (\347)3303 2336 w (\347)3303 2436 w (\347)3303 2536 w (\347)3303 2636 w (\347)3303 2736 w (\347)3303 2836 w (\347)3303 2936 w (\347)3303 3036 w (\347)3303 3136 w (\347)3303 3236 w (\347)3218 2320 w (\347)3218 2336 w (\347)3218 2436 w (\347)3218 2536 w (\347)3218 2636 w (\347)3218 2736 w (\347)3218 2836 w (\347)3218 2936 w (\347)3218 3036 w (\347)3218 3136 w (\347)3218 3236 w (\347)3134 2320 w (\347)3134 2336 w (\347)3134 2436 w (\347)3134 2536 w (\347)3134 2636 w (\347)3134 2736 w (\347)3134 2836 w (\347)3134 2936 w (\347)3134 3036 w (\347)3134 3136 w (\347)3134 3236 w (\347)3049 2320 w (\347)3049 2336 w (\347)3049 2436 w (\347)3049 2536 w (\347)3049 2636 w (\347)3049 2736 w (\347)3049 2836 w (\347)3049 2936 w (\347)3049 3036 w (\347)3049 3136 w (\347)3049 3236 w (\347)2964 2320 w (\347)2964 2336 w (\347)2964 2436 w (\347)2964 2536 w (\347)2964 2636 w (\347)2964 2736 w (\347)2964 2836 w (\347)2964 2936 w (\347)2964 3036 w (\347)2964 3136 w (\347)2964 3236 w (\347)2880 2320 w (\347)2880 2336 w (\347)2880 2436 w (\347)2880 2536 w (\347)2880 2636 w (\347)2880 2736 w (\347)2880 2836 w (\347)2880 2936 w (\347)2880 3036 w (\347)2880 3136 w (\347)2880 3236 w (\347)2795 2320 w (\347)2795 2336 w (\347)2795 2436 w (\347)2795 2536 w (\347)2795 2636 w (\347)2795 2736 w (\347)2795 2836 w (\347)2795 2936 w (\347)2795 3036 w (\347)2795 3136 w (\347)2795 3236 w (\347)2710 2320 w (\347)2710 2336 w (\347)2710 2436 w (\347)2710 2536 w (\347)2710 2636 w (\347)2710 2736 w (\347)2710 2836 w (\347)2710 2936 w (\347)2710 3036 w (\347)2710 3136 w (\347)2710 3236 w (\347)2625 2320 w (\347)2625 2336 w (\347)2625 2436 w (\347)2625 2536 w (\347)2625 2636 w (\347)2625 2736 w (\347)2625 2836 w (\347)2625 2936 w (\347)2625 3036 w (\347)2625 3136 w (\347)2625 3236 w (\347)2541 2320 w (\347)2541 2336 w (\347)2541 2436 w (\347)2541 2536 w (\347)2541 2636 w (\347)2541 2736 w (\347)2541 2836 w (\347)2541 2936 w (\347)2541 3036 w (\347)2541 3136 w (\347)2541 3236 w (\347)2456 2320 w (\347)2456 2336 w (\347)2456 2436 w (\347)2456 2536 w (\347)2456 2636 w (\347)2456 2736 w (\347)2456 2836 w (\347)2456 2936 w (\347)2456 3036 w (\347)2456 3136 w (\347)2456 3236 w (\347)2371 2320 w (\347)2371 2336 w (\347)2371 2436 w (\347)2371 2536 w (\347)2371 2636 w (\347)2371 2736 w (\347)2371 2836 w (\347)2371 2936 w (\347)2371 3036 w (\347)2371 3136 w (\347)2371 3236 w (\347)2287 2320 w (\347)2287 2336 w (\347)2287 2436 w (\347)2287 2536 w (\347)2287 2636 w (\347)2287 2736 w (\347)2287 2836 w (\347)2287 2936 w (\347)2287 3036 w (\347)2287 3136 w (\347)2287 3236 w (\347)2202 2320 w (\347)2202 2336 w (\347)2202 2436 w (\347)2202 2536 w (\347)2202 2636 w (\347)2202 2736 w (\347)2202 2836 w (\347)2202 2936 w (\347)2202 3036 w (\347)2202 3136 w (\347)2202 3236 w (\347)2117 2320 w (\347)2117 2336 w (\347)2117 2436 w (\347)2117 2536 w (\347)2117 2636 w (\347)2117 2736 w (\347)2117 2836 w (\347)2117 2936 w (\347)2117 3036 w (\347)2117 3136 w (\347)2117 3236 w (\347)2032 2320 w (\347)2032 2336 w (\347)2032 2436 w (\347)2032 2536 w (\347)2032 2636 w (\347)2032 2736 w (\347)2032 2836 w (\347)2032 2936 w (\347)2032 3036 w (\347)2032 3136 w (\347)2032 3236 w (\347)1948 2320 w (\347)1948 2336 w (\347)1948 2436 w (\347)1948 2536 w (\347)1948 2636 w (\347)1948 2736 w (\347)1948 2836 w (\347)1948 2936 w (\347)1948 3036 w (\347)1948 3136 w (\347)1948 3236 w (\347)1863 2320 w (\347)1863 2336 w (\347)1863 2436 w (\347)1863 2536 w (\347)1863 2636 w (\347)1863 2736 w (\347)1863 2836 w (\347)1863 2936 w (\347)1863 3036 w (\347)1863 3136 w (\347)1863 3236 w (\347)1778 2320 w (\347)1778 2336 w (\347)1778 2436 w (\347)1778 2536 w (\347)1778 2636 w (\347)1778 2736 w (\347)1778 2836 w (\347)1778 2936 w (\347)1778 3036 w (\347)1778 3136 w (\347)1778 3236 w (\347)1694 2320 w (\347)1694 2336 w (\347)1694 2436 w (\347)1694 2536 w (\347)1694 2636 w (\347)1694 2736 w (\347)1694 2836 w (\347)1694 2936 w (\347)1694 3036 w (\347)1694 3136 w (\347)1694 3236 w (\347)1609 2320 w (\347)1609 2336 w (\347)1609 2436 w (\347)1609 2536 w (\347)1609 2636 w (\347)1609 2736 w (\347)1609 2836 w (\347)1609 2936 w (\347)1609 3036 w (\347)1609 3136 w (\347)1609 3236 w (\347)1524 2320 w (\347)1524 2336 w (\347)1524 2436 w (\347)1524 2536 w (\347)1524 2636 w (\347)1524 2736 w (\347)1524 2836 w (\347)1524 2936 w (\347)1524 3036 w (\347)1524 3136 w (\347)1524 3236 w (\347)1440 2320 w (\347)1440 2336 w (\347)1440 2436 w (\347)1440 2536 w (\347)1440 2636 w (\347)1440 2736 w (\347)1440 2836 w (\347)1440 2936 w (\347)1440 3036 w (\347)1440 3136 w (\347)1440 3236 w 1638 2456 1609 2474 Dl 4121 2492 4150 2474 Dl 1638 2492 1609 2474 Dl 4121 2456 4150 2474 Dl 1668 2448 1639 2455 Dl 4091 2499 4120 2492 Dl 1668 2499 1639 2492 Dl 4091 2448 4120 2455 Dl 1697 2443 1668 2448 Dl 4062 2504 4091 2499 Dl 1697 2504 1668 2499 Dl 4062 2443 4091 2448 Dl 1727 2438 1698 2442 Dl 4032 2509 4061 2505 Dl 1727 2509 1698 2505 Dl 4032 2438 4061 2442 Dl 1757 2435 1728 2438 Dl 4003 2512 4032 2509 Dl 1757 2512 1728 2509 Dl 4003 2435 4032 2438 Dl 1786 2431 1757 2434 Dl 3973 2516 4002 2513 Dl 1786 2516 1757 2513 Dl 3973 2431 4002 2434 Dl 1816 2427 1787 2430 Dl 3943 2520 3972 2517 Dl 1816 2520 1787 2517 Dl 3943 2427 3972 2430 Dl 1845 2425 1816 2427 Dl 3914 2522 3943 2520 Dl 1845 2522 1816 2520 Dl 3914 2425 3943 2427 Dl 1875 2422 1846 2424 Dl 3884 2525 3913 2523 Dl 1875 2525 1846 2523 Dl 3884 2422 3913 2424 Dl 1905 2420 1876 2422 Dl 3854 2528 3883 2526 Dl 1905 2528 1876 2526 Dl 3854 2420 3883 2422 Dl 1934 2417 1905 2419 Dl 3825 2530 3854 2528 Dl 1934 2530 1905 2528 Dl 3825 2417 3854 2419 Dl 1964 2415 1935 2417 Dl 3795 2532 3824 2530 Dl 1964 2532 1935 2530 Dl 3795 2415 3824 2417 Dl 1994 2414 1965 2415 Dl 3765 2533 3794 2532 Dl 1994 2533 1965 2532 Dl 3765 2414 3794 2415 Dl 2023 2412 1994 2413 Dl 3736 2535 3765 2534 Dl 2023 2535 1994 2534 Dl 3736 2412 3765 2413 Dl 2053 2410 2024 2411 Dl 3706 2537 3735 2536 Dl 2053 2537 2024 2536 Dl 3706 2410 3735 2411 Dl 2083 2408 2054 2409 Dl 3676 2539 3705 2538 Dl 2083 2539 2054 2538 Dl 3676 2408 3705 2409 Dl 2112 2407 2083 2408 Dl 3647 2541 3676 2540 Dl 2112 2541 2083 2540 Dl 3647 2407 3676 2408 Dl 2142 2405 2113 2406 Dl 3617 2542 3646 2541 Dl 2142 2542 2113 2541 Dl 3617 2405 3646 2406 Dl 2172 2404 2143 2405 Dl 3587 2544 3616 2543 Dl 2172 2544 2143 2543 Dl 3587 2404 3616 2405 Dl 2201 2402 2172 2403 Dl 3558 2545 3587 2544 Dl 2201 2545 2172 2544 Dl 3558 2402 3587 2403 Dl 2231 2401 2202 2402 Dl 3528 2546 3557 2545 Dl 2231 2546 2202 2545 Dl 3528 2401 3557 2402 Dl 2261 2400 2232 2401 Dl 3499 2547 3528 2546 Dl 2261 2547 2232 2546 Dl 3499 2400 3528 2401 Dl 2290 2399 2261 2400 Dl 3469 2549 3498 2548 Dl 2290 2549 2261 2548 Dl 3469 2399 3498 2400 Dl 2320 2398 2291 2399 Dl 3439 2550 3468 2549 Dl 2320 2550 2291 2549 Dl 3439 2398 3468 2399 Dl 2349 2398 2320 2398 Dl 3410 2550 3439 2550 Dl 2349 2550 2320 2550 Dl 3410 2398 3439 2398 Dl 2379 2397 2350 2397 Dl 3380 2551 3409 2551 Dl 2379 2551 2350 2551 Dl 3380 2397 3409 2397 Dl 2409 2396 2380 2396 Dl 3350 2551 3379 2551 Dl 2409 2551 2380 2551 Dl 3350 2396 3379 2396 Dl 2438 2395 2409 2395 Dl 3321 2552 3350 2552 Dl 2438 2552 2409 2552 Dl 3321 2395 3350 2395 Dl 2468 2394 2439 2394 Dl 3291 2553 3320 2553 Dl 2468 2553 2439 2553 Dl 3291 2394 3320 2394 Dl 2498 2393 2469 2393 Dl 3261 2554 3290 2554 Dl 2498 2554 2469 2554 Dl 3261 2393 3290 2393 Dl 2527 2393 2498 2393 Dl 3232 2554 3261 2554 Dl 2527 2554 2498 2554 Dl 3232 2393 3261 2393 Dl 2557 2392 2528 2392 Dl 3202 2555 3231 2555 Dl 2557 2555 2528 2555 Dl 3202 2392 3231 2392 Dl 2587 2392 2558 2392 Dl 3172 2556 3201 2556 Dl 2587 2556 2558 2556 Dl 3172 2392 3201 2392 Dl 2616 2391 2587 2391 Dl 3143 2556 3172 2556 Dl 2616 2556 2587 2556 Dl 3143 2391 3172 2391 Dl 2646 2391 2617 2391 Dl 3113 2556 3142 2556 Dl 2646 2556 2617 2556 Dl 3113 2391 3142 2391 Dl 2676 2390 2647 2390 Dl 3083 2557 3112 2557 Dl 2676 2557 2647 2557 Dl 3083 2390 3112 2390 Dl 2705 2390 2676 2390 Dl 3054 2557 3083 2557 Dl 2705 2557 2676 2557 Dl 3054 2390 3083 2390 Dl 2735 2390 2706 2390 Dl 3024 2558 3053 2558 Dl 2735 2558 2706 2558 Dl 3024 2390 3053 2390 Dl 2765 2389 2736 2389 Dl 2995 2558 3024 2558 Dl 2765 2558 2736 2558 Dl 2995 2389 3024 2389 Dl 2794 2389 2765 2389 Dl 2965 2558 2994 2558 Dl 2794 2558 2765 2558 Dl 2965 2389 2994 2389 Dl 9 R f (Proper tails and fishy tails,)4 965 1 1476 3512 t 9 I f (a)2464 3512 w 9 S f (=)2554 3512 w 9 R f (1 ,)1 75 1 2649 3512 t 9 I f (b)2761 3512 w 9 S f (=)2851 3512 w 9 R f ( rotated 90)2 386(15. Figure)1 394 2 2946 3512 t 9 S f (\260)3726 3512 w 9 R f (to save space.)2 499 1 3785 3512 t 10 R f (We close this essay with the remark that values of)9 2071 1 720 3752 t 10 I f (h)2824 3752 w 10 R f (may become quite large and that therefore overflow)7 2132 1 2908 3752 t ( most)1 220( Unfortunately,)1 635(may occur when the algorithm is interpreted by computers with insufficient word size.)12 3465 3 720 3872 t ( 32-bit arithmetic, ellipses with values of)6 1641( Using)1 290(computer systems do not indicate integer overflow!)6 2064 3 720 3992 t 10 I f (a)4742 3992 w 10 R f (and)4819 3992 w 10 I f (b)4990 3992 w 10 R f (up to 1000 can be drawn without failure.)7 1625 1 720 4112 t 9 R f ( that I can)3 395( computer hardware)2 743( All)1 172(The exclamation directs attention away from software to hardware.)8 2510 4 970 4268 t ( code for languages such as Pas-)6 1186( Compiled)1 404( indicates integer overflow, although not by trapping.)7 1943(think of)1 287 4 970 4388 t (cal almost universally ignores the indication, however.)6 1974 1 970 4508 t ( the discriminator)2 640( the slope of the ellipse is near zero,)8 1305( Where)1 287(The claim of a range up to 1000 is too rosy.)10 1588 4 970 4664 t 9 I f (g)970 4784 w 9 R f ( away from)2 413(may be evaluated at points up to 2 units)8 1450 2 1040 4784 t 9 I f (g)2927 4784 w 9 S f (=)3017 4784 w 9 R f ( algorithm visits points as much as 1/2)7 1393(0. \(The)1 285 2 3112 4784 t (unit off the ellipse, and)4 839 1 970 4904 t 9 I f (g)1833 4904 w 9 S f (=)1923 4904 w 9 R f ( with)1 185( such a point)3 462( At)1 137(0 is displaced 3/2 units from the ellipse.\))7 1476 4 2018 4904 t 9 I f (a)4303 4904 w 9 S f (=)4393 4904 w 9 I f (b)4488 4904 w 9 R f (,)4533 4904 w 9 I f (x)4581 4904 w 9 S f (~)4658 4904 w 9 R f (0)4745 4904 w (and)970 5024 w 9 I f (y)1138 5024 w 9 S f (~)1215 5024 w 9 I f (a)1302 5024 w 9 R f (, the magnitude of)3 702 1 1347 5024 t 9 I f (g)2087 5024 w 9 R f (, estimated as)2 519 1 2132 5024 t 9 S f (\357)2688 5039 w 9 R f (\()2733 5024 w 9 S f (\266)2770 5024 w 9 I f (g /)1 77 1 2821 5024 t 9 S f (\266)2905 5024 w 9 I f (y)2956 5024 w 9 R f (\))3003 5024 w 9 S f (D)3048 5024 w 9 I f (y)3110 5024 w 9 S f (\357)3150 5039 w 9 R f (, is approximately 4)3 759 1 3210 5024 t 9 I f (a)3976 5024 w 6 R f (3)4031 4988 w 9 R f ( overflow is)2 459(. Thus)1 263 2 4068 5024 t ( 2)1 52( parameter values around \()4 1002(liable to occur at)3 631 3 970 5144 t 6 R f (31)2660 5108 w 9 I f (/)2734 5144 w 9 R f (4 \))1 82 1 2766 5144 t 6 R f (1)2853 5108 w 6 I f (/)2888 5108 w 6 R f (3)2910 5108 w 9 R f ( confirms this)2 516( Testing)1 326(, not much more than 800.)5 1001 3 2947 5144 t (estimate.)970 5264 w 9 S f (\347)2190 5420 w 9 R f (Big-oh estimates are not quantitative.)4 1350 1 2205 5420 t 9 S f ( ______________________________)1 1335( _)1 -1335( ``````````````````````````````)1 1335(\347 `)1 -1335 4 3570 5420 t 9 R f ( particular, double plotting is self-)5 1308( In)1 137( is bad practice to draw points twice.)7 1436( It)1 117( to say.)2 291(There is more)2 531 6 970 5576 t ( plotting at the beginnings of the arcs)7 1418( Double)1 322( into a bitmap.)3 555(nullifying when drawing by exclusive or)5 1525 4 970 5696 t (points can be averted by proper coding of the)8 1658 1 970 5816 t 9 I f (Mark)2654 5816 w 9 R f ( double plotting can also occur)5 1130(procedure. However,)1 785 2 2875 5816 t ( in the poorly closing example mentioned above \()8 1813( example,)1 355( For)1 175(where the two branches meet.)4 1091 4 970 5936 t 9 I f (a)4404 5936 w 9 S f (=)4494 5936 w 9 R f (2 and)1 201 1 4589 5936 t 9 I f (b)970 6056 w 9 S f (=)1060 6056 w 9 R f (3\),)1155 6056 w 9 I f (Mark)1276 6056 w 9 R f ( will be called in the second loop as well as in the first.)13 1982(\( 0 , 3 \))4 201 2 1478 6056 t ( the two branches always)4 939( Will)1 213(Other important properties of the algorithm are left to be taken on faith.)12 2668 3 970 6212 t ( is)1 84( \(This)1 237( not, color would leak out on attempting to shade the inside of an ellipse.)14 2656( If)1 108(meet without a gap?)3 735 5 970 6332 t ( omitted algorithm for circles)4 1101( The)1 197(not an idle question.)3 765 3 970 6452 t 9 I f (can)3067 6452 w 9 R f ( circles drawn by the)4 800( Will)1 217(produce gaps.\))1 542 3 3231 6452 t ( diagonal,)1 361(algorithm be symmetric about the)4 1241 2 970 6572 t 9 I f (y)2600 6572 w 9 S f (=)2685 6572 w 9 I f (x)2780 6572 w 9 R f ( answer is not immediately obvious, because in)7 1739(? The)1 231 2 2820 6572 t (all but the smallest circles, the juncture of the two branches lies off the diagonal.)14 2908 1 970 6692 t 9 S f (\347)2140 6848 w 9 R f (Formulate and confirm proper behavior.)4 1450 1 2155 6848 t 9 S f ( ________________________________)1 1435( _)1 -1435( ````````````````````````````````)1 1435(\347 `)1 -1435 4 3620 6848 t 9 R f ( from fixed points on the shores of the first)9 1556(The ellipse-drawing algorithm works like two ships setting out)8 2264 2 970 7004 t ( leaving)1 288( most difficult sailing will be experienced in)7 1603( The)1 187(quadrant to rendezvous near the octant juncture.)6 1742 4 970 7124 t ( and most confined turns must be navigated, and at the meeting point,)12 2656(the harbors, where the sharpest)4 1164 2 970 7244 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 52 514 764 %%EndPage: 28 28 %%Page: 29 29 %%PageBoundingBox: (atend) /saveobj save def mark 29 pagesetup 10 R f (- 29 -)2 216 1 2772 480 t 9 R f ( the steering of a ship may be trusted to an)10 1630( as at sea, where)4 634( Just)1 197(where precision docking is required.)4 1359 4 970 840 t ( for close navigation, so ellipse-drawing)5 1468(apprentice seaman in open water, but needs an experienced pilot)9 2352 2 970 960 t ( to simple homework-assignment code only in the open and needs more attention in the)14 3222(can be entrusted)2 598 2 970 1080 t ( present algorithm has not earned a pilot's license.)8 1812( The)1 186(critical stretches.)1 611 3 970 1200 t 10 R f (REFERENCES [for Wirth])2 1094 1 720 1356 t ( Cossitt.)1 332([1] N.)1 347 2 720 1512 t 10 I f ( Drawing Circles with the NS32CG16.)5 1566(Line Drawing with the NS32CG16 and)5 1575 2 1452 1512 t 10 R f (Technical)4647 1512 w (Report AN-522 and AN-523, National Semiconductor Corp., 1988)7 2669 1 970 1632 t ( commands, non-determinacy, and the formal derivation of programs.)8 2997( Guarded)1 419( W. Dijkstra.)2 568([2] E.)1 336 4 720 1788 t 10 I f (Comm. ACM,)1 547 1 970 1908 t 10 R f (18\(8\):453-457, August 1975.)2 1166 1 1542 1908 t ( Foley and A Van Dam.)5 1019( D.)1 134([3] J.)1 314 3 720 2064 t 10 I f (Fundamentals of Interactive Computer Graphics.)4 2031 1 2250 2064 t 10 R f (Addison-Wesley,)4344 2064 w (1982.)970 2184 w 10 B f (References [for McIlroy])2 1056 1 720 2424 t 10 R f ( N., ``Drawing lines, circles and ellipses in a raster,'' pp. 427-434 in)12 2737([1] Wirth,)1 449 2 889 2640 t 10 I f (Beauty is our Business)3 914 1 4101 2640 t 10 R f (,)5015 2640 w ( D., and Misra, J. \(Eds.\), Springer-Verlag, New)7 1974(Feijen, W. H. J., van Gasteren, A. J. M., Gries,)9 1986 2 1080 2760 t (York \(1990\).)1 521 1 1080 2880 t ( N., ``From Modula to Oberon,'')5 1434([2] Wirth,)1 449 2 889 3060 t 10 I f ( Experience)1 499(Software\320Practice and)1 983 2 2822 3060 t 10 B f (18)4355 3060 w 10 R f (, pp. 661-670)2 585 1 4455 3060 t (\(1988\). Acknowledgements.)1 1159 1 1080 3180 t ( M. L. V., ``Algorithms for drawing ellipses or hyperbolae with a digital plotter,'')13 3333([3] Pitteway,)1 566 2 889 3360 t 10 I f (Com-)4818 3360 w (puter J.)1 305 1 1080 3480 t 10 B f (10)1410 3480 w 10 R f (, pp. 282-289 \(1967\).)3 849 1 1510 3480 t ( ``A linear algorithm for incremental digital display of circular arcs,'')10 2814( J.,)1 117([4] Bresenham,)1 665 3 889 3660 t 10 I f (Comm. ACM)1 526 1 4514 3660 t 10 B f (20)1080 3780 w 10 R f (, pp. 100-106 \(1977\).)3 849 1 1180 3780 t ( D., ``Best approximate circles on integer grids,'')7 2073( M.)1 153([5] McIlroy,)1 543 3 889 3960 t 10 I f (ACM Trans. on Graphics)3 1062 1 3698 3960 t 10 B f (2)4800 3960 w 10 R f (, pp.)1 190 1 4850 3960 t (237-264 \(Oct. 1983\).)2 843 1 1080 4080 t cleartomark showpage saveobj restore %%PageBoundingBox: 61 368 514 764 %%EndPage: 29 29 %%Trailer done %%Pages: 29 %%DocumentFonts: Courier Times-Bold Times-Italic Times-Roman Times-Roman Symbol %%BoundingBox: 61 45 535 764