2852:
1983:
2847:{\displaystyle {\begin{aligned}b(q)&={\frac {4}{3\pi }}\int _{0}^{1}{\biggl \{}3{\sqrt {q^{2}-u^{2}}}{\sqrt {1-u^{2}}}+{\frac {(1-u^{2})^{\tfrac {3}{2}}}{\sqrt {q^{2}-u^{2}}}}{\biggr \}}{\textrm {d}}u\\&={\frac {4}{3\pi }}\int _{0}^{1}{\biggl \{}4{\sqrt {q^{2}-u^{2}}}{\sqrt {1-u^{2}}}-{\frac {q^{2}-1}{q}}{\frac {\sqrt {1-u^{2}}}{\sqrt {q^{2}-u^{2}}}}{\biggr \}}{\textrm {d}}u\\&={\frac {4}{3\pi }}{\biggl \{}{\frac {4q}{3}}{\biggl (}(q^{2}+1)E({\tfrac {1}{q^{2}}})-(q^{2}-1)K({\tfrac {1}{q^{2}}}){\biggr )}-(q^{2}-1){\biggl (}qE({\tfrac {1}{q^{2}}})-{\frac {q^{2}-1}{q}}K({\tfrac {1}{q^{2}}}){\biggr )}{\biggr \}}\\&={\frac {4}{9\pi }}{\biggl \{}q(q^{2}+7)E({\tfrac {1}{q^{2}}})-{\frac {q^{2}-1}{q}}(q^{2}+3)K({\tfrac {1}{q^{2}}}){\biggr \}}\end{aligned}}}
45:
1967:
1726:
1444:
945:
848:
1204:
1618:
855:
A uniform distribution on a unit circular disk is occasionally encountered in statistics. It most commonly occurs in operations research in the mathematics of urban planning, where it may be used to model a population within a city. Other uses may take advantage of the fact that it is a distribution
1027:
1962:{\displaystyle b(q)={\frac {4}{3\pi }}\int _{0}^{{\textrm {sin}}^{-1}{\tfrac {1}{q}}}{\biggl \{}3q^{2}{\textrm {cos}}^{2}\theta {\sqrt {1-q^{2}{\textrm {sin}}^{2}\theta }}+{\Bigl (}1-q^{2}{\textrm {sin}}^{2}\theta {\Bigr )}^{\tfrac {3}{2}}{\biggl \}}{\textrm {d}}\theta .}
1456:
1376:
591:
447:
1988:
780:
3179:
In higher dimensions, the Euler characteristic of a closed ball remains equal to +1, but the Euler characteristic of an open ball is +1 for even-dimensional balls and −1 for odd-dimensional balls. See
1721:
2951:
1199:{\displaystyle b(q)={\frac {1}{\pi }}\int _{0}^{2\pi }{\textrm {d}}\theta \int _{0}^{s(\theta )}r^{2}{\textrm {d}}r={\frac {1}{3\pi }}\int _{0}^{2\pi }s(\theta )^{3}{\textrm {d}}\theta .}
3252:
3.155.7 and 3.169.9, taking due account of the difference in notation from
Abramowitz and Stegun. (Compare A&S 17.3.11 with G&R 8.113.) This article follows A&S's notation.
782:
which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle
268:
1232:
3288:
838:
204:
1613:{\displaystyle b(q)={\frac {2}{3\pi }}\int _{0}^{{\textrm {sin}}^{-1}{\tfrac {1}{q}}}{\biggl \{}s_{+}(\theta )^{3}-s_{-}(\theta )^{3}{\biggr \}}{\textrm {d}}\theta }
462:
316:
235:
170:
143:
3281:
1649:
328:
2882:
3274:
1229:. The steps needed to evaluate the integral, together with several references, will be found in the paper by Lew et al.; the result is that
696:
914:
from points in the distribution to this location and the average square of such distances. The latter value can be computed directly as
3164:
3137:
3107:
3078:
3051:
3024:
857:
3561:
3427:
687:
671:
3571:
3463:
240:
3536:
3467:
856:
for which it is easy to compute the probability that a given set of linear inequalities will be satisfied. (
785:
3266:
977:
If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of
175:
3403:
3345:
3249:
3237:
2967:
861:
279:
38:
3525:
3500:
2990:
963:
we need to look separately at the cases in which the location is internal or external, i.e. in which
660:
631:), as they have different topological properties from each other. For instance, every closed disk is
3495:
3489:
3159:, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, p. 79,
644:
3068:
1451:
Turning to an external location, we can set up the integral in a similar way, this time obtaining
1013:, integrating in polar coordinates centered on the fixed location for which the area of a cell is
2973:
1371:{\displaystyle b(q)={\frac {4}{9\pi }}{\biggl \{}4(q^{2}-1)K(q^{2})+(q^{2}+7)E(q^{2}){\biggr \}}}
971:
868:
640:
636:
3369:
3160:
3133:
3127:
3103:
3074:
3047:
3020:
3014:
648:
621:
586:{\displaystyle {\overline {D}}=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}\leq R^{2}\}.}
3154:
3041:
3566:
3351:
2985:
107:
100:
289:
213:
148:
3453:
3183:
627:
The open disk and the closed disk are not topologically equivalent (that is, they are not
597:
3415:
1226:
667:
652:
609:
128:
3096:
903:
from the center of the disk, it is also of interest to determine the average distance
890:, while direct integration in polar coordinates shows the mean squared distance to be
3555:
3516:
3472:
3458:
3356:
632:
628:
54:
3363:
2979:
44:
17:
970:, and we find that in both cases the result can only be expressed in terms of
679:
442:{\displaystyle D=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}<R^{2}\}}
3531:
3420:
2961:
675:
3408:
207:
92:
3228:
J. S. Lew et al., "On the
Average Distances in a Circular Disc" (1977).
3129:
New
Foundations for Physical Geometry: The Theory of Linear Structures
3073:, Dover Books on Mathematics, Courier Dover Publications, p. 44,
3046:, Dover Books on Mathematics, Courier Dover Publications, p. 58,
1443:
944:
775:{\displaystyle f(x,y)=\left({\frac {x+{\sqrt {1-y^{2}}}}{2}},y\right)}
635:
whereas every open disk is not compact. However from the viewpoint of
3316:
111:
847:
663:
of a point (and therefore also that of a closed or open disk) is 1.
3322:
1442:
943:
846:
43:
31:
1716:{\displaystyle s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.}
867:"An ingenious argument via elementary functions" shows the mean
3270:
1390:
are complete elliptic integrals of the first and second kinds.
118:
if it contains the circle that constitutes its boundary, and
3190:, Lezioni Lincee, Cambridge University Press, pp. 46–50
2946:{\displaystyle \lim _{q\to \infty }b(q)=q+{\tfrac {1}{8q}}.}
2976:, the usual term for the 3-dimensional analogue of a disk
851:
The average distance to a location from points on a disc
3297:
Compact topological surfaces and their immersions in 3D
655:
are trivial except the 0th one, which is isomorphic to
30:"2-ball" redirects here. For the basketball event, see
2924:
2812:
2734:
2635:
2579:
2509:
2456:
2127:
1927:
1788:
1518:
2885:
1986:
1729:
1652:
1459:
1447:
The average distance from a disk to an external point
1235:
1030:
948:
The average distance from a disk to an internal point
788:
699:
465:
331:
292:
243:
216:
178:
151:
131:
897:
If we are given an arbitrary location at a distance
3509:
3481:
3446:
3437:
3383:
3338:
3309:
3302:
3070:
3156:Combinatorial Group Theory: A Topological Approach
3095:
2945:
2846:
1961:
1715:
1612:
1370:
1198:
832:
774:
585:
441:
310:
262:
229:
198:
164:
137:
2835:
2697:
2665:
2658:
2564:
2532:
2422:
2400:
2358:
2226:
2169:
2042:
1941:
1921:
1880:
1803:
1703:
1699:
1595:
1533:
1363:
1268:
2887:
670:from the closed disk to itself has at least one
3013:Clapham, Christopher; Nicholson, James (2014),
1439:Average distance to an arbitrary external point
940:Average distance to an arbitrary internal point
3282:
639:they share many properties: both of them are
8:
3016:The Concise Oxford Dictionary of Mathematics
690:. The statement is false for the open disk:
577:
479:
456:of the same center and radius is given by:
436:
338:
2970:, the region between two concentric circles
647:to a single point. This implies that their
3443:
3306:
3289:
3275:
3267:
2993:, containing certain centers of a triangle
3043:Intuitive Concepts in Elementary Topology
2923:
2890:
2884:
2834:
2833:
2821:
2811:
2790:
2765:
2758:
2743:
2733:
2712:
2696:
2695:
2680:
2664:
2663:
2657:
2656:
2644:
2634:
2610:
2603:
2588:
2578:
2563:
2562:
2547:
2531:
2530:
2518:
2508:
2487:
2465:
2455:
2434:
2421:
2420:
2405:
2399:
2398:
2383:
2364:
2363:
2357:
2356:
2347:
2334:
2322:
2309:
2291:
2284:
2273:
2261:
2253:
2240:
2234:
2225:
2224:
2218:
2213:
2194:
2175:
2174:
2168:
2167:
2158:
2145:
2126:
2116:
2100:
2089:
2077:
2069:
2056:
2050:
2041:
2040:
2034:
2029:
2010:
1987:
1985:
1947:
1946:
1940:
1939:
1926:
1920:
1919:
1909:
1903:
1902:
1895:
1879:
1878:
1864:
1858:
1857:
1850:
1838:
1829:
1823:
1822:
1815:
1802:
1801:
1787:
1778:
1772:
1771:
1769:
1764:
1745:
1728:
1693:
1677:
1676:
1675:
1657:
1651:
1601:
1600:
1594:
1593:
1587:
1571:
1558:
1542:
1532:
1531:
1517:
1508:
1502:
1501:
1499:
1494:
1475:
1458:
1362:
1361:
1352:
1327:
1308:
1283:
1267:
1266:
1251:
1234:
1184:
1183:
1177:
1155:
1150:
1131:
1119:
1118:
1112:
1093:
1088:
1075:
1074:
1065:
1060:
1046:
1029:
806:
793:
787:
747:
735:
726:
698:
571:
558:
533:
507:
503:
502:
500:
466:
464:
430:
417:
392:
366:
362:
361:
359:
330:
291:
263:{\displaystyle \operatorname {int} D^{2}}
254:
242:
221:
215:
185:
179:
177:
156:
150:
130:
3132:, Oxford University Press, p. 339,
3019:, Oxford University Press, p. 138,
3002:
1620:where the law of cosines tells us that
3261:Abramowitz and Stegun, 17.3.11 et seq.
3224:
3222:
3213:
3201:
983:under a distribution whose density is
210:the closed disk is usually denoted as
3188:Introduction to Geometric Probability
3008:
3006:
871:between two points in the disk to be
833:{\displaystyle x^{2}+y^{2}=1,x>0.}
145:, an open disk is usually denoted as
7:
600:of a closed or open disk of radius
199:{\displaystyle {\overline {D_{r}}}}
2897:
693:Consider for example the function
25:
2982:, a space of functions on a disk
674:(we don't require the map to be
27:Plane figure, bounded by circle
2911:
2905:
2894:
2830:
2808:
2802:
2783:
2752:
2730:
2724:
2705:
2653:
2631:
2597:
2575:
2559:
2540:
2527:
2505:
2499:
2480:
2474:
2452:
2446:
2427:
2123:
2103:
2000:
1994:
1739:
1733:
1584:
1577:
1555:
1548:
1469:
1463:
1358:
1345:
1339:
1320:
1314:
1301:
1295:
1276:
1245:
1239:
1174:
1167:
1103:
1097:
1040:
1034:
715:
703:
555:
542:
530:
517:
494:
482:
414:
401:
389:
376:
353:
341:
305:
293:
1:
843:As a statistical distribution
471:
191:
3102:. Oxford University Press.
972:complete elliptic integrals
688:Brouwer fixed point theorem
3588:
3216:, Ex. 1, p. 135.
3094:Altmann, Simon L. (1992).
3067:Rotman, Joseph J. (2013),
2854:using standard integrals.
206:. However in the field of
36:
29:
3153:Cohen, Daniel E. (1989),
1215:can be found in terms of
322:is given by the formula:
2964:, a disk with radius one
3428:Sphere with three holes
3114:disc circular symmetry.
237:while the open disk is
114:. A disk is said to be
83: center or origin
3040:Arnold, B. H. (2013),
2947:
2848:
1963:
1717:
1614:
1448:
1372:
1200:
949:
858:Gaussian distributions
852:
834:
776:
587:
443:
312:
264:
231:
200:
166:
139:
88:
3346:Real projective plane
3331:Pretzel (genus 3) ...
3250:Gradshteyn and Ryzhik
3238:Abramowitz and Stegun
3126:Maudlin, Tim (2014),
2968:Annulus (mathematics)
2948:
2849:
1964:
1718:
1615:
1446:
1373:
1201:
947:
860:in the plane require
850:
835:
777:
651:are trivial, and all
588:
444:
313:
311:{\displaystyle (a,b)}
280:Cartesian coordinates
265:
232:
230:{\displaystyle D^{2}}
201:
172:and a closed disk is
167:
165:{\displaystyle D_{r}}
140:
106:) is the region in a
47:
3501:Euler characteristic
3098:Icons and Symmetries
2991:Orthocentroidal disk
2883:
1984:
1727:
1650:
1457:
1233:
1028:
862:numerical quadrature
786:
697:
682:); this is the case
661:Euler characteristic
463:
329:
290:
241:
214:
176:
149:
129:
37:For other uses, see
2223:
2039:
1800:
1530:
1163:
1107:
1073:
645:homotopy equivalent
3562:Euclidean geometry
3328:Number 8 (genus 2)
3182:Klain, Daniel A.;
2974:Ball (mathematics)
2943:
2938:
2901:
2844:
2842:
2828:
2750:
2651:
2595:
2525:
2472:
2209:
2136:
2025:
1969:We may substitute
1959:
1936:
1797:
1760:
1713:
1640:are the roots for
1610:
1527:
1490:
1449:
1368:
1196:
1146:
1084:
1056:
950:
869:Euclidean distance
853:
830:
772:
649:fundamental groups
637:algebraic topology
583:
439:
308:
260:
227:
196:
162:
135:
89:
3549:
3548:
3545:
3544:
3379:
3378:
2937:
2886:
2827:
2781:
2749:
2693:
2650:
2626:
2594:
2524:
2471:
2418:
2396:
2367:
2354:
2353:
2328:
2307:
2279:
2259:
2207:
2178:
2165:
2164:
2135:
2095:
2075:
2023:
1950:
1935:
1906:
1873:
1861:
1826:
1796:
1775:
1758:
1680:
1604:
1526:
1505:
1488:
1264:
1187:
1144:
1122:
1078:
1054:
759:
753:
622:circular symmetry
474:
194:
138:{\displaystyle r}
16:(Redirected from
3579:
3464:Triangulatedness
3444:
3307:
3303:Without boundary
3291:
3284:
3277:
3268:
3262:
3259:
3253:
3247:
3241:
3235:
3229:
3226:
3217:
3211:
3205:
3199:
3193:
3191:
3184:Rota, Gian-Carlo
3177:
3171:
3169:
3150:
3144:
3142:
3123:
3117:
3116:
3101:
3091:
3085:
3083:
3064:
3058:
3056:
3037:
3031:
3029:
3010:
2986:Circular segment
2952:
2950:
2949:
2944:
2939:
2936:
2925:
2900:
2878:
2877:
2875:
2874:
2871:
2868:
2853:
2851:
2850:
2845:
2843:
2839:
2838:
2829:
2826:
2825:
2813:
2795:
2794:
2782:
2777:
2770:
2769:
2759:
2751:
2748:
2747:
2735:
2717:
2716:
2701:
2700:
2694:
2692:
2681:
2673:
2669:
2668:
2662:
2661:
2652:
2649:
2648:
2636:
2627:
2622:
2615:
2614:
2604:
2596:
2593:
2592:
2580:
2568:
2567:
2552:
2551:
2536:
2535:
2526:
2523:
2522:
2510:
2492:
2491:
2473:
2470:
2469:
2457:
2439:
2438:
2426:
2425:
2419:
2414:
2406:
2404:
2403:
2397:
2395:
2384:
2376:
2369:
2368:
2365:
2362:
2361:
2355:
2352:
2351:
2339:
2338:
2329:
2327:
2326:
2311:
2310:
2308:
2303:
2296:
2295:
2285:
2280:
2278:
2277:
2262:
2260:
2258:
2257:
2245:
2244:
2235:
2230:
2229:
2222:
2217:
2208:
2206:
2195:
2187:
2180:
2179:
2176:
2173:
2172:
2166:
2163:
2162:
2150:
2149:
2140:
2139:
2138:
2137:
2128:
2121:
2120:
2101:
2096:
2094:
2093:
2078:
2076:
2074:
2073:
2061:
2060:
2051:
2046:
2045:
2038:
2033:
2024:
2022:
2011:
1979:
1968:
1966:
1965:
1960:
1952:
1951:
1948:
1945:
1944:
1938:
1937:
1928:
1925:
1924:
1914:
1913:
1908:
1907:
1904:
1900:
1899:
1884:
1883:
1874:
1869:
1868:
1863:
1862:
1859:
1855:
1854:
1839:
1834:
1833:
1828:
1827:
1824:
1820:
1819:
1807:
1806:
1799:
1798:
1789:
1786:
1785:
1777:
1776:
1773:
1768:
1759:
1757:
1746:
1722:
1720:
1719:
1714:
1698:
1697:
1682:
1681:
1678:
1662:
1661:
1646:of the equation
1645:
1639:
1629:
1619:
1617:
1616:
1611:
1606:
1605:
1602:
1599:
1598:
1592:
1591:
1576:
1575:
1563:
1562:
1547:
1546:
1537:
1536:
1529:
1528:
1519:
1516:
1515:
1507:
1506:
1503:
1498:
1489:
1487:
1476:
1434:
1432:
1430:
1429:
1426:
1423:
1411:
1410:
1408:
1407:
1404:
1401:
1389:
1383:
1377:
1375:
1374:
1369:
1367:
1366:
1357:
1356:
1332:
1331:
1313:
1312:
1288:
1287:
1272:
1271:
1265:
1263:
1252:
1224:
1220:
1214:
1205:
1203:
1202:
1197:
1189:
1188:
1185:
1182:
1181:
1162:
1154:
1145:
1143:
1132:
1124:
1123:
1120:
1117:
1116:
1106:
1092:
1080:
1079:
1076:
1072:
1064:
1055:
1047:
1023:
1012:
1000:
999:
997:
996:
993:
990:
982:
969:
962:
935:
934:
932:
931:
928:
925:
913:
902:
893:
889:
887:
885:
884:
881:
878:
839:
837:
836:
831:
811:
810:
798:
797:
781:
779:
778:
773:
771:
767:
760:
755:
754:
752:
751:
736:
727:
592:
590:
589:
584:
576:
575:
563:
562:
538:
537:
513:
512:
511:
506:
475:
467:
448:
446:
445:
440:
435:
434:
422:
421:
397:
396:
372:
371:
370:
365:
317:
315:
314:
309:
269:
267:
266:
261:
259:
258:
236:
234:
233:
228:
226:
225:
205:
203:
202:
197:
195:
190:
189:
180:
171:
169:
168:
163:
161:
160:
144:
142:
141:
136:
122:if it does not.
82:
73:
64:
52:
21:
3587:
3586:
3582:
3581:
3580:
3578:
3577:
3576:
3572:Planar surfaces
3552:
3551:
3550:
3541:
3505:
3482:Characteristics
3477:
3439:
3433:
3375:
3334:
3298:
3295:
3265:
3260:
3256:
3248:
3244:
3236:
3232:
3227:
3220:
3212:
3208:
3200:
3196:
3181:
3178:
3174:
3167:
3152:
3151:
3147:
3140:
3125:
3124:
3120:
3110:
3093:
3092:
3088:
3081:
3066:
3065:
3061:
3054:
3039:
3038:
3034:
3027:
3012:
3011:
3004:
3000:
2958:
2929:
2881:
2880:
2872:
2869:
2866:
2865:
2863:
2858:
2841:
2840:
2817:
2786:
2761:
2760:
2739:
2708:
2685:
2671:
2670:
2640:
2606:
2605:
2584:
2543:
2514:
2483:
2461:
2430:
2407:
2388:
2374:
2373:
2343:
2330:
2318:
2287:
2286:
2269:
2249:
2236:
2199:
2185:
2184:
2154:
2141:
2122:
2112:
2102:
2085:
2065:
2052:
2015:
2003:
1982:
1981:
1970:
1918:
1901:
1891:
1856:
1846:
1821:
1811:
1770:
1750:
1725:
1724:
1689:
1653:
1648:
1647:
1641:
1637:
1631:
1627:
1621:
1583:
1567:
1554:
1538:
1500:
1480:
1455:
1454:
1441:
1427:
1424:
1421:
1420:
1418:
1413:
1405:
1402:
1399:
1398:
1396:
1391:
1385:
1379:
1348:
1323:
1304:
1279:
1256:
1231:
1230:
1222:
1216:
1209:
1173:
1136:
1108:
1026:
1025:
1014:
1002:
994:
991:
988:
987:
985:
984:
978:
964:
953:
942:
929:
926:
923:
922:
920:
915:
904:
898:
891:
882:
879:
876:
875:
873:
872:
845:
802:
789:
784:
783:
743:
728:
725:
721:
695:
694:
653:homology groups
618:
567:
554:
529:
501:
461:
460:
426:
413:
388:
360:
327:
326:
288:
287:
276:
250:
239:
238:
217:
212:
211:
181:
174:
173:
152:
147:
146:
127:
126:
87:
80:
78:
71:
69:
65: diameter
62:
60:
50:
42:
35:
28:
23:
22:
18:Disc (geometry)
15:
12:
11:
5:
3585:
3583:
3575:
3574:
3569:
3564:
3554:
3553:
3547:
3546:
3543:
3542:
3540:
3539:
3534:
3528:
3522:
3519:
3513:
3511:
3507:
3506:
3504:
3503:
3498:
3493:
3485:
3483:
3479:
3478:
3476:
3475:
3470:
3461:
3456:
3450:
3448:
3441:
3435:
3434:
3432:
3431:
3425:
3424:
3423:
3413:
3412:
3411:
3406:
3398:
3397:
3396:
3387:
3385:
3381:
3380:
3377:
3376:
3374:
3373:
3370:Dyck's surface
3367:
3361:
3360:
3359:
3354:
3342:
3340:
3339:Non-orientable
3336:
3335:
3333:
3332:
3329:
3326:
3320:
3313:
3311:
3304:
3300:
3299:
3296:
3294:
3293:
3286:
3279:
3271:
3264:
3263:
3254:
3242:
3230:
3218:
3206:
3204:, p. 132.
3194:
3172:
3165:
3145:
3138:
3118:
3108:
3086:
3079:
3059:
3052:
3032:
3025:
3001:
2999:
2996:
2995:
2994:
2988:
2983:
2977:
2971:
2965:
2957:
2954:
2942:
2935:
2932:
2928:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2899:
2896:
2893:
2889:
2837:
2832:
2824:
2820:
2816:
2810:
2807:
2804:
2801:
2798:
2793:
2789:
2785:
2780:
2776:
2773:
2768:
2764:
2757:
2754:
2746:
2742:
2738:
2732:
2729:
2726:
2723:
2720:
2715:
2711:
2707:
2704:
2699:
2691:
2688:
2684:
2679:
2676:
2674:
2672:
2667:
2660:
2655:
2647:
2643:
2639:
2633:
2630:
2625:
2621:
2618:
2613:
2609:
2602:
2599:
2591:
2587:
2583:
2577:
2574:
2571:
2566:
2561:
2558:
2555:
2550:
2546:
2542:
2539:
2534:
2529:
2521:
2517:
2513:
2507:
2504:
2501:
2498:
2495:
2490:
2486:
2482:
2479:
2476:
2468:
2464:
2460:
2454:
2451:
2448:
2445:
2442:
2437:
2433:
2429:
2424:
2417:
2413:
2410:
2402:
2394:
2391:
2387:
2382:
2379:
2377:
2375:
2372:
2360:
2350:
2346:
2342:
2337:
2333:
2325:
2321:
2317:
2314:
2306:
2302:
2299:
2294:
2290:
2283:
2276:
2272:
2268:
2265:
2256:
2252:
2248:
2243:
2239:
2233:
2228:
2221:
2216:
2212:
2205:
2202:
2198:
2193:
2190:
2188:
2186:
2183:
2171:
2161:
2157:
2153:
2148:
2144:
2134:
2131:
2125:
2119:
2115:
2111:
2108:
2105:
2099:
2092:
2088:
2084:
2081:
2072:
2068:
2064:
2059:
2055:
2049:
2044:
2037:
2032:
2028:
2021:
2018:
2014:
2009:
2006:
2004:
2002:
1999:
1996:
1993:
1990:
1989:
1958:
1955:
1943:
1934:
1931:
1923:
1917:
1912:
1898:
1894:
1890:
1887:
1882:
1877:
1872:
1867:
1853:
1849:
1845:
1842:
1837:
1832:
1818:
1814:
1810:
1805:
1795:
1792:
1784:
1781:
1767:
1763:
1756:
1753:
1749:
1744:
1741:
1738:
1735:
1732:
1712:
1709:
1706:
1702:
1696:
1692:
1688:
1685:
1674:
1671:
1668:
1665:
1660:
1656:
1635:
1625:
1609:
1597:
1590:
1586:
1582:
1579:
1574:
1570:
1566:
1561:
1557:
1553:
1550:
1545:
1541:
1535:
1525:
1522:
1514:
1511:
1497:
1493:
1486:
1483:
1479:
1474:
1471:
1468:
1465:
1462:
1440:
1437:
1365:
1360:
1355:
1351:
1347:
1344:
1341:
1338:
1335:
1330:
1326:
1322:
1319:
1316:
1311:
1307:
1303:
1300:
1297:
1294:
1291:
1286:
1282:
1278:
1275:
1270:
1262:
1259:
1255:
1250:
1247:
1244:
1241:
1238:
1227:Law of cosines
1195:
1192:
1180:
1176:
1172:
1169:
1166:
1161:
1158:
1153:
1149:
1142:
1139:
1135:
1130:
1127:
1115:
1111:
1105:
1102:
1099:
1096:
1091:
1087:
1083:
1071:
1068:
1063:
1059:
1053:
1050:
1045:
1042:
1039:
1036:
1033:
1024: ; hence
941:
938:
844:
841:
829:
826:
823:
820:
817:
814:
809:
805:
801:
796:
792:
770:
766:
763:
758:
750:
746:
742:
739:
734:
731:
724:
720:
717:
714:
711:
708:
705:
702:
668:continuous map
617:
614:
610:area of a disk
594:
593:
582:
579:
574:
570:
566:
561:
557:
553:
550:
547:
544:
541:
536:
532:
528:
525:
522:
519:
516:
510:
505:
499:
496:
493:
490:
487:
484:
481:
478:
473:
470:
450:
449:
438:
433:
429:
425:
420:
416:
412:
409:
406:
403:
400:
395:
391:
387:
384:
381:
378:
375:
369:
364:
358:
355:
352:
349:
346:
343:
340:
337:
334:
307:
304:
301:
298:
295:
275:
272:
257:
253:
249:
246:
224:
220:
193:
188:
184:
159:
155:
134:
125:For a radius,
79:
70:
61:
49:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3584:
3573:
3570:
3568:
3565:
3563:
3560:
3559:
3557:
3538:
3535:
3533:
3529:
3527:
3523:
3521:Making a hole
3520:
3518:
3517:Connected sum
3515:
3514:
3512:
3508:
3502:
3499:
3497:
3494:
3491:
3487:
3486:
3484:
3480:
3474:
3473:Orientability
3471:
3469:
3465:
3462:
3460:
3457:
3455:
3454:Connectedness
3452:
3451:
3449:
3445:
3442:
3436:
3429:
3426:
3422:
3419:
3418:
3417:
3414:
3410:
3407:
3405:
3402:
3401:
3399:
3394:
3393:
3392:
3389:
3388:
3386:
3384:With boundary
3382:
3372:(genus 3) ...
3371:
3368:
3365:
3362:
3358:
3357:Roman surface
3355:
3353:
3352:Boy's surface
3349:
3348:
3347:
3344:
3343:
3341:
3337:
3330:
3327:
3324:
3321:
3318:
3315:
3314:
3312:
3308:
3305:
3301:
3292:
3287:
3285:
3280:
3278:
3273:
3272:
3269:
3258:
3255:
3251:
3246:
3243:
3239:
3234:
3231:
3225:
3223:
3219:
3215:
3214:Arnold (2013)
3210:
3207:
3203:
3202:Arnold (2013)
3198:
3195:
3189:
3185:
3176:
3173:
3168:
3166:9780521349369
3162:
3158:
3157:
3149:
3146:
3141:
3139:9780191004551
3135:
3131:
3130:
3122:
3119:
3115:
3111:
3109:9780198555995
3105:
3100:
3099:
3090:
3087:
3082:
3080:9780486151687
3076:
3072:
3071:
3063:
3060:
3055:
3053:9780486275765
3049:
3045:
3044:
3036:
3033:
3028:
3026:9780199679591
3022:
3018:
3017:
3009:
3007:
3003:
2997:
2992:
2989:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2959:
2955:
2953:
2940:
2933:
2930:
2926:
2920:
2917:
2914:
2908:
2902:
2891:
2879:, while also
2861:
2855:
2822:
2818:
2814:
2805:
2799:
2796:
2791:
2787:
2778:
2774:
2771:
2766:
2762:
2755:
2744:
2740:
2736:
2727:
2721:
2718:
2713:
2709:
2702:
2689:
2686:
2682:
2677:
2675:
2645:
2641:
2637:
2628:
2623:
2619:
2616:
2611:
2607:
2600:
2589:
2585:
2581:
2572:
2569:
2556:
2553:
2548:
2544:
2537:
2519:
2515:
2511:
2502:
2496:
2493:
2488:
2484:
2477:
2466:
2462:
2458:
2449:
2443:
2440:
2435:
2431:
2415:
2411:
2408:
2392:
2389:
2385:
2380:
2378:
2370:
2348:
2344:
2340:
2335:
2331:
2323:
2319:
2315:
2312:
2304:
2300:
2297:
2292:
2288:
2281:
2274:
2270:
2266:
2263:
2254:
2250:
2246:
2241:
2237:
2231:
2219:
2214:
2210:
2203:
2200:
2196:
2191:
2189:
2181:
2159:
2155:
2151:
2146:
2142:
2132:
2129:
2117:
2113:
2109:
2106:
2097:
2090:
2086:
2082:
2079:
2070:
2066:
2062:
2057:
2053:
2047:
2035:
2030:
2026:
2019:
2016:
2012:
2007:
2005:
1997:
1991:
1977:
1973:
1956:
1953:
1932:
1929:
1915:
1910:
1896:
1892:
1888:
1885:
1875:
1870:
1865:
1851:
1847:
1843:
1840:
1835:
1830:
1816:
1812:
1808:
1793:
1790:
1782:
1779:
1765:
1761:
1754:
1751:
1747:
1742:
1736:
1730:
1710:
1707:
1704:
1700:
1694:
1690:
1686:
1683:
1672:
1669:
1666:
1663:
1658:
1654:
1644:
1634:
1624:
1607:
1588:
1580:
1572:
1568:
1564:
1559:
1551:
1543:
1539:
1523:
1520:
1512:
1509:
1495:
1491:
1484:
1481:
1477:
1472:
1466:
1460:
1452:
1445:
1438:
1436:
1416:
1394:
1388:
1382:
1353:
1349:
1342:
1336:
1333:
1328:
1324:
1317:
1309:
1305:
1298:
1292:
1289:
1284:
1280:
1273:
1260:
1257:
1253:
1248:
1242:
1236:
1228:
1219:
1212:
1206:
1193:
1190:
1178:
1170:
1164:
1159:
1156:
1151:
1147:
1140:
1137:
1133:
1128:
1125:
1113:
1109:
1100:
1094:
1089:
1085:
1081:
1069:
1066:
1061:
1057:
1051:
1048:
1043:
1037:
1031:
1021:
1017:
1010:
1006:
981:
975:
973:
967:
960:
956:
946:
939:
937:
918:
911:
907:
901:
895:
870:
865:
863:
859:
849:
842:
840:
827:
824:
821:
818:
815:
812:
807:
803:
799:
794:
790:
768:
764:
761:
756:
748:
744:
740:
737:
732:
729:
722:
718:
712:
709:
706:
700:
691:
689:
685:
681:
677:
673:
669:
664:
662:
658:
654:
650:
646:
642:
638:
634:
630:
625:
623:
620:The disk has
615:
613:
611:
607:
603:
599:
580:
572:
568:
564:
559:
551:
548:
545:
539:
534:
526:
523:
520:
514:
508:
497:
491:
488:
485:
476:
468:
459:
458:
457:
455:
431:
427:
423:
418:
410:
407:
404:
398:
393:
385:
382:
379:
373:
367:
356:
350:
347:
344:
335:
332:
325:
324:
323:
321:
302:
299:
296:
285:
281:
273:
271:
255:
251:
247:
244:
222:
218:
209:
186:
182:
157:
153:
132:
123:
121:
117:
113:
110:bounded by a
109:
105:
102:
98:
94:
86:
77:
74: radius
68:
59:
56:
55:circumference
46:
40:
33:
19:
3416:Möbius strip
3390:
3364:Klein bottle
3257:
3245:
3233:
3209:
3197:
3187:
3175:
3155:
3148:
3128:
3121:
3113:
3097:
3089:
3069:
3062:
3042:
3035:
3015:
2980:Disk algebra
2859:
2857:Hence again
2856:
1975:
1971:
1642:
1632:
1622:
1453:
1450:
1414:
1392:
1386:
1380:
1217:
1210:
1207:
1019:
1015:
1008:
1004:
979:
976:
965:
958:
954:
951:
916:
909:
905:
899:
896:
866:
854:
692:
683:
665:
656:
641:contractible
629:homeomorphic
626:
619:
605:
601:
595:
453:
451:
319:
283:
277:
124:
119:
115:
103:
101:also spelled
96:
90:
84:
75:
66:
57:
3459:Compactness
672:fixed point
643:and so are
454:closed disk
318:and radius
3556:Categories
3510:Operations
3492:components
3488:Number of
3468:smoothness
3447:Properties
3395:Semisphere
3310:Orientable
2998:References
1225:using the
1003:0 ≤
686:=2 of the
680:surjective
616:Properties
452:while the
286:of center
48:Disk with
3537:Immersion
3532:cross-cap
3530:Gluing a
3524:Gluing a
3421:Cross-cap
3366:(genus 2)
3350:genus 1;
3325:(genus 1)
3319:(genus 0)
2962:Unit disk
2898:∞
2895:→
2772:−
2756:−
2690:π
2617:−
2601:−
2554:−
2538:−
2494:−
2478:−
2393:π
2341:−
2316:−
2298:−
2282:−
2267:−
2247:−
2211:∫
2204:π
2152:−
2110:−
2083:−
2063:−
2027:∫
2020:π
1954:θ
1916:θ
1889:−
1871:θ
1844:−
1836:θ
1780:−
1762:∫
1755:π
1701:−
1684:θ
1664:−
1608:θ
1581:θ
1573:−
1565:−
1552:θ
1510:−
1492:∫
1485:π
1433:≈ 1.13177
1290:−
1261:π
1191:θ
1171:θ
1160:π
1148:∫
1141:π
1101:θ
1086:∫
1082:θ
1070:π
1058:∫
1052:π
888:≈ 0.90541
741:−
676:bijective
565:≤
549:−
524:−
498:∈
472:¯
408:−
383:−
357:∈
284:open disk
248:
192:¯
3490:boundary
3409:Cylinder
3186:(1997),
2956:See also
1980:to get
1007:≤
952:To find
678:or even
274:Formulas
208:topology
93:geometry
3567:Circles
3440:notions
3438:Related
3404:Annulus
3400:Ribbon
3240:, 17.3.
2876:
2864:
1431:
1419:
1409:
1397:
998:
986:
933:
921:
886:
874:
633:compact
3526:handle
3317:Sphere
3163:
3136:
3106:
3077:
3050:
3023:
2862:(1) =
1723:Hence
1417:(1) =
1395:(0) =
1378:where
666:Every
659:. The
282:, the
116:closed
112:circle
81:
72:
63:
53:
51:
3496:Genus
3323:Torus
1978:sinθ
1208:Here
608:(see
108:plane
32:2Ball
3391:Disk
3161:ISBN
3134:ISBN
3104:ISBN
3075:ISBN
3048:ISBN
3021:ISBN
1630:and
1384:and
1221:and
1001:for
825:>
604:is π
598:area
596:The
424:<
120:open
104:disc
97:disk
95:, a
39:Disc
3466:or
3430:...
2888:lim
1905:sin
1860:sin
1825:cos
1774:sin
1679:cos
1638:(θ)
1628:(θ)
1504:sin
1213:(θ)
1011:(θ)
968:≶ 1
883:45π
877:128
864:.)
612:).
278:In
245:int
91:In
3558::
3221:^
3112:.
3005:^
2873:9π
2867:32
1974:=
1711:0.
1435:.
1428:9π
1422:32
1412:;
1022:dθ
974:.
936:.
894:.
828:0.
624:.
270:.
3290:e
3283:t
3276:v
3192:.
3170:.
3143:.
3084:.
3057:.
3030:.
2941:.
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2906:(
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2870:/
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2836:}
2831:)
2823:2
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2809:(
2806:K
2803:)
2800:3
2797:+
2792:2
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2784:(
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2725:)
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2500:)
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2475:)
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2401:{
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2381:=
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2366:d
2359:}
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2008:=
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1998:q
1995:(
1992:b
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1957:.
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1942:}
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1930:3
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1911:2
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1168:(
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1098:(
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1038:q
1035:(
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1020:r
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1016:r
1009:s
1005:r
995:π
992:/
989:1
980:r
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959:q
957:(
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930:2
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602:R
581:.
578:}
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521:x
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492:y
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437:}
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386:a
380:x
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300:,
297:a
294:(
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187:r
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158:r
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99:(
85:O
76:R
67:D
58:C
41:.
34:.
20:)
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