155:
53:
2609:
2404:
900:
over concatenation of curves, so it suffices to prove the formula for a single line segment. Since the right-hand side does not depend on the positioning of the line segment, it must equal some function of the segment's length. Because, again, the formula is additive over concatenation of line
887:
1768:
419:
1972:
743:
748:
1648:
1440:
line either twice or not at all, the unoriented
Crofton formula for convex curves can be stated without numerical factors: the measure of the set of straight lines which intersect a convex curve is equal to its length.
2604:{\displaystyle {\frac {|\partial S|}{E}}={\frac {|{\text{unit sphere in }}\mathbb {R} ^{n}|}{|{\text{unit ball in }}\mathbb {R} ^{n-1}|}}=2{\sqrt {\pi }}{\frac {\Gamma ({\frac {n+1}{2}})}{\Gamma ({\frac {n}{2}})}}}
1435:
of the space of unoriented lines. The
Crofton formula is often stated in terms of the corresponding density in the latter space, in which the numerical factor is not 1/4 but 1/2. Since a convex curve intersects
1069:
1395:
995:
901:
segments, the integral must be a constant times the length of the line segment. It remains only to determine the factor of 1/4; this is easily done by computing both sides when γ is the
1855:
1444:
The same formula (with the same multiplicative constants) apply for hyperbolic spaces and spherical spaces, when the kinematic measure is suitably scaled. The proof is essentially the same.
1203:
2236:
1810:
148:
334:
658:
604:
460:
1423:, thus the formula is correct with an undetermined multiplicative constant. Then explicitly calculate this constant, using the simplest possible case: two circles of radius 1.
2353:
2075:
633:
535:
496:
1110:
2132:
1515:
222:
80:
2399:
1130:
274:
572:
322:
246:
184:
100:
1643:
1616:
1569:
1542:
1257:
1230:
1162:
941:
2661:
2635:
2101:
1421:
1286:
2376:
2681:
2324:
2304:
2284:
2152:
2046:
2026:
2006:
1850:
1830:
1589:
653:
294:
1291:
2904:
882:{\displaystyle C_{n}={\frac {1}{2\cdot |{\text{unit ball in }}\mathbb {R} ^{n-1}|}}={\frac {\Gamma {({\frac {n+1}{2}})}}{2\pi ^{\frac {n-1}{2}}}}}
1763:{\displaystyle Pr(l{\text{ intersects }}S_{1}|l{\text{ intersects }}S_{2})={\frac {\operatorname {area} (S_{1})}{\operatorname {area} (S_{2})}}}
1474:
Given two nested, convex, closed curves, the inner one is shorter. In general, for two such codimension 1 surfaces, the inner one has less area.
2706:
can be viewed as a measure-theoretic generalization of the
Crofton formula and the Crofton formula is used in the inversion formula of the
1000:
2885:
2742:
2840:
Izrail
Moiseevich Gel'fand; Mark Iosifovich Graev (1991), "Crofton's function and inversion formulas in real integral geometry",
2792:
2950:
498:, so it is a natural integration measure for speaking of an "average" number of intersections. It is usually called the
2684:
950:
2940:
1771:
1167:
2206:
2178:
2104:
1780:
105:
2918:
2945:
2161:
414:{\displaystyle \operatorname {length} (\gamma )={\frac {1}{4}}\iint n_{\gamma }(\varphi ,p)\;d\varphi \;dp.}
159:
2924:
1967:{\displaystyle Pr(l{\text{ intersects }}P|l,P{\text{ intersects }}S)={\frac {|S|}{|\partial S|\cdot E}}}
577:
433:
2329:
2051:
609:
511:
472:
2713:
2157:
1074:
297:
2663:
gives the classic example "the average shadow of a convex body is 1/4 of its surface area". General
2110:
1448:
154:
2873:
2857:
1480:
200:
41:
2697:
1397:
The proof is done similarly as above. First note that both sides of the formula are additive in
59:
2381:
2881:
2807:
2738:
1437:
1115:
425:
259:
187:
33:
544:
307:
231:
169:
85:
2900:
2849:
2766:
1452:
301:
2778:
1621:
1594:
1547:
1520:
1235:
1208:
1135:
919:
738:{\displaystyle \operatorname {area} (S)=C_{n}\iint n_{\gamma }(\varphi ,p)\;d\varphi \;dp.}
2774:
2703:
944:
466:
2640:
2614:
2181:: Among all closed curves with a given perimeter, the circle has the unique maximum area.
2080:
1400:
1265:
2358:
2666:
2309:
2289:
2241:
2137:
2031:
2011:
1977:
1835:
1815:
1574:
638:
538:
279:
37:
25:
2934:
2815:
2861:
1432:
897:
505:
The right-hand side in the
Crofton formula is sometimes called the Favard length.
2757:
Ueno, Seitarô (1955), "On the densities in a two-dimensional generalized space",
2185:
1470:
Crofton's formula yields elegant proofs of the following results, among others:
902:
17:
2811:
52:
2770:
1455:; the integral is then performed with the natural measure on the space of
1456:
325:
2853:
1462:
More general forms exist, such as the kinematic formula of Chern.
1064:{\displaystyle (\varphi ,x,y)\in [0,2\pi )\times \mathbb {R} ^{2}}
190:
153:
908:
The proof for the generalized version proceeds exactly as above.
2203:
Cauchy's surface area formula: Given any convex compact subset
2028:, that is, the expected length of the orthogonal projection of
1390:{\displaystyle \int _{T\in E^{2}}|C\cap T(D)|dT=4|C|\cdot |D|}
1770:
This is the justification for the surface area heuristic in
2326:
is the orthogonal projection to a random hyperplane of
1262:
Given rectifiable simple (no self-intersection) curves
1132:
counterclockwise around the origin, then translate by
606:
on it, which is also invariant under rigid motions of
2793:"On the Kinematic Formula in the Lives of the Saints"
2669:
2643:
2617:
2407:
2384:
2361:
2332:
2312:
2292:
2244:
2209:
2140:
2113:
2083:
2054:
2034:
2014:
1980:
1858:
1838:
1818:
1783:
1651:
1624:
1597:
1577:
1550:
1523:
1483:
1403:
1294:
1268:
1238:
1211:
1170:
1138:
1118:
1077:
1003:
953:
922:
751:
661:
641:
612:
580:
547:
514:
475:
436:
337:
310:
282:
262:
234:
203:
172:
108:
88:
62:
2927:, a visualization of Cauchy's surface area formula.
1618:, conditional on it intersecting the outer surface
1232:on itself, thus we obtained a kinematic measure on
2675:
2655:
2629:
2603:
2393:
2370:
2355:), then by integrating Crofton formula first over
2347:
2318:
2298:
2278:
2230:
2146:
2126:
2095:
2069:
2040:
2020:
2000:
1966:
1844:
1824:
1804:
1762:
1637:
1610:
1583:
1563:
1536:
1509:
1415:
1389:
1280:
1251:
1224:
1197:
1156:
1124:
1104:
1063:
989:
935:
881:
737:
647:
627:
598:
574:, and we can similarly define a kinematic measure
566:
529:
490:
454:
413:
316:
288:
268:
240:
216:
178:
142:
94:
74:
990:{\displaystyle [0,2\pi )\times \mathbb {R} ^{2}}
252:intersect. We can parametrize the general line
2710:-plane Radon transform of Gel'fand and Graev
2683:gives generalization of Barbier's theorem for
1451:surface or more generally to two-dimensional
8:
2925:Alice, Bob, and the average shadow of a cube
2800:Notices of the American Mathematical Society
2735:Integral geometry and geometric probability
1198:{\displaystyle dx\wedge dy\wedge d\varphi }
508:In general, the space of oriented lines in
276:in which it points and its signed distance
2188:of every bounded rectifiable closed curve
1477:Given two nested, convex, closed surfaces
912:Poincare’s formula for intersecting curves
725:
718:
401:
394:
2668:
2642:
2616:
2585:
2556:
2547:
2540:
2526:
2514:
2510:
2509:
2503:
2498:
2491:
2485:
2481:
2480:
2474:
2469:
2466:
2452:
2435:
2422:
2411:
2408:
2406:
2383:
2360:
2339:
2335:
2334:
2331:
2311:
2291:
2268:
2251:
2243:
2231:{\displaystyle S\subset \mathbb {R} ^{n}}
2222:
2218:
2217:
2208:
2139:
2114:
2112:
2082:
2061:
2057:
2056:
2053:
2033:
2013:
1987:
1979:
1950:
1936:
1925:
1918:
1910:
1907:
1893:
1879:
1871:
1857:
1837:
1817:
1805:{\displaystyle S\subset \mathbb {R} ^{n}}
1796:
1792:
1791:
1782:
1748:
1724:
1708:
1696:
1687:
1679:
1673:
1664:
1650:
1629:
1623:
1602:
1596:
1576:
1555:
1549:
1528:
1522:
1501:
1488:
1482:
1402:
1382:
1374:
1366:
1358:
1341:
1318:
1310:
1299:
1293:
1267:
1243:
1237:
1216:
1210:
1169:
1137:
1117:
1076:
1055:
1051:
1050:
1002:
981:
977:
976:
952:
927:
921:
857:
826:
822:
816:
805:
793:
789:
788:
782:
777:
765:
756:
750:
697:
684:
660:
640:
619:
615:
614:
611:
579:
552:
546:
521:
517:
516:
513:
482:
478:
477:
474:
435:
373:
356:
336:
309:
281:
261:
233:
208:
202:
171:
143:{\displaystyle n_{\gamma }(\varphi ,p)=2}
113:
107:
87:
61:
2842:Functional Analysis and Its Applications
1431:The space of oriented lines is a double
947:on the plane. It can be parametrized as
158:Application of the Crofton formula in a
51:
2725:
2107:, this probability is upper bounded by
1447:The Crofton formula generalizes to any
896:Both sides of the Crofton formula are
328:over the space of all oriented lines:
36:relating the length of a curve to the
7:
2192:has perimeter at most the length of
300:. The Crofton formula expresses the
1571:, the probability of a random line
635:. Then for any rectifiable surface
228:) be the number of points at which
2579:
2550:
2416:
1930:
819:
599:{\displaystyle d\varphi \wedge dp}
455:{\displaystyle d\varphi \wedge dp}
14:
2919:Cauchy–Crofton formula page
2897:Introduction to Integral Geometry
2759:Memoirs of the Faculty of Science
2348:{\displaystyle \mathbb {R} ^{n}}
2070:{\displaystyle \mathbb {R} ^{n}}
628:{\displaystyle \mathbb {R} ^{n}}
530:{\displaystyle \mathbb {R} ^{n}}
491:{\displaystyle \mathbb {R} ^{2}}
2286:be the expected shadow area of
2048:to a random linear subspace of
1591:intersecting the inner surface
1105:{\displaystyle T(\varphi ,x,y)}
56:The line defined by choices of
2595:
2582:
2574:
2553:
2527:
2499:
2492:
2470:
2457:
2453:
2449:
2443:
2436:
2432:
2423:
2412:
2273:
2269:
2265:
2259:
2252:
2248:
2127:{\displaystyle {\frac {1}{2}}}
1995:
1984:
1958:
1947:
1937:
1926:
1919:
1911:
1901:
1880:
1865:
1754:
1741:
1730:
1717:
1702:
1680:
1658:
1383:
1375:
1367:
1359:
1342:
1338:
1332:
1319:
1151:
1139:
1099:
1081:
1043:
1028:
1022:
1004:
969:
954:
844:
823:
806:
778:
715:
703:
674:
668:
391:
379:
350:
344:
131:
119:
1:
2685:bodies of constant brightness
1852:be a random hyperplane, then
1205:is invariant under action of
2921:, with demonstration applets
1777:Given compact convex subset
1510:{\displaystyle S_{1},S_{2}}
217:{\displaystyle n_{\gamma }}
40:number of times a "random"
2967:
2200:is already a convex curve.
2196:, with equality only when
655:of codimension 1, we have
75:{\displaystyle \varphi ,p}
2637:gives Barbier's theorem,
2394:{\displaystyle d\varphi }
1772:bounding volume hierarchy
193:. Given an oriented line
32:) is a classic result of
28:(1826–1915), (also
2791:Calegari, Danny (2020).
2179:isoperimetric inequality
2105:isoperimetric inequality
2008:is the average width of
1125:{\displaystyle \varphi }
269:{\displaystyle \varphi }
2895:Santalo, L. A. (1953).
2880:. AMS. pp. 36–40.
2611:In particular, setting
2162:curve of constant width
567:{\displaystyle S^{n-1}}
317:{\displaystyle \gamma }
241:{\displaystyle \gamma }
179:{\displaystyle \gamma }
95:{\displaystyle \gamma }
2899:. pp. 12–13, 54.
2878:Geometry and Billiards
2771:10.2206/kyushumfs.9.65
2677:
2657:
2631:
2605:
2395:
2372:
2349:
2320:
2300:
2280:
2232:
2148:
2128:
2097:
2071:
2042:
2022:
2002:
1968:
1895: intersects
1873: intersects
1846:
1832:be a random line, and
1826:
1806:
1764:
1689: intersects
1666: intersects
1639:
1612:
1585:
1565:
1538:
1511:
1417:
1391:
1282:
1253:
1226:
1199:
1158:
1126:
1106:
1065:
991:
937:
883:
739:
649:
629:
600:
568:
531:
492:
456:
415:
318:
290:
270:
242:
218:
180:
163:
160:Monte-Carlo simulation
151:
144:
96:
76:
30:Cauchy-Crofton formula
2951:Differential geometry
2733:Luis Santaló (1976),
2678:
2658:
2632:
2606:
2396:
2373:
2350:
2321:
2301:
2281:
2233:
2149:
2129:
2098:
2072:
2043:
2023:
2003:
1969:
1847:
1827:
1807:
1765:
1640:
1638:{\displaystyle S_{2}}
1613:
1611:{\displaystyle S_{1}}
1586:
1566:
1564:{\displaystyle S_{2}}
1539:
1537:{\displaystyle S_{1}}
1512:
1418:
1392:
1283:
1254:
1252:{\displaystyle E^{2}}
1227:
1225:{\displaystyle E^{2}}
1200:
1159:
1157:{\displaystyle (x,y)}
1127:
1107:
1066:
992:
938:
936:{\displaystyle E^{2}}
884:
740:
650:
630:
601:
569:
532:
493:
457:
416:
319:
291:
271:
243:
219:
181:
157:
145:
97:
82:intersects the curve
77:
55:
2714:Steinhaus longimeter
2667:
2641:
2615:
2476:unit sphere in
2405:
2382:
2359:
2330:
2310:
2290:
2242:
2207:
2138:
2134:, with equality iff
2111:
2081:
2052:
2032:
2012:
1978:
1856:
1836:
1816:
1781:
1649:
1622:
1595:
1575:
1548:
1521:
1481:
1401:
1292:
1266:
1236:
1209:
1168:
1136:
1116:
1075:
1001:
951:
920:
749:
659:
639:
610:
578:
545:
512:
473:
434:
335:
308:
280:
260:
232:
201:
170:
106:
86:
60:
2821:on 20 November 2020
2656:{\displaystyle n=3}
2630:{\displaystyle n=2}
2096:{\displaystyle n=2}
1416:{\displaystyle C,D}
1288:in the plane, then
1281:{\displaystyle C,D}
465:is invariant under
2874:Tabachnikov, Serge
2854:10.1007/BF01090671
2737:, Addison-Wesley,
2673:
2653:
2627:
2601:
2505:unit ball in
2391:
2371:{\displaystyle dp}
2368:
2345:
2316:
2296:
2276:
2228:
2144:
2124:
2093:
2067:
2038:
2018:
1998:
1964:
1842:
1822:
1802:
1760:
1635:
1608:
1581:
1561:
1534:
1507:
1413:
1387:
1278:
1249:
1222:
1195:
1154:
1122:
1102:
1061:
987:
933:
879:
784:unit ball in
735:
645:
625:
596:
564:
527:
488:
452:
411:
314:
286:
266:
238:
214:
176:
164:
152:
140:
102:twice, therefore,
92:
72:
2941:Integral geometry
2676:{\displaystyle n}
2599:
2593:
2572:
2545:
2532:
2506:
2477:
2461:
2319:{\displaystyle T}
2299:{\displaystyle S}
2279:{\displaystyle E}
2158:Barbier's theorem
2147:{\displaystyle S}
2122:
2041:{\displaystyle S}
2021:{\displaystyle S}
2001:{\displaystyle E}
1990:
1962:
1953:
1896:
1874:
1845:{\displaystyle P}
1825:{\displaystyle l}
1758:
1690:
1667:
1584:{\displaystyle l}
1453:Finsler manifolds
997:, such that each
877:
873:
842:
811:
785:
648:{\displaystyle S}
500:kinematic measure
426:differential form
364:
289:{\displaystyle p}
256:by the direction
34:integral geometry
2958:
2908:
2891:
2865:
2864:
2837:
2831:
2830:
2828:
2826:
2820:
2814:. Archived from
2806:(7): 1042–1044.
2797:
2788:
2782:
2781:
2754:
2748:
2747:
2730:
2682:
2680:
2679:
2674:
2662:
2660:
2659:
2654:
2636:
2634:
2633:
2628:
2610:
2608:
2607:
2602:
2600:
2598:
2594:
2586:
2577:
2573:
2568:
2557:
2548:
2546:
2541:
2533:
2531:
2530:
2525:
2524:
2513:
2507:
2504:
2502:
2496:
2495:
2490:
2489:
2484:
2478:
2475:
2473:
2467:
2462:
2460:
2456:
2439:
2427:
2426:
2415:
2409:
2400:
2398:
2397:
2392:
2377:
2375:
2374:
2369:
2354:
2352:
2351:
2346:
2344:
2343:
2338:
2325:
2323:
2322:
2317:
2305:
2303:
2302:
2297:
2285:
2283:
2282:
2277:
2272:
2255:
2237:
2235:
2234:
2229:
2227:
2226:
2221:
2170:
2153:
2151:
2150:
2145:
2133:
2131:
2130:
2125:
2123:
2115:
2102:
2100:
2099:
2094:
2076:
2074:
2073:
2068:
2066:
2065:
2060:
2047:
2045:
2044:
2039:
2027:
2025:
2024:
2019:
2007:
2005:
2004:
1999:
1991:
1988:
1973:
1971:
1970:
1965:
1963:
1961:
1954:
1951:
1940:
1929:
1923:
1922:
1914:
1908:
1897:
1894:
1883:
1875:
1872:
1851:
1849:
1848:
1843:
1831:
1829:
1828:
1823:
1811:
1809:
1808:
1803:
1801:
1800:
1795:
1769:
1767:
1766:
1761:
1759:
1757:
1753:
1752:
1733:
1729:
1728:
1709:
1701:
1700:
1691:
1688:
1683:
1678:
1677:
1668:
1665:
1644:
1642:
1641:
1636:
1634:
1633:
1617:
1615:
1614:
1609:
1607:
1606:
1590:
1588:
1587:
1582:
1570:
1568:
1567:
1562:
1560:
1559:
1543:
1541:
1540:
1535:
1533:
1532:
1516:
1514:
1513:
1508:
1506:
1505:
1493:
1492:
1422:
1420:
1419:
1414:
1396:
1394:
1393:
1388:
1386:
1378:
1370:
1362:
1345:
1322:
1317:
1316:
1315:
1314:
1287:
1285:
1284:
1279:
1258:
1256:
1255:
1250:
1248:
1247:
1231:
1229:
1228:
1223:
1221:
1220:
1204:
1202:
1201:
1196:
1163:
1161:
1160:
1155:
1131:
1129:
1128:
1123:
1111:
1109:
1108:
1103:
1070:
1068:
1067:
1062:
1060:
1059:
1054:
996:
994:
993:
988:
986:
985:
980:
942:
940:
939:
934:
932:
931:
888:
886:
885:
880:
878:
876:
875:
874:
869:
858:
848:
847:
843:
838:
827:
817:
812:
810:
809:
804:
803:
792:
786:
783:
781:
766:
761:
760:
744:
742:
741:
736:
702:
701:
689:
688:
654:
652:
651:
646:
634:
632:
631:
626:
624:
623:
618:
605:
603:
602:
597:
573:
571:
570:
565:
563:
562:
536:
534:
533:
528:
526:
525:
520:
497:
495:
494:
489:
487:
486:
481:
461:
459:
458:
453:
420:
418:
417:
412:
378:
377:
365:
357:
323:
321:
320:
315:
295:
293:
292:
287:
275:
273:
272:
267:
247:
245:
244:
239:
223:
221:
220:
215:
213:
212:
185:
183:
182:
177:
149:
147:
146:
141:
118:
117:
101:
99:
98:
93:
81:
79:
78:
73:
2966:
2965:
2961:
2960:
2959:
2957:
2956:
2955:
2931:
2930:
2915:
2894:
2888:
2872:
2869:
2868:
2839:
2838:
2834:
2824:
2822:
2818:
2795:
2790:
2789:
2785:
2756:
2755:
2751:
2745:
2732:
2731:
2727:
2722:
2704:Radon transform
2698:Buffon's noodle
2694:
2665:
2664:
2639:
2638:
2613:
2612:
2578:
2558:
2549:
2508:
2497:
2479:
2468:
2428:
2410:
2403:
2402:
2380:
2379:
2357:
2356:
2333:
2328:
2327:
2308:
2307:
2288:
2287:
2240:
2239:
2216:
2205:
2204:
2168:
2136:
2135:
2109:
2108:
2079:
2078:
2055:
2050:
2049:
2030:
2029:
2010:
2009:
1976:
1975:
1924:
1909:
1854:
1853:
1834:
1833:
1814:
1813:
1790:
1779:
1778:
1744:
1734:
1720:
1710:
1692:
1669:
1647:
1646:
1625:
1620:
1619:
1598:
1593:
1592:
1573:
1572:
1551:
1546:
1545:
1524:
1519:
1518:
1497:
1484:
1479:
1478:
1468:
1429:
1399:
1398:
1306:
1295:
1290:
1289:
1264:
1263:
1239:
1234:
1233:
1212:
1207:
1206:
1166:
1165:
1134:
1133:
1114:
1113:
1073:
1072:
1049:
999:
998:
975:
949:
948:
945:Euclidean group
923:
918:
917:
914:
894:
859:
853:
849:
828:
818:
787:
770:
752:
747:
746:
693:
680:
657:
656:
637:
636:
613:
608:
607:
576:
575:
548:
543:
542:
515:
510:
509:
476:
471:
470:
432:
431:
369:
333:
332:
324:in terms of an
306:
305:
278:
277:
258:
257:
230:
229:
204:
199:
198:
168:
167:
109:
104:
103:
84:
83:
58:
57:
50:
44:intersects it.
22:Crofton formula
12:
11:
5:
2964:
2962:
2954:
2953:
2948:
2946:Measure theory
2943:
2933:
2932:
2929:
2928:
2922:
2914:
2913:External links
2911:
2910:
2909:
2892:
2886:
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2258:
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2250:
2247:
2225:
2220:
2215:
2212:
2201:
2182:
2175:
2167:has perimeter
2155:
2143:
2121:
2118:
2092:
2089:
2086:
2064:
2059:
2037:
2017:
1997:
1994:
1989:width of
1986:
1983:
1960:
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1663:
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1628:
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1601:
1580:
1558:
1554:
1544:nested inside
1531:
1527:
1504:
1500:
1496:
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1188:
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926:
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893:
890:
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841:
837:
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755:
734:
731:
728:
724:
721:
717:
714:
711:
708:
705:
700:
696:
692:
687:
683:
679:
676:
673:
670:
667:
664:
644:
622:
617:
595:
592:
589:
586:
583:
561:
558:
555:
551:
539:tangent bundle
524:
519:
485:
480:
463:
462:
451:
448:
445:
442:
439:
422:
421:
410:
407:
404:
400:
397:
393:
390:
387:
384:
381:
376:
372:
368:
363:
360:
355:
352:
349:
346:
343:
340:
313:
285:
265:
237:
211:
207:
175:
139:
136:
133:
130:
127:
124:
121:
116:
112:
91:
71:
68:
65:
49:
46:
26:Morgan Crofton
24:, named after
13:
10:
9:
6:
4:
3:
2:
2963:
2952:
2949:
2947:
2944:
2942:
2939:
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2926:
2923:
2920:
2917:
2916:
2912:
2906:
2902:
2898:
2893:
2889:
2887:0-8218-3919-5
2883:
2879:
2875:
2871:
2870:
2863:
2859:
2855:
2851:
2847:
2843:
2836:
2833:
2817:
2813:
2809:
2805:
2801:
2794:
2787:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2753:
2750:
2746:
2744:0-201-13500-0
2740:
2736:
2729:
2726:
2719:
2715:
2712:
2709:
2705:
2701:
2699:
2696:
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2691:
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2670:
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2647:
2644:
2624:
2621:
2618:
2590:
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2542:
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2534:
2521:
2518:
2515:
2486:
2463:
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2429:
2419:
2388:
2385:
2365:
2362:
2340:
2313:
2293:
2262:
2256:
2245:
2223:
2213:
2210:
2202:
2199:
2195:
2191:
2187:
2183:
2180:
2176:
2173:
2166:
2163:
2159:
2156:
2141:
2119:
2116:
2106:
2090:
2087:
2084:
2062:
2035:
2015:
1992:
1981:
1955:
1944:
1941:
1933:
1915:
1904:
1898:
1890:
1887:
1884:
1876:
1868:
1862:
1859:
1839:
1819:
1797:
1787:
1784:
1776:
1773:
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1738:
1735:
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1630:
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1578:
1556:
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1529:
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1502:
1498:
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1471:
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1410:
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1404:
1379:
1371:
1363:
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1335:
1329:
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1323:
1311:
1307:
1303:
1300:
1296:
1275:
1272:
1269:
1260:
1244:
1240:
1217:
1213:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1148:
1145:
1142:
1119:
1096:
1093:
1090:
1087:
1084:
1078:
1071:defines some
1056:
1046:
1040:
1037:
1034:
1031:
1025:
1019:
1016:
1013:
1010:
1007:
982:
972:
966:
963:
960:
957:
946:
928:
924:
911:
909:
906:
904:
899:
891:
889:
870:
866:
863:
860:
854:
850:
839:
835:
832:
829:
813:
800:
797:
794:
774:
771:
767:
762:
757:
753:
732:
729:
726:
722:
719:
712:
709:
706:
698:
694:
690:
685:
681:
677:
671:
665:
662:
642:
620:
593:
590:
587:
584:
581:
559:
556:
553:
549:
540:
522:
506:
503:
501:
483:
468:
467:rigid motions
449:
446:
443:
440:
437:
430:
429:
428:
427:
408:
405:
402:
398:
395:
388:
385:
382:
374:
370:
366:
361:
358:
353:
347:
341:
338:
331:
330:
329:
327:
311:
304:of the curve
303:
299:
283:
263:
255:
251:
235:
227:
209:
205:
196:
192:
189:
173:
161:
156:
137:
134:
128:
125:
122:
114:
110:
89:
69:
66:
63:
54:
47:
45:
43:
39:
35:
31:
27:
23:
19:
2896:
2877:
2845:
2841:
2835:
2823:. Retrieved
2816:the original
2803:
2799:
2786:
2762:
2758:
2752:
2734:
2728:
2707:
2378:, then over
2197:
2193:
2189:
2171:
2164:
1469:
1466:Applications
1461:
1446:
1443:
1438:almost every
1430:
1261:
1112:: rotate by
915:
907:
895:
892:Proof sketch
507:
504:
499:
464:
423:
253:
249:
225:
194:
165:
29:
21:
15:
2186:convex hull
1427:Other forms
903:unit circle
191:plane curve
188:rectifiable
18:mathematics
2935:Categories
2720:References
2306:(that is,
2154:is a disk.
1449:Riemannian
302:arc length
2812:0002-9920
2765:: 65–77,
2580:Γ
2551:Γ
2543:π
2519:−
2417:∂
2389:φ
2214:⊂
2103:, by the
1942:⋅
1931:∂
1788:⊂
1739:
1715:
1457:geodesics
1372:⋅
1327:∩
1304:∈
1297:∫
1193:φ
1187:∧
1178:∧
1120:φ
1085:φ
1047:×
1041:π
1026:∈
1008:φ
973:×
967:π
864:−
855:π
820:Γ
798:−
775:⋅
723:φ
707:φ
699:γ
691:∬
666:
588:∧
585:φ
557:−
444:∧
441:φ
399:φ
383:φ
375:γ
367:∬
348:γ
342:
312:γ
296:from the
264:φ
236:γ
210:γ
174:γ
123:φ
115:γ
90:γ
64:φ
48:Statement
2905:QA641.S3
2876:(2005).
2862:24484682
2692:See also
2401:, we get
2160:: Every
898:additive
326:integral
166:Suppose
38:expected
2848:: 1–5,
2779:0071801
2077:. When
1517:, with
1164:. Then
943:be the
537:is the
2903:
2884:
2860:
2825:7 June
2810:
2777:
2741:
2238:, let
1974:where
1812:, let
339:length
298:origin
197:, let
20:, the
2858:S2CID
2819:(PDF)
2796:(PDF)
1433:cover
745:where
186:is a
2882:ISBN
2827:2022
2808:ISSN
2739:ISBN
2702:The
2184:The
2177:The
1736:area
1712:area
1645:, is
916:Let
663:area
424:The
248:and
42:line
2901:LCC
2850:doi
2767:doi
541:of
469:of
16:In
2937::
2856:,
2846:25
2844:,
2804:67
2802:.
2798:.
2775:MR
2773:,
2761:,
1459:.
1259:.
905:.
502:.
2907:.
2890:.
2852::
2829:.
2769::
2763:9
2708:k
2687:.
2671:n
2651:3
2648:=
2645:n
2625:2
2622:=
2619:n
2596:)
2591:2
2588:n
2583:(
2575:)
2570:2
2566:1
2563:+
2560:n
2554:(
2538:2
2535:=
2528:|
2522:1
2516:n
2511:R
2500:|
2493:|
2487:n
2482:R
2471:|
2464:=
2458:]
2454:|
2450:)
2447:S
2444:(
2441:T
2437:|
2433:[
2430:E
2424:|
2420:S
2413:|
2386:d
2366:p
2363:d
2341:n
2336:R
2314:T
2294:S
2274:]
2270:|
2266:)
2263:S
2260:(
2257:T
2253:|
2249:[
2246:E
2224:n
2219:R
2211:S
2198:C
2194:C
2190:C
2174:.
2172:w
2169:π
2165:w
2142:S
2120:2
2117:1
2091:2
2088:=
2085:n
2063:n
2058:R
2036:S
2016:S
1996:]
1993:S
1985:[
1982:E
1959:]
1956:S
1948:[
1945:E
1938:|
1934:S
1927:|
1920:|
1916:S
1912:|
1905:=
1902:)
1899:S
1891:P
1888:,
1885:l
1881:|
1877:P
1869:l
1866:(
1863:r
1860:P
1840:P
1820:l
1798:n
1793:R
1785:S
1774:.
1755:)
1750:2
1746:S
1742:(
1731:)
1726:1
1722:S
1718:(
1706:=
1703:)
1698:2
1694:S
1685:l
1681:|
1675:1
1671:S
1662:l
1659:(
1656:r
1653:P
1631:2
1627:S
1604:1
1600:S
1579:l
1557:2
1553:S
1530:1
1526:S
1503:2
1499:S
1495:,
1490:1
1486:S
1411:D
1408:,
1405:C
1384:|
1380:D
1376:|
1368:|
1364:C
1360:|
1356:4
1353:=
1350:T
1347:d
1343:|
1339:)
1336:D
1333:(
1330:T
1324:C
1320:|
1312:2
1308:E
1301:T
1276:D
1273:,
1270:C
1245:2
1241:E
1218:2
1214:E
1190:d
1184:y
1181:d
1175:x
1172:d
1152:)
1149:y
1146:,
1143:x
1140:(
1100:)
1097:y
1094:,
1091:x
1088:,
1082:(
1079:T
1057:2
1052:R
1044:)
1038:2
1035:,
1032:0
1029:[
1023:)
1020:y
1017:,
1014:x
1011:,
1005:(
983:2
978:R
970:)
964:2
961:,
958:0
955:[
929:2
925:E
871:2
867:1
861:n
851:2
845:)
840:2
836:1
833:+
830:n
824:(
814:=
807:|
801:1
795:n
790:R
779:|
772:2
768:1
763:=
758:n
754:C
733:.
730:p
727:d
720:d
716:)
713:p
710:,
704:(
695:n
686:n
682:C
678:=
675:)
672:S
669:(
643:S
621:n
616:R
594:p
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450:p
447:d
438:d
409:.
406:p
403:d
396:d
392:)
389:p
386:,
380:(
371:n
362:4
359:1
354:=
351:)
345:(
284:p
254:ℓ
250:ℓ
226:ℓ
224:(
206:n
195:ℓ
162:.
150:.
138:2
135:=
132:)
129:p
126:,
120:(
111:n
70:p
67:,
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