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1811:
Fenchel's
Theorem Theorem: The total curvature of a regular closed space curve C is greater than or equal to 2π.
1033:
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64:
1352:
650:{\displaystyle \textstyle 0=(\alpha (1)-\alpha (0))\cdot v=\int _{0}^{L}\gamma (t)\cdot v\,\mathrm {d} t>0}
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are symmetric about the axis through the poles. By the previous paragraph, at least one of the two curves
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is the length of the curve. The only curves of this type whose total absolute curvature equals
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1761:(Third edition of 1975 original ed.). Wilmington, DE: Publish or Perish, Inc.
265:
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60:
1567:(Revised & updated second edition of 1976 original ed.). Mineola, NY:
1720:
156:
1623:
39:
1502:{\displaystyle l(\gamma )=2l(\gamma _{0})=l(\gamma _{1})\geq 2\pi }
19:
This article is about the concept in geometry. For the concept in
1759:
A comprehensive introduction to differential geometry. Vol. III
1711:(Revised second edition of 1966 original ed.). Amsterdam:
1605:
412:{\displaystyle \gamma ={\dot {\alpha }}:\to \mathbb {S} ^{2}}
843:
have the same length. By rotating the sphere, we may assume
16:
Gives the average curvature of any closed convex plane curve
279:, which says that if a closed smooth simple space curve is
475:
does not lie in an open hemisphere. If so, then there is
92:, with equality if and only if it is a convex plane curve
1326:, which is the length of the great semicircle between
549:
1660:"On the differential geometry of closed space curves"
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72:
1606:"Über Krümmung und Windung geschlossener Raumkurven"
283:, then the total absolute curvature is greater than
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1565:Differential geometry of curves & surfaces
1276:{\displaystyle l(\gamma _{1})=2l(\gamma _{0})}
423:). The total absolute curvature is its length
1665:Bulletin of the American Mathematical Society
1172:, and the north pole, forming a closed curve
8:
341:{\displaystyle \alpha :\to \mathbb {R} ^{3}}
30:
657:, a contradiction. This also shows that if
1077:{\displaystyle l(\gamma )=2l(\gamma _{0})}
983:intersects with the equator at some point
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1387:{\displaystyle l(\gamma _{1})\geq 2\pi }
275:The Fenchel theorem is enhanced by the
1529:lies in a closed hemisphere, and thus
702:{\displaystyle \gamma \cdot v\equiv 0}
419:is also a closed smooth curve (called
1700:; see especially equation 13, page 49
503:{\displaystyle v\in \mathbb {S} ^{2}}
132:, stating that it is always at least
7:
1822:"2. Curvature and Fenchel's Theorem"
535:{\displaystyle \gamma \cdot v>0}
233:and whose average curvature equals
677:lies in a closed hemisphere, then
633:
14:
1850:Theorems in differential geometry
1826:Brown University Math Department
1801:Brown University Math Department
1709:Elementary differential geometry
1679:10.1090/S0002-9904-1951-09440-9
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1:
1509:, and if equality holds then
1394:, and if equality holds then
268:. The theorem is named after
1421:does not cross the equator.
1199:containing antipodal points
295:Given a closed smooth curve
272:, who published it in 1929.
155:. Equivalently, the average
1414:{\displaystyle \gamma _{0}}
1192:{\displaystyle \gamma _{1}}
1107:{\displaystyle \gamma _{0}}
1023:{\displaystyle \gamma _{0}}
1881:
1855:Theorems in plane geometry
1165:{\displaystyle \gamma (T)}
1136:{\displaystyle \gamma (0)}
1003:. We denote this curve by
894:{\displaystyle \gamma (T)}
865:{\displaystyle \gamma (0)}
754:{\displaystyle \gamma (T)}
445:{\displaystyle l(\gamma )}
85:{\displaystyle \geq 2\pi }
18:
1114:across the plane through
976:{\displaystyle \gamma ()}
935:{\displaystyle \gamma ()}
836:{\displaystyle \gamma ()}
795:{\displaystyle \gamma ()}
25:Fenchel's duality theorem
21:mathematical optimization
1569:Dover Publications, Inc.
126:total absolute curvature
124:is an inequality on the
65:total absolute curvature
1865:Curvature (mathematics)
1797:"Differential Geometry"
1721:10.1016/C2009-0-05241-6
1542:{\displaystyle \alpha }
1522:{\displaystyle \gamma }
722:{\displaystyle \alpha }
670:{\displaystyle \gamma }
468:{\displaystyle \gamma }
257:{\displaystyle 2\pi /L}
183:{\displaystyle 2\pi /L}
1543:
1523:
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469:
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227:
204:
184:
149:
86:
1860:Theorems about curves
1611:Mathematische Annalen
1561:do Carmo, Manfredo P.
1544:
1524:
1504:
1416:
1389:
1344:
1342:{\displaystyle \pm p}
1321:
1301:
1299:{\displaystyle \pm p}
1283:. A curve connecting
1278:
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1215:{\displaystyle \pm p}
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348:with unit speed, the
343:
259:
228:
226:{\displaystyle 2\pi }
205:
185:
150:
148:{\displaystyle 2\pi }
118:differential geometry
87:
50:Differential geometry
1533:
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1398:
1353:
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1319:{\displaystyle \pi }
1310:
1306:has length at least
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281:nontrivially knotted
237:
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194:
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70:
612:
277:Fáry–Milnor theorem
128:of a closed smooth
32:
1818:Thomas F. Banchoff
1793:Thomas F. Banchoff
1624:10.1007/bf01454836
1549:is a plane curve.
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751:
729:is a plane curve.
719:
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532:
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421:tangent indicatrix
409:
338:
254:
223:
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180:
145:
82:
1730:978-0-12-088735-4
1577:978-0-486-80699-0
996:{\displaystyle p}
761:such that curves
732:Consider a point
373:
203:{\displaystyle L}
122:Fenchel's theorem
114:
113:
31:Fenchel's theorem
1872:
1835:
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1705:O'Neill, Barrett
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59:A smooth closed
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1791:
1769:
1755:Spivak, Michael
1753:
1731:
1703:
1656:Fenchel, Werner
1654:
1602:Fenchel, Werner
1600:
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97:First stated by
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11:
5:
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1857:
1852:
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1814:
1789:
1767:
1751:
1729:
1713:Academic Press
1701:
1652:
1618:(1): 238–252.
1598:
1576:
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1222:, with length
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270:Werner Fenchel
264:are the plane
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107:First proof in
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101:Werner Fenchel
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1768:0-914098-72-1
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1614:(in German).
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266:convex curves
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34:
26:
22:
1829:. Retrieved
1825:
1810:
1804:. Retrieved
1800:
1758:
1708:
1672:(1): 44–54.
1669:
1663:
1615:
1609:
1564:
1423:
1086:
731:
454:
294:
274:
159:is at least
121:
115:
1424:Therefore,
1087:We reflect
130:space curve
61:space curve
1844:Categories
1831:2024-05-26
1806:2024-05-26
1785:1213.53001
1747:1208.53003
1696:0042.40006
1632:55.0394.06
1594:1352.53002
1553:References
510:such that
455:The curve
1648:119908321
1537:α
1517:γ
1497:π
1491:≥
1479:γ
1457:γ
1438:γ
1403:γ
1382:π
1376:≥
1364:γ
1334:±
1314:π
1291:±
1262:γ
1237:γ
1207:±
1181:γ
1151:γ
1122:γ
1096:γ
1063:γ
1044:γ
1012:γ
950:γ
909:γ
880:γ
851:γ
810:γ
769:γ
740:γ
717:α
694:≡
688:⋅
685:γ
665:γ
626:⋅
614:γ
600:∫
590:⋅
575:α
572:−
560:α
521:⋅
518:γ
486:∈
463:γ
437:γ
395:→
371:˙
368:α
359:γ
324:→
303:α
244:π
221:π
170:π
157:curvature
143:π
80:π
74:≥
56:Statement
1757:(1999).
1707:(2006).
1658:(1951).
1604:(1929).
1563:(2016).
350:velocity
190:, where
1777:0532832
1739:2351345
1688:0040040
1640:1512528
1586:3837152
1030:. Then
285:4π
40:Theorem
1783:
1775:
1765:
1745:
1737:
1727:
1694:
1686:
1646:
1638:
1630:
1592:
1584:
1574:
23:, see
1644:S2CID
1349:. So
709:, so
542:, so
291:Proof
46:Field
1763:ISBN
1725:ISBN
1572:ISBN
942:and
872:and
802:and
641:>
527:>
110:1929
63:has
36:Type
1781:Zbl
1743:Zbl
1717:doi
1692:Zbl
1674:doi
1628:JFM
1620:doi
1616:101
1590:Zbl
116:In
1846::
1824:.
1820:.
1809:.
1799:.
1795:.
1779:.
1773:MR
1771:.
1741:.
1735:MR
1733:.
1723:.
1715:.
1690:.
1684:MR
1682:.
1670:57
1668:.
1662:.
1642:.
1636:MR
1634:.
1626:.
1608:.
1588:.
1582:MR
1580:.
1143:,
1084:.
452:.
287:.
120:,
1834:.
1787:.
1749:.
1719::
1698:.
1676::
1650:.
1622::
1596:.
1494:2
1488:)
1483:1
1475:(
1472:l
1469:=
1466:)
1461:0
1453:(
1450:l
1447:2
1444:=
1441:)
1435:(
1432:l
1407:0
1379:2
1373:)
1368:1
1360:(
1357:l
1337:p
1294:p
1271:)
1266:0
1258:(
1255:l
1252:2
1249:=
1246:)
1241:1
1233:(
1230:l
1210:p
1185:1
1160:)
1157:T
1154:(
1131:)
1128:0
1125:(
1100:0
1072:)
1067:0
1059:(
1056:l
1053:2
1050:=
1047:)
1041:(
1038:l
1016:0
991:p
971:)
968:]
965:L
962:,
959:T
956:[
953:(
930:)
927:]
924:T
921:,
918:0
915:[
912:(
889:)
886:T
883:(
860:)
857:0
854:(
831:)
828:]
825:L
822:,
819:T
816:[
813:(
790:)
787:]
784:T
781:,
778:0
775:[
772:(
749:)
746:T
743:(
697:0
691:v
644:0
638:t
634:d
629:v
623:)
620:t
617:(
609:L
604:0
596:=
593:v
587:)
584:)
581:0
578:(
569:)
566:1
563:(
557:(
554:=
551:0
530:0
524:v
496:2
491:S
483:v
440:)
434:(
431:l
405:2
400:S
392:]
389:L
386:,
383:0
380:[
377::
362:=
334:3
329:R
321:]
318:L
315:,
312:0
309:[
306::
252:L
248:/
241:2
218:2
198:L
178:L
174:/
167:2
140:2
77:2
27:.
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