a145a174a89 a77a34a88
2001a3510a21926a134
1 a178a193a77a34
a183a46a89a1a39a203a76a27a89a24a154a135a232a73a243a1a91a222a123a97a126,a89a52a58a158a7a155a123a21a48.
a0a143,a44a33a59a92a123a60a89a13a41,a183a162a210a89a178a193a222a123a77a34,a34a247a178a193a222a139a24a30a77
a34x(t) = (x1(t),x2(t)),a253a243a89a199a67a222a196a243a123a60a89,a126a135a232a73a123a155a17,a210a52a89
a30a77a34a139a30a35a34,a35a34a139a199a35a37a222,a233a149a35a37a222,a183a162a82a89a199a37a222a52a92a160a37
a222,a199a123a159a221a521,a15a210a52a82a143a92a160a35a37a222,a149a52a183a162a127a119a34a38a178a43a41xa44a43
a196a140a159sa123a60a106a123a155,a89a210a44a43a89a199a92a160a35a37a222a210a52x a233s a123a135a73dxds,a253a92a160
a35a37a222a143
e1 = (dx1ds,dx2ds ),(e1,e1) = 1,sa52a140a159,eqno(3.1)
a0a143a14a143a248a207a196a89a30a35a34a17a218a105a59a92,a0a210a52a139a234a24a199a92a160a35a37a222a131a128,a63
a34a247a152a42a178a193a52a189a53a123,a253a139a24a199a221a196a123a48a53,a0a143a199a210a139a24a199a92a160a27a53a222,
a15a210a52a203a199a18a134a123a0a199a92a160a37a222,a25a243,a183a119e1a52a92a160a35a37a222,e2 a52a92a160a27
a37a222,a149a52a7a122a116a89a30a35a34a123a117a159,a145a24a71a47a60a210a52a178e1a89a199a60a106a233a149sa242a70
a135a73,a0a143a242a70a135a73a131a128,a104a108a89a52a24a199a99a123a37a222,a79a143e1a52a24a199a92a160a37a222,
a196a49(e1,e1) = 1,a0a143a178a199a135a73a24a5a6,a183a162a210a122a116de1ds a51e1a18a134,a196a49a199a24
a189a243a27a34a123a48a53,a79a36,a183a162a210a139de1ds a127a149a92a160a27a37a222e2a123a245a106,a89a199a245a106a52
a140a159a123a24a199a60a106,a183a162a119k(s),a89a199a245a106a119a22
de1
ds = ke2,e2 a52a92a160a27a37a222,(e1,e2) = 0,(3.2)
ka89a199a60a106a24a196a119a41a77a71,a52a89a30a77a34a243a89a199a178a193a176a62a33a7a155a123a24a199a117a159,
a52a140a159a123a24a199a60a106.
a243a11:
a233a149a201a189a123a77a34a48a199,a201a241a77a71k a123a218a42,[a10a43a213ka52a77a34a48a199a123a24a126a90
a19a126a135a73a123a24a199a60a106]
1
2 a56a45a77a34
a44a178a193a77a34a242a159a24a90a14a143a248a17a218a183a162a23a23a56a45a123a60a111,a34a247a183a162a25a243a139a24
a30a77a34a52a56a45a123a77a34x(t) = (x1(t),x2(t),x3(t)),a2433a145a56a45a176a139a35a34,a196a49
a89a199a56a45a123a43a41x,y a210a44a43a143a107a106t a123a60a106,a89a74a192a220a28a162a76a150a209a127a119,a183
a242a167a243a24a5a6,a196a49x(t) a52a24a199a37a222,a199a123a73a222a210a52a154a123a43a41,a13a154a52t a123
a60a106,a149a52a199a210a42a24a30a35a34,a893a199a73a222a119a41xi(t)(i = 1,2,3),x(t)a52a24a199a37
a222,a52a107a106ta123a60a106,a199a1233a199a73a222a210a52a178a154a123a43a41a44a43a143t a123a60a106,a0a143a14a143
a159a113a17a218a51a248a123a48a27a210a52a233a89a30a77a34a126a135a232a73,a34a189a77a34a139a35a34,a63a37a243a35
a34a222a193a139a92a160a37a222,a253a139a92a160a35a37a222,a233a77a34a139a199a48a53,a24a134a89a143a18,a215a248a24
a199a48a53,a7a152a128a107a106t a52a15a28a123a24a199a48a53,a15a13a211a131,a183a162a210a139a24a199a92a160a35
a37a222,a119a199e1,a0a143a203a178a193a51a248a123a60a89,a178e1a89a199a60a106a233s a242a70a135a73,a210a127a149
a128a233s a70a19a126a123a135a73,a79a143e1 a52a92a160a37a222,a196a49a122a116a123a89a199a135a73a203e1 a52a18a134
a123,a13a233a149a24a30a56a45a165a123a77a34,a199a123a35a34a52a24a30,a199a123a27a34a139a195a106a199,a217a34
a178a44a89a24a154a203a35a34a18a134a123a27a34a139a195a106a199,a0a143a217a165a139a24a199a52de1ds,a199a52a88a113a92
a104a189a123,a132a149a51a248a123a174a132,a183a178a89a199a48a53a123a92a160a37a222a119a41e2,a0a143a199a210a52ke2,
a253a44a49a85a196 de
1
ds = ke2,(3.3)
e2a52a217a165a24a30a27a34,a183a162a119a199a143a204a27a34(principal normal),a13ka89a199a60a106a166a178
a193a123a60a111a24a248,a119a41a77a34a123a77a71,a196a49a25a243,a183a139a24a199a35a34a123a48a53a90a24a199a204
a27a34a48a53,a243a174a145a56a45a210a139a12a105a24a199a48a53a203a89a220a199a48a53a18a134,a183a162a47a158a119
a199binormal,a15a13a211a131,a63a243a12a105a24a199a27a34,a196a49a199a139a220a199a27a34,a0a143a89a220a199
a27a34a196a196a123a178a193a210a52a27a178a193,a77a34a521a145a123,a35a34a521a145a123,a196a49a27a34a522a145a123,
a52a199a178a193,a0a143a139a24a199e3 = e1 ×e2,a217a165e1 ×e2 a52a37a222a232,a233a149a220a199a48a53a123
a155,a139a24a199a104a189a123a145a174a199a48a53a52a203a199a162a18a134a123,a63a37a38a122e1,e2,e3a196a24a199a116
a98a248a58,a183a162a34a189a89a199a56a45a52a189a53a123,a141a67,a38a67a209a139a24a199a104a189a123a48a53,a47
a158a183a162a209a126a141a67,a196a49a210a139a24a199a104a189a123e3a119a22
(ei,ej) = δij,(3.4)
2
a79a36a243a207a196a174a145a1a91a123a30a127,a89a248a123a174a199a144a35a18a134a123a92a160a37a222a196a196a123a67a111
a58a158a7a155,a89a52a143a31a143a17a218a89a248a24a199a192a220a183a119a199a41a37,a28a178a24a199a41a37a194a116a12
a24a199a41a37a123a228a196a52a113a92a104a189a123,a0a143a174a145a56a45a33a7a155a123a117a159a210a52a174a145a56a45
a123a228a196,a183a162a7a207a196a123a1a91a117a159a52a178a44a228a196a88a35a123,a196a49a210a7a127a119a31a143a30a127
a28a44a49a178a89a199a192a220a194a116a12a105a24a199a160a152,a31a143a30a127a199a123a160a152a35a139a243a149a24a199a228
a196,a13a41a37a210a52a89a199a228a196a137a219a123a44a43a123a48a27,a28a7a21a234a194a44a86a210a44a43a89a199a41
a37a194a116a12a105a24a199a41a37a123a228a196a52a113a92a104a189a123,a23a108,a144a139a24a199a228a196a63a37a24a189
a139a24a199a228a196a178a24a199a41a37a35a143a12a105a24a199a41a37,a79a143a7a207a196a56a45a178a44a228a196a88a35
a123a117a159,a196a49a137a219a123a48a27a210a52a188a126a41a37,a0a143a34a38a183a25a243a139a24a30a77a34,a183a88
a144a139a24a199a41a37,a89a74a41a37a52a30a45a123a60a106,a243a0a176a228a196,a79a36eia89a174a199a42a143a41
a37a123a37a222a209a52a30a45t a123a60a106,a149a52a183a44a49a70a199a123a135a73deidt,deidt a52a199a37a222,a79a143
a52(e1,e2,e3)a52a199a41a37,a196a49a127a91a24a199a37a222a210a44a49a85a196e1,e2,e3a893a199a37a222a123
a34a117a28a92,a0a143a183a178a199a227a135a77a18a24a154,a210a178a199a85a196deia127a149eja123a34a117a28a92:
dei = ωijek,(3.5)
a89a199a28a92a123a248a106a52a24a39a135a73a42,a89a52a79a143a183a25a243a41a234a24a5a135a73,a132a89a60a106a34
a122a116a24a39a135a73a42,a0a143a89a248a123a24a39a135a73a42a183a119a199ωij,a89a210a44a43a220a199a35a249a123
a41a37a123a7a248,a28a139a24a199ei a123a41a37,a107a30a139a199a35a249a123a41a37ei + dei,a0a143dei a44
a143eia123a60a106a123a30a127,a199a123a28a92a123a248a106a210a52a24a39a135a73a42,ωij a52a24a39a135a73a42,a243
a89a176,a183a123a141a41i,ja209a52a441 a1163,3 a52a183a162a56a45a123a145a106,a196a49ωij a24a225a1399a199:
i,j a154a24a199a441a1163,a196a49a24a225a1399a199.a899a199a24a39a135a73a42a52a139a7a248a123,a199a88a52a113
a92a127a63a123,a89a199a7a248a210a52
ωij + ωji = 0,(3.6)
a196a49a34a38a28a178ωij a23a196a24a1993 × 3 a123a48a106a123a195a163a123a155,a89a199a48a106a52a39a233a193a123.
a89a24a28a48a199a105a142a52a44ei,ej a123a19a232a127a149δija122a116,a178a89a199a7a248(a48a199(3.4))a135a73
a123a155,a210a193a47a122a116a89a24a28a117a159,a89a210a52a128(ωij) a52a24a1993 × 3 a48a106a123a195a163,a89
a199a48a106a123a195a163a209a52a24a39a135a73a42,a63a37a89a199a48a106a114a199a52a39a233a193a123,a166a233a155a128,
a89a199a48a106a204a110a34a222a123a195a163a520,a217a199a17,a132a149a39a233a193,a139ω12 =?ω21a127,a52a39
a233a193a123,a79a36a183a162a25a243a139a24a199a92a160a35a37a222,a139a24a199a204a27a37a222,a164a139a24a199a166a199
3
a162a18a134a123a196a24a199a41a37a123e3 = e1 × e2(a37a222a232),a233a149a89a199a41a37,a183a178a199a123a174a199
a48a199a209a233a140a159a70a60a106( a135a73),a210a44a49a178a89a199a60a106a44a143e1,e2,e3 a123a24a199a34a117a28
a92,a89a199a28a92a52a89a248a123a24a28a48a199a213a145a24a199a48a199a52de1 = ke2,a79a143a183a162a123a48a106
a52a39a233a193a123,a204a233a110a34a222a123a195a163a209a520,a196a49ω13 = 0,a98a52a217a198a123a195a163a7a213
a63e3a123a160a152,a132a149e1,e2 a160a10a123a7a248,ω13 a52a127a1490a123,a79a36ω31 a15a127a1490,a196a49
a89a199a48a106a85a241a117,a210a52a183a5a193a1233a199a48a199:
de1
ds = ke2
de2
ds =?ke1 + ωe3 (3.7)
de3
ds =?ωe2
a89a28a48a199a52a104a35a77a34a88a21a48a123a30a127a33a0a123a24a28a48a199,a183a162a47a158a119a199Frenet
a48a199,Fren eta52a27a41a123a24a199a106a166a27,a183a46a89a52a198a123a123a70a44a88a169,a198a88a24a189a52
a62a24a199a201a241a89a199a48a199,a104a30a139a1a199a124a41a241a89a199a211a42,a44a183a89a123,a28a162a44a49a23
a241a117a122a89a199a48a199a63a88a212a104a10,a79a36a183a248a234a77a71a49a105,a164a139a12a24a199a60a106ω,ω
a210a52a48a199a165a123a12a71(torison),a15a52a140a159a123a60a106,a52a44a43a56a45a123a77a34a243a228a196a107
a5a123a117a159,a196a49a56a45a77a34a139a220a199a60a106,a24a199a52a77a71,a12a24a199a52a12a71,a12a71a210a52
a195a85a199a14a143a248a88a52a24a30a178a193a77a34,a199a52a243a56a45a107a77a123a24a199a222,a196a49a56a45a77a34
a52a126a220a199a60a106a117a195a85a123,a199a162a137a219a143a195a85a89a56a45a123a117a159,a89a220a199a60a106a23a108a105
a173a7,a79a143a199a162a7a52a127a1490a123a30a127,a210a44a43a234a77a34a105a59a92a123a117a159,a7a52k = 0
a123a155,a89a77a34a210a196a143a134a34,a89a105a142a52a121a210,a183a5a193a201a241a121a210a213a79a143k = 0,a196
a49de1 = 0,a79a36a92a160a35a37a222e1a52a24a199a158a106,a79a143a89a248a199a123a135a73a98a127a1490,a0a143
a28a178a89a199a158a106e1 = Ca83a156a116e1 = dxdsa165,a242a178a199a232a73a24a5a6a210a122a116xa52s a123a24
a39a60a106a213
x = Cs,(3.8)
a196a49a89a210a52a24a30a134a34,a39a44a117a15a105a142a52a21a121a210a152a42a28a139a24a30a134a34a123a155,a199a123
a77a71k a52a127a1490a123,a79a36k = 0 a83a44a77a34a123a33a59a92a123a117a159,a210a52a134a34,a0a143a7
a213a63a123a52a243a189a12a71a123a30a127,a24a189a7k negationslash= 0,a165a52k = 0 a123,a149a52de1ds = 0,a15a210
a52a134a34a234,a89a30a199a210a150a139a27a6a189a204a27a34,a24a30a134a34a203a199a18a134a123a52a24a199a178a193,
4
a89a199a178a193a176a62a196a139a203a36a134a34a18a134a123a48a53a209a52a139a51a248a123a117a159,a196a49a210a150a139a204
a27a34,a79a36a15a88a21a189a12a71,a12a71a24a189a7a243a89a154a123a77a71k negationslash= 0a123a30a127a98a139a63a66.
a13a104a12a71a127a1490a123a30a127,a104a108a210a52a44a43a89a30a77a34a52a243a178a193a222a24a30a77a34,a5a193
a183a15a201a234a24a199a59a92a123a121a210,a79a143a12a71a89a199a60a106a52a243Frenet a218a42a123a145a174a199a218
a42a176,a196a49a132ω = 0 a44a127a36a30e3a52a199a158a106a123a37a222,a233a149e3 a89a199a158a106a37a222,a132
a149e3a52a24a199a27a34,a63a37a79a143a27a34a203a35a34a52a123a207a18a134a123a199a196a49e3 a203dxds a123a19a232
a52a123a207a127a1490,a79a36e3a7a52a127a149a158a106a123a155,a183a210a44a49a178a48a199
(e3,dxds) = 0 (3.9)
a232a73,a79a143e3a52a24a199a158a106a199a196a49a89a232a73a210a52e3 a203x a123a19a232a127a149a24a199a158a106,a79
a36a199a52a24a199a178a193a77a34,a149a52ω = 0a52a44a43a77a34a52a199a178a193a77a34,a39a44a117,a44a49
a105a142a52a121a210a178a193a77a34a123a12a71a520,a196a49a77a71a90a12a71a220a199a60a106a209a139a59a92a123a1a91
a117a159,a12a105a24a171a105a139a63a135a123a77a34a52ka166ωa209a88a1430,a98a209a52a158a106,a0a143a243a89a199
a60a111a131a5,a44a49a121a210a77a34a52a199a90a34,a210a52a89a248a59a92a123a135a232a73a123a97a126a243a9a212
a154a166a222a139a173a7a123a63a66,a79a143a183a162a127a119a9a212a154a166a123a24a199a204a7a123a154a92a212a52DNA.
DNAa52a220a30a90a34,a52a199a118a90a34,a89a52a9a212a222a58a158a228a253a131a25a54,a243a9a212a222,a89
a248a123a77a34a210a115a241a117a234,a79a36,a77a34a88a243a9a212a154a166a139a173a7a123a97a126,a15a210a52a76a27
a7a127a119a77a34a123a117a159a152a91a107a47DNAa123a154a166a117a159,a196a49a106a166a210a105a173a7a234,a77a34
a123a117a159a107a47a116a154a166a123a117a159,a131a217a52a243a154a166a176a62,a139a30a127a28a178DNAa35a228a234,a199
a123a117a159a210a149a35a234,a196a49a35a228a131a128,a106a166a117a159a210a21a9a149a35a199a199a123a154a166a117a159a15a149
a35,a255a88a199a162a152a91a149a35a199a89a52a243DNAa123a207a196a249a243a9a212a154a166a48a193a52a58a158a228a253
a123a175a11,a76a27a41a234a105a245a123a211a42,a25a243a252a24a253a135a9a212a154a166a123a86a7a32a13a117a123a155,
a210a23a68a139a24a199a228a253a123a218a42,a119a41Whitea218a42,Whitea52a183a123a24a199a166a9a123a166a9,
a198a41a89a199a211a42a52a198a70a44a88a169a123a24a199a136a42,a198a228a237a105a80,a198a89a199a136a42a35a143a9a212
a154a166a123a24a199a228a253a123a218a42,a183a25a243a88a21a89a89a199a136a42.
5
3 Feuchlla88a127a42,Croftona232a73a218a42a90Fary-Milnor
a189a174
a41a89a199a1a91a207a196a123a30a127,a139a74a173a7a117a159a7a255a88,a47a158a173a7a123a117a159a128a128a52a114a14
a1a91a123a117a159,a243a77a34a123a60a89a15a88a52a207a196a24a66a227a123a1a91,a183a162a25a243a178a222a193a123a89
a199a119a41a219a92a123a1a91a199a253a207a196a24a66a227a123a1a91a117a159,a89a30k,ωa209a52a140a159a123a60a106,
a126a199a162a207a196a199a123a117a159,a52a14a143a248a107a77,a88a44a205a173a7a123a52a207a196a114a30a77a34,a34a38
a139a24a30a77a34a52a85a20a123,a23a89a30a77a34a139a31a143a248a123a117a159,a89a176a139a199a58a158a7a155a123a218
a42,a233a149a89a30a85a20a123a77a34,a154a154a139a199a77a71,a79a143a85a20a234,a183a210a207a196a92a77a71,a253
a178a89a199a77a71a215a114a30a77a34a70a199a123a232a73,a149a52a210a139a196a162Feuchlla218a42
integraldisplay
c
kds ≥ 2pi,(3.10)
a210a52a128a89a199a232a73a139a199a5a142,a24a189a522pi,a89a199a134a250a89a229a117a210a52a152a42a28a24a154a77
a71a209a150a139,a89a30a77a34a210a134a34a18a5a86,a52a88a204a85a20a229a117,a28a7a134a34a18a5a86a21a234
a85a20a123a155,a15a7a178a199a107a44a117,a107a44a117a210a7a139a77a71,a0a143a152a42a21a234a107a44a117,a166a198
a117a143a48a233a222a117a123a155,a0a143a77a71a243a114a199a77a34a222a123a232a73a139a24a199a251a189a123a5a142,a89a199
a5a142a522pi,a89a210a52a196a162a123Feuchlla218a42,a183a7a121a210a89a199a218a42,a143a31a143a89a92a77a71
a7a139a24a199a5a142? a121a210a89a199a218a42a123a48a27a210a52a255a88a89a30a77a34a123a92a160a35a37a222,a253
a243a35a34a123a48a53a123a24a199a92a160a37a222,a204a7a10a39a210a52a188a126a176a132a110a243(Gau ss map).
a233a149a89a248a24a199a92a160a35a37a222,a243a251a189a154a82a203a89a199a35a37a222a52a178a113a123a0a74a92a160a37
a222,a196a49a28a139a142a245a92a160a37a222a52e1(s),a199a52a159a127a1491 a123a24a199a37a222,a196a49a28a34a38
a178a199a23a196a56a45a24a199a154,a199a210a52a243a92a160a69a222a123a24a199a154,a79a36a28a82a24a199a92a160a69,
a82e1(s)a23a196a92a160a69a222a123a24a199a154,a0a143a104a28a89a199a154a215a198a117a77a34a18a24a87a123a30a127,
e1(s)a243a92a160a69a222a196a24a30a77a34,a89a52a176a132a110a243a123a63a135,a253a196a162a123a176a132a110a243,a89
a30a77a34e1(s)a243a92a160a69a222a196a196a123a123a77a34a88a52a198a117a123a77a34,a166a234a24a171a234,a79a143a199
a123a48a199a52e1(s),a88a52x(s),e1(s) a119a22:
dσ2
ds2 = (
de1
ds,
de1
ds ) = k
2,σa52e1(s)a123a140a159,(3.11)
a92a160a69a193a152a42a139a24a199a76a203,a0a143a89a199a76a203a139a24a199a244a154,a183a119a199y,a15a210a52a128,
6
a76a203a52a6a204a203,y a52a89a199a76a203a123a244a154.a178a89a199a244a154a104a196a186a222a123a24a154,a0a143a77a34
a210a139a199a176a221,a89a199a176a221a250a82a196a52y(x),a52a199a19a232,a89a199a176a221a44a49a23a196a52a77a34
a222a123a24a199a60a106,a0a143a77a34a243a56a45a176a62a210a139a234a24a199a176a221,a183a162a89a77a34a52a85a20a123,
a104a108a139a199a33a176a154,a139a199a33a132a154,a243a33a176a154,a79a143a199a52a33a176a123,a23a108a199a123a35a34a52
a178a123,a243a33a132a154,a35a34a15a52a178a123,a196a49a210a139a220a199a248a142a154,a24a199a52a83a44a176a221a33
a176a123,a24a199a52a33a132a123,a13a166ya89a199a154a18a134a123a0a199a76a203,a24a189a203a92a160a69a222a123a77a34
a35a98,a79a143(y,dx) a89a199a19a232a520,a196a49ya203a69a193a123e1(s)a243a24a154a35a98,a79a36a243a92
a160a69a222a123a77a34a203a89a199a92a160a203a35a98,a98a52ya52a127a63a48a53,a196a49a183a162a69a222a123a176a132
a110a243a123a77a34a52a203a69a222a123a31a143a203a209a35a98,a89a52a79a143ya52a127a63a24a199a154,a149a52ya123a0
a199a6a204a203a98a183a162a89a30a77a34,a183a46a176a132a77a34a183a119a199γ,a199a203ya123a6a204a203a35a98,a79
a36a203a127a63a92a160a203a35a98,a89a52a105a7a155a123a24a199a117a159,a196a49γa89a30a77a34a24a189a7a35a104
a159,a199a243a69a222a88a52a24a30a127a91a123a77a34,a199a243a69a222a52a24a30a203a196a139a123a203a177(a76a203)a209
a35a98a123a77a34,a0a143a183a164a52a137a7a121a210a199a1392pi a89a199a5a142,a5a193a117a128a210a89a199a175a11.
a183a89a30a77a34γa52e1(s),a119a51a160a203,e1(s) a210a52a56a45a77a34a123a24a199a176a132a110a243,a199a52
a35a34a123a48a53a243a92a160a69a222a196a196a123a77a34,a13a89a30a77a34a119a41γ,a199a203a69a222a123a127a91a92
a160a203a209a35a98,a0a143a24a196a143,a89a30a77a34a35a104a159a234,a152a42a212a225,a199a150a139a89a199a117a159.
a128a199a139a24a1992pi a123a5a142,a52Croftona218a42a213
integraldisplay integraldisplay
= 4L,(3.12)
Croftona123a169a57a15a105a139a63a135,a88a52a24a199a116a42a123a106a166a169a57,a93a41a123a186a41a92
a86(Encyclopaedi a)a62a198a85a24a159a169a57,a52a7a149a1a91a150a71a123,a198a52a243a85(a1a91)a150
a71a123a169a57a30a127a178a198a123a136a42a85a243a176a62a234,a89a199a136a42a105a7a155,a166a233a155a128,a210a243a178
a193a222a89,a77a34a123a159a221a44a49a44a143a203a199a35a98a123a134a34a123a221a222,a159a221,a210a52a134a34a123a221
a222,a89a199a63a135a243a230a212a166a176,a163a83a243a26a166a209a139a97a126,a139a30a127,a28a252a14a222a123a192a220,
a7a175a199a139a245a76,a15a88a21a178a124a35a13,a52a88a80a222a123,a149a52a210a52a126a23a35a98a89a199a192a220
a123a34a42a143a221a222,a117a222a89a199a252a14a222a123a236a24a92a73a123a76a66,a89a24a92a73a106a166a24a196a119a41
a232a73a1a91,a88a52a135a73a1a91,a232a73a1a91a210a52a207a196a89a74a232a73a123a7a248a166a117a159,a232
a73a1a91a243a26a166a222a139a105a76a123a97a126,a139a105a245a229a236a102a126a102a126a89a174a88,a79a143a28a7a23
a124a123a62a14,a15a88a21a252a89a199a62a14a117a222,a210a252a34a243a199,a149a243a199a123a72a42a117a23a62a14a123
7
a76a66a249a217a199a123a117a159,a89a210a52a232a73a1a91,a0a143a183a25a243a128,(3.12)a52Croftona218a42,
a210a52a128a69a193a222a30a15a139a1a91,a52a69a193a1a91,a199a123a154a183a209a127a119,a199a123a134a34a210a52a76
a203,a196a49a25a243a183a243a69a193a222a139a30a77a34,a203a196a139a76a203a209a35a98,a0a143,a89a171a134a34a123a199
a106a139a24a199a222a221,Croftona218a42a89,a199a52a127a1494a245a199a123a222a221,Feuchlla218a42a52a69a193
a222Croftona218a42a123a24a199a136a42,a79a143Croftona218a42a128a199a123a222a221a524a245a149a199a123a143a30
a52a159,a196a49a210a188a126a89a30a77a34a203a154a24a199a76a203a209a35a98a123a117a159a210a122a116Feuchlla218a42.
a79a36a175a11a210a52a128a31a143a52a69a193a222a76a203a123a222a221,a69a193a222a123a76a203a52a69a193a222a58a78a1
a91a123a134a34,a13a233a149a134a34,a183a15a139a199a222a221(measure).a243a69a193a222a123a222a221a105a59a92,a89
a52a79a143a69a193a222a123a134a34a203a244a154a139a199a59a92a123a233a83a7a248,a79a143a28a139a24a199a76a203,a199a139
a24a199a244a154,a79a36a89a30a134a34a203a89a199a244a154a139a199a233a83a7a248,a149a52a89a199a76a203a123a222a221a210
a82a143a89a199a244a154a123a222a221,a196a49a183a25a243a76a150a89a24a89a14a143a248a121a210a69a193a222a123Crofton
a218a42,a89a199a121a210a217a34a105a59a92,a88a44a7a66a101a24a154,a210a52a46a27a6a166a166a43a41a210a113a234.
a25a243a7a48a128,γa89a30a77a34a203a24a199a76a203a35a98,a0a143a89a199a76a203a44a49a166a43a41,a152a42a166
a43a41,a132a149e1(s)a89a30a77a34a203a89a199a76a203a35a98,a13a76a203a139a199a244a154y,a196a49a7a166ya123
a43a41,a0a143a166a43a41a166a31a143a17a218a89a199a76a203a166a134a34a35a98,a13a76a203a522a145a123a56a45,a196
a49a7a139a220a199a43a41,a183a210a82a89a30a77a34a123a140a159sa42a143a24a199a43a41,a0a143,a89a199a76a203a203
a134a34a35a98a123a155,a210a139a67a222a123a89a199a60a89a213a164a98a24a199a43a41a52a31a143a17a218a76a203a139a24
a199a244a154,a89a199a244a154a52y,a23a108ya52a166e1a18a134a123,a196a49y a24a189a243e2,e3 a196a196a123a203a177
a222,a89a52a79a143e2,e3a52a203e1 a18a134a123,a79a36ya210a52a243e2,e3a123a203a177a222,a149a52ya210a44a49
a44a143
y = cosθe2(s) + sinθe3(s),(3.13)
a34a19a222,ya52e2,e3 a123a24a199a34a117a28a92,a145a79a143a199a52a199a92a160a37a222,a196a49a44a49a85a196
a89a248a123a111a231,a79a36ya89a154a139a220a199a43a41,a183a44a49a82a143s,θ,a0a143a25a243a123a175a11a52a70y
a243a89a199a193a232a195a163(element area)a123a221a222,a52a7a178y a89a154a123a221a222a23a196a52a76a203a123
a221a222,a47a158a89a88a10a70,y a52a24a199a154,a28a210a70dy,ya52a243a92a160a69a222a123a24a154,a149a52a5
a193a117a70dy,a178a89a199dya85a143a203y a18a134a123a220a199a48a53a123a60a106,a0a143a89a220a199a92a160a37a222
a248a106a123a105a232a210a52y a123a221a222,a196a49a183a25a243a89a143a41,e1,e2,e3a52a41a37,a199a162a174a199a37
8
a222a52a144a35a18a134a123a92a160a37a222,a199a162a15a209a52sa123a60a106,a196a49a183a178a199a123a218a42a85a241a117:
de1
ds = a2e2 + a3e3
de2
ds =?a2e1 + a1e3 (3.14)
de3
ds =?a3e1?a1e2
a195a88a152a91,a89a218a42a176a62a123a248a106a196a24a199a39a233a193a123a48a106,a89a210a41a5a86,a105a59a92a143,
a152a42a174a24a174dy,a210a122a116dya24a199a218a42:
dy = (?sinθe2 + cosθe3)(dθ + a3ds)?e1(a1 cosθ + a3 sinθ)ds,(3.15)
a149a52a183a162a139
a193a232a195a163= (a2 cosθ + a3 sinθ)dθ ∧ds,(3.16)
a165a215
a2 = cosτ,a3 = sinτ,(3.17)
a11
a193a232a195a163= cos(τ?θ)dθ ∧ds,(3.18)
a253a33a128cos(τ? θ)dθ ∧ ds a52a69a193a222a123a193a232a195a163,a143a234a7a70a89a199a69a193a222a123a193
a232,a70a89a199a192a220a123a173a232a73,a15a210a52a70a89a199a42a6a123a173a232a73,a0a143a183a106a89a199a154a123
a30a127,a52a31a68a113a92a253a233a138,a117a70a89a199a253a233a138a123a89a199a173a232a73,a183a166a98a34a247a139a220
a199a35a106s a203θ,a0a143a210a233a199a162a117a70a232a73,a232a73a123a30a127a11a251a189s,a82cosa123a253a233a138
a117a70a232a73,a34a38a110a221a221a24a87a123a155,|cos|a24a225a35a234a245a232a17a218cosa123a253a233a138a123a232
a73a524,a89a52a79a143a104cosa440a116pia30a52a441a116-1,a24a225a524a391,a196a49a524,a79a36a89
a248a28a210a70a116a203a77a34a35a98a123a76a203a123a221a222,a253a139a245a232a199a76a203,a89a199a221a222a52a70a89a199
a173a232a73,a178a199a174a241a117,a174a241a117a131a128,a28a144a7a70a24a39a233θa123a232a73a210a44a49a234,a16a5
a123a52a77a34a123a140a159,a196a49Cr oftona175a11a210a21a44a89a199a15a174a122a116,a28a162a168a5a117,a195
a86a46a24a46,a210a44a49a122a116,a79a36Croftona218a42a128γa89a30a77a34a150a232a139a245a159,a199a178a89
a199a77a34a203a127a91a76a203a35a98a123a117a159a44a143a24a199a221a222a123a117a159,a253a139a245a159,a132a89a199a183
a210a122a116Feuchlla189a174,a89a52a105a164a224a123a24a199a189a174,a183a162a34a38a89a199a22a111a52a85a20a123,
9
a2432a145a123a60a111,a210a52a85a20a123a77a34,a233a149a85a20a123a77a34,a199a123a15a77a71a139a24a199a24a189a123
a5a33,a31a143a30a127a89a199a5a33a21a72a116? a23a108a104a89a30a77a34a52a178a193a77a34a63a37a52a24a30
a113a247(complete)a77a34a30a52a44a49a72a116a123,a202a226a222a193a255a88a15a44a49a121a210a89a199a136a88.
a205a139a63a135a123a24a199a136a42a52a196a162a123Fray-Milnor a189a174a213a152a42a77a34Ca139a136,a11
integraldisplay
C
kds ≥ 4pi,(3.19)
a183a46Milnora52a163a74a35a117a155a41a33a128a132a123a24a199a95a192a166a27a86,a198a41a89a199a189a174a123a30
a127,a210a203a28a162a24a248,a222a49a30,a144a19a225a116a89a199a175a11,a198a201a241a24a199a30a71a31a143a30a127
a24a30a85a20a123a77a34a21a75a196a24a199a136,a34a38a139a136a123a155,a23a108a183a162a77a127a137a7a77a71a205a245
a24a74,a79a143a75a136a210a122a221a221,a88a44Fray -Milnora189a174a210a52a89a34a38a89a199a85a20a123a77
a34a139a24a199a136a123a155,a199a123a92a77a71a123a232a73a150a232a1434,a196a49a92a77a71a24a189a210a76a1494,a89
a199a121a210a105a59a92,a79a143a152a42a199a66a1494a123a155,a24a189a139a199a48a53,a243a89a199a48a53a222a123a176a221
a144a139a24a199a244a76a203a24a199a244a66a154,a196a49a183a89a30a77a34a24a189a52a203a122a89a199a244a76a154a166a244
a66a154a123a220a199a140,a0a143a165a45a210a150a139a244a76a154a90a244a66a154,a196a49a165a45a0a74a178a113a123a178a193
a203a77a34a35a98a123a155,a209a35a98a149a220a154,a0a143a126a24a30a34a227a178a89a220a154a203a229a117,a149a52a199
a89a77a34a210a140a196a24a199a75a173,a23a108a77a34a210a44a49a192a196a24a154,a196a49a210a88a52a136,a196a49a152
a42a89a92a77a71a66a1494 a123a155,a199a89a30a77a34a123a136a210a44a49a137a13,a79a36a152a42a77a34a139a199a136
a123a155,a199a123a92a77a71a24a189≥ 4,a149a52a193a47a122a241a117Fray-Milnor a189a174,a5a193a183a89a89
a166a189a41a98a123a211a42a123a7a248,a189a41a98a123a211a42a52a255a88a89a199a136,a159a113a88a144a24a30a77a34,
a44a49a139a80a1a30a77a34,a253a196a162linka123a207a196,a255a88a31a143a30a127a89a199link a21a119a44a117a77
a71a52a12a71,a89a199a175a11a212a174a166a27a105a161a108a85,a33a163,a41a234a105a245a89a48a193a123a211a42,a88
a44,a77a18Fary-Milnor a189a174a139a199a44a21a117,a210a52a255a88a35a34a123a77a71a203a12a71a233a149a189
a41a98a123a88a35a42a139a31a143a107a47,a189a41a98a52a207a196a77a34a123a233a83a51a174a123a117a159a203a135a73a42
a123a7a248,a196a49a93a116a126a243a89a199a48a193,a28a88a68a122a21a178a89a199a136a137a13,a98a52a139a44a21a104
a77a34a123a77a71a166a12a71a139a236a171a117a159a123a30a127,a198a196a189a123a89a199a88a35a42a204a127a1490,a196a49
a139a74a175a11a207a196a123a30a127a44a49a46a24a46,a152a42a139a124a38a234Harrisa115a71a123a49a123a155,a198a210
a243a89a189a41a98a123a211a42,a189a41a98a52a163a1a35a117a165a41a151a9a123a24a199a58a158a128a132a123a106a166a27.
a198a150a139a214,a88a209a166a152a144a139a31a143a7a248,a24a199a124a41a198a123a11a173,a41a58a158a1a79,a58a158a13
a99a123a211a42.
10
2001a3510a21926a134
1 a178a193a77a34
a183a46a89a1a39a203a76a27a89a24a154a135a232a73a243a1a91a222a123a97a126,a89a52a58a158a7a155a123a21a48.
a0a143,a44a33a59a92a123a60a89a13a41,a183a162a210a89a178a193a222a123a77a34,a34a247a178a193a222a139a24a30a77
a34x(t) = (x1(t),x2(t)),a253a243a89a199a67a222a196a243a123a60a89,a126a135a232a73a123a155a17,a210a52a89
a30a77a34a139a30a35a34,a35a34a139a199a35a37a222,a233a149a35a37a222,a183a162a82a89a199a37a222a52a92a160a37
a222,a199a123a159a221a521,a15a210a52a82a143a92a160a35a37a222,a149a52a183a162a127a119a34a38a178a43a41xa44a43
a196a140a159sa123a60a106a123a155,a89a210a44a43a89a199a92a160a35a37a222a210a52x a233s a123a135a73dxds,a253a92a160
a35a37a222a143
e1 = (dx1ds,dx2ds ),(e1,e1) = 1,sa52a140a159,eqno(3.1)
a0a143a14a143a248a207a196a89a30a35a34a17a218a105a59a92,a0a210a52a139a234a24a199a92a160a35a37a222a131a128,a63
a34a247a152a42a178a193a52a189a53a123,a253a139a24a199a221a196a123a48a53,a0a143a199a210a139a24a199a92a160a27a53a222,
a15a210a52a203a199a18a134a123a0a199a92a160a37a222,a25a243,a183a119e1a52a92a160a35a37a222,e2 a52a92a160a27
a37a222,a149a52a7a122a116a89a30a35a34a123a117a159,a145a24a71a47a60a210a52a178e1a89a199a60a106a233a149sa242a70
a135a73,a0a143a242a70a135a73a131a128,a104a108a89a52a24a199a99a123a37a222,a79a143e1a52a24a199a92a160a37a222,
a196a49(e1,e1) = 1,a0a143a178a199a135a73a24a5a6,a183a162a210a122a116de1ds a51e1a18a134,a196a49a199a24
a189a243a27a34a123a48a53,a79a36,a183a162a210a139de1ds a127a149a92a160a27a37a222e2a123a245a106,a89a199a245a106a52
a140a159a123a24a199a60a106,a183a162a119k(s),a89a199a245a106a119a22
de1
ds = ke2,e2 a52a92a160a27a37a222,(e1,e2) = 0,(3.2)
ka89a199a60a106a24a196a119a41a77a71,a52a89a30a77a34a243a89a199a178a193a176a62a33a7a155a123a24a199a117a159,
a52a140a159a123a24a199a60a106.
a243a11:
a233a149a201a189a123a77a34a48a199,a201a241a77a71k a123a218a42,[a10a43a213ka52a77a34a48a199a123a24a126a90
a19a126a135a73a123a24a199a60a106]
1
2 a56a45a77a34
a44a178a193a77a34a242a159a24a90a14a143a248a17a218a183a162a23a23a56a45a123a60a111,a34a247a183a162a25a243a139a24
a30a77a34a52a56a45a123a77a34x(t) = (x1(t),x2(t),x3(t)),a2433a145a56a45a176a139a35a34,a196a49
a89a199a56a45a123a43a41x,y a210a44a43a143a107a106t a123a60a106,a89a74a192a220a28a162a76a150a209a127a119,a183
a242a167a243a24a5a6,a196a49x(t) a52a24a199a37a222,a199a123a73a222a210a52a154a123a43a41,a13a154a52t a123
a60a106,a149a52a199a210a42a24a30a35a34,a893a199a73a222a119a41xi(t)(i = 1,2,3),x(t)a52a24a199a37
a222,a52a107a106ta123a60a106,a199a1233a199a73a222a210a52a178a154a123a43a41a44a43a143t a123a60a106,a0a143a14a143
a159a113a17a218a51a248a123a48a27a210a52a233a89a30a77a34a126a135a232a73,a34a189a77a34a139a35a34,a63a37a243a35
a34a222a193a139a92a160a37a222,a253a139a92a160a35a37a222,a233a77a34a139a199a48a53,a24a134a89a143a18,a215a248a24
a199a48a53,a7a152a128a107a106t a52a15a28a123a24a199a48a53,a15a13a211a131,a183a162a210a139a24a199a92a160a35
a37a222,a119a199e1,a0a143a203a178a193a51a248a123a60a89,a178e1a89a199a60a106a233s a242a70a135a73,a210a127a149
a128a233s a70a19a126a123a135a73,a79a143e1 a52a92a160a37a222,a196a49a122a116a123a89a199a135a73a203e1 a52a18a134
a123,a13a233a149a24a30a56a45a165a123a77a34,a199a123a35a34a52a24a30,a199a123a27a34a139a195a106a199,a217a34
a178a44a89a24a154a203a35a34a18a134a123a27a34a139a195a106a199,a0a143a217a165a139a24a199a52de1ds,a199a52a88a113a92
a104a189a123,a132a149a51a248a123a174a132,a183a178a89a199a48a53a123a92a160a37a222a119a41e2,a0a143a199a210a52ke2,
a253a44a49a85a196 de
1
ds = ke2,(3.3)
e2a52a217a165a24a30a27a34,a183a162a119a199a143a204a27a34(principal normal),a13ka89a199a60a106a166a178
a193a123a60a111a24a248,a119a41a77a34a123a77a71,a196a49a25a243,a183a139a24a199a35a34a123a48a53a90a24a199a204
a27a34a48a53,a243a174a145a56a45a210a139a12a105a24a199a48a53a203a89a220a199a48a53a18a134,a183a162a47a158a119
a199binormal,a15a13a211a131,a63a243a12a105a24a199a27a34,a196a49a199a139a220a199a27a34,a0a143a89a220a199
a27a34a196a196a123a178a193a210a52a27a178a193,a77a34a521a145a123,a35a34a521a145a123,a196a49a27a34a522a145a123,
a52a199a178a193,a0a143a139a24a199e3 = e1 ×e2,a217a165e1 ×e2 a52a37a222a232,a233a149a220a199a48a53a123
a155,a139a24a199a104a189a123a145a174a199a48a53a52a203a199a162a18a134a123,a63a37a38a122e1,e2,e3a196a24a199a116
a98a248a58,a183a162a34a189a89a199a56a45a52a189a53a123,a141a67,a38a67a209a139a24a199a104a189a123a48a53,a47
a158a183a162a209a126a141a67,a196a49a210a139a24a199a104a189a123e3a119a22
(ei,ej) = δij,(3.4)
2
a79a36a243a207a196a174a145a1a91a123a30a127,a89a248a123a174a199a144a35a18a134a123a92a160a37a222a196a196a123a67a111
a58a158a7a155,a89a52a143a31a143a17a218a89a248a24a199a192a220a183a119a199a41a37,a28a178a24a199a41a37a194a116a12
a24a199a41a37a123a228a196a52a113a92a104a189a123,a0a143a174a145a56a45a33a7a155a123a117a159a210a52a174a145a56a45
a123a228a196,a183a162a7a207a196a123a1a91a117a159a52a178a44a228a196a88a35a123,a196a49a210a7a127a119a31a143a30a127
a28a44a49a178a89a199a192a220a194a116a12a105a24a199a160a152,a31a143a30a127a199a123a160a152a35a139a243a149a24a199a228
a196,a13a41a37a210a52a89a199a228a196a137a219a123a44a43a123a48a27,a28a7a21a234a194a44a86a210a44a43a89a199a41
a37a194a116a12a105a24a199a41a37a123a228a196a52a113a92a104a189a123,a23a108,a144a139a24a199a228a196a63a37a24a189
a139a24a199a228a196a178a24a199a41a37a35a143a12a105a24a199a41a37,a79a143a7a207a196a56a45a178a44a228a196a88a35
a123a117a159,a196a49a137a219a123a48a27a210a52a188a126a41a37,a0a143a34a38a183a25a243a139a24a30a77a34,a183a88
a144a139a24a199a41a37,a89a74a41a37a52a30a45a123a60a106,a243a0a176a228a196,a79a36eia89a174a199a42a143a41
a37a123a37a222a209a52a30a45t a123a60a106,a149a52a183a44a49a70a199a123a135a73deidt,deidt a52a199a37a222,a79a143
a52(e1,e2,e3)a52a199a41a37,a196a49a127a91a24a199a37a222a210a44a49a85a196e1,e2,e3a893a199a37a222a123
a34a117a28a92,a0a143a183a178a199a227a135a77a18a24a154,a210a178a199a85a196deia127a149eja123a34a117a28a92:
dei = ωijek,(3.5)
a89a199a28a92a123a248a106a52a24a39a135a73a42,a89a52a79a143a183a25a243a41a234a24a5a135a73,a132a89a60a106a34
a122a116a24a39a135a73a42,a0a143a89a248a123a24a39a135a73a42a183a119a199ωij,a89a210a44a43a220a199a35a249a123
a41a37a123a7a248,a28a139a24a199ei a123a41a37,a107a30a139a199a35a249a123a41a37ei + dei,a0a143dei a44
a143eia123a60a106a123a30a127,a199a123a28a92a123a248a106a210a52a24a39a135a73a42,ωij a52a24a39a135a73a42,a243
a89a176,a183a123a141a41i,ja209a52a441 a1163,3 a52a183a162a56a45a123a145a106,a196a49ωij a24a225a1399a199:
i,j a154a24a199a441a1163,a196a49a24a225a1399a199.a899a199a24a39a135a73a42a52a139a7a248a123,a199a88a52a113
a92a127a63a123,a89a199a7a248a210a52
ωij + ωji = 0,(3.6)
a196a49a34a38a28a178ωij a23a196a24a1993 × 3 a123a48a106a123a195a163a123a155,a89a199a48a106a52a39a233a193a123.
a89a24a28a48a199a105a142a52a44ei,ej a123a19a232a127a149δija122a116,a178a89a199a7a248(a48a199(3.4))a135a73
a123a155,a210a193a47a122a116a89a24a28a117a159,a89a210a52a128(ωij) a52a24a1993 × 3 a48a106a123a195a163,a89
a199a48a106a123a195a163a209a52a24a39a135a73a42,a63a37a89a199a48a106a114a199a52a39a233a193a123,a166a233a155a128,
a89a199a48a106a204a110a34a222a123a195a163a520,a217a199a17,a132a149a39a233a193,a139ω12 =?ω21a127,a52a39
a233a193a123,a79a36a183a162a25a243a139a24a199a92a160a35a37a222,a139a24a199a204a27a37a222,a164a139a24a199a166a199
3
a162a18a134a123a196a24a199a41a37a123e3 = e1 × e2(a37a222a232),a233a149a89a199a41a37,a183a178a199a123a174a199
a48a199a209a233a140a159a70a60a106( a135a73),a210a44a49a178a89a199a60a106a44a143e1,e2,e3 a123a24a199a34a117a28
a92,a89a199a28a92a52a89a248a123a24a28a48a199a213a145a24a199a48a199a52de1 = ke2,a79a143a183a162a123a48a106
a52a39a233a193a123,a204a233a110a34a222a123a195a163a209a520,a196a49ω13 = 0,a98a52a217a198a123a195a163a7a213
a63e3a123a160a152,a132a149e1,e2 a160a10a123a7a248,ω13 a52a127a1490a123,a79a36ω31 a15a127a1490,a196a49
a89a199a48a106a85a241a117,a210a52a183a5a193a1233a199a48a199:
de1
ds = ke2
de2
ds =?ke1 + ωe3 (3.7)
de3
ds =?ωe2
a89a28a48a199a52a104a35a77a34a88a21a48a123a30a127a33a0a123a24a28a48a199,a183a162a47a158a119a199Frenet
a48a199,Fren eta52a27a41a123a24a199a106a166a27,a183a46a89a52a198a123a123a70a44a88a169,a198a88a24a189a52
a62a24a199a201a241a89a199a48a199,a104a30a139a1a199a124a41a241a89a199a211a42,a44a183a89a123,a28a162a44a49a23
a241a117a122a89a199a48a199a63a88a212a104a10,a79a36a183a248a234a77a71a49a105,a164a139a12a24a199a60a106ω,ω
a210a52a48a199a165a123a12a71(torison),a15a52a140a159a123a60a106,a52a44a43a56a45a123a77a34a243a228a196a107
a5a123a117a159,a196a49a56a45a77a34a139a220a199a60a106,a24a199a52a77a71,a12a24a199a52a12a71,a12a71a210a52
a195a85a199a14a143a248a88a52a24a30a178a193a77a34,a199a52a243a56a45a107a77a123a24a199a222,a196a49a56a45a77a34
a52a126a220a199a60a106a117a195a85a123,a199a162a137a219a143a195a85a89a56a45a123a117a159,a89a220a199a60a106a23a108a105
a173a7,a79a143a199a162a7a52a127a1490a123a30a127,a210a44a43a234a77a34a105a59a92a123a117a159,a7a52k = 0
a123a155,a89a77a34a210a196a143a134a34,a89a105a142a52a121a210,a183a5a193a201a241a121a210a213a79a143k = 0,a196
a49de1 = 0,a79a36a92a160a35a37a222e1a52a24a199a158a106,a79a143a89a248a199a123a135a73a98a127a1490,a0a143
a28a178a89a199a158a106e1 = Ca83a156a116e1 = dxdsa165,a242a178a199a232a73a24a5a6a210a122a116xa52s a123a24
a39a60a106a213
x = Cs,(3.8)
a196a49a89a210a52a24a30a134a34,a39a44a117a15a105a142a52a21a121a210a152a42a28a139a24a30a134a34a123a155,a199a123
a77a71k a52a127a1490a123,a79a36k = 0 a83a44a77a34a123a33a59a92a123a117a159,a210a52a134a34,a0a143a7
a213a63a123a52a243a189a12a71a123a30a127,a24a189a7k negationslash= 0,a165a52k = 0 a123,a149a52de1ds = 0,a15a210
a52a134a34a234,a89a30a199a210a150a139a27a6a189a204a27a34,a24a30a134a34a203a199a18a134a123a52a24a199a178a193,
4
a89a199a178a193a176a62a196a139a203a36a134a34a18a134a123a48a53a209a52a139a51a248a123a117a159,a196a49a210a150a139a204
a27a34,a79a36a15a88a21a189a12a71,a12a71a24a189a7a243a89a154a123a77a71k negationslash= 0a123a30a127a98a139a63a66.
a13a104a12a71a127a1490a123a30a127,a104a108a210a52a44a43a89a30a77a34a52a243a178a193a222a24a30a77a34,a5a193
a183a15a201a234a24a199a59a92a123a121a210,a79a143a12a71a89a199a60a106a52a243Frenet a218a42a123a145a174a199a218
a42a176,a196a49a132ω = 0 a44a127a36a30e3a52a199a158a106a123a37a222,a233a149e3 a89a199a158a106a37a222,a132
a149e3a52a24a199a27a34,a63a37a79a143a27a34a203a35a34a52a123a207a18a134a123a199a196a49e3 a203dxds a123a19a232
a52a123a207a127a1490,a79a36e3a7a52a127a149a158a106a123a155,a183a210a44a49a178a48a199
(e3,dxds) = 0 (3.9)
a232a73,a79a143e3a52a24a199a158a106a199a196a49a89a232a73a210a52e3 a203x a123a19a232a127a149a24a199a158a106,a79
a36a199a52a24a199a178a193a77a34,a149a52ω = 0a52a44a43a77a34a52a199a178a193a77a34,a39a44a117,a44a49
a105a142a52a121a210a178a193a77a34a123a12a71a520,a196a49a77a71a90a12a71a220a199a60a106a209a139a59a92a123a1a91
a117a159,a12a105a24a171a105a139a63a135a123a77a34a52ka166ωa209a88a1430,a98a209a52a158a106,a0a143a243a89a199
a60a111a131a5,a44a49a121a210a77a34a52a199a90a34,a210a52a89a248a59a92a123a135a232a73a123a97a126a243a9a212
a154a166a222a139a173a7a123a63a66,a79a143a183a162a127a119a9a212a154a166a123a24a199a204a7a123a154a92a212a52DNA.
DNAa52a220a30a90a34,a52a199a118a90a34,a89a52a9a212a222a58a158a228a253a131a25a54,a243a9a212a222,a89
a248a123a77a34a210a115a241a117a234,a79a36,a77a34a88a243a9a212a154a166a139a173a7a123a97a126,a15a210a52a76a27
a7a127a119a77a34a123a117a159a152a91a107a47DNAa123a154a166a117a159,a196a49a106a166a210a105a173a7a234,a77a34
a123a117a159a107a47a116a154a166a123a117a159,a131a217a52a243a154a166a176a62,a139a30a127a28a178DNAa35a228a234,a199
a123a117a159a210a149a35a234,a196a49a35a228a131a128,a106a166a117a159a210a21a9a149a35a199a199a123a154a166a117a159a15a149
a35,a255a88a199a162a152a91a149a35a199a89a52a243DNAa123a207a196a249a243a9a212a154a166a48a193a52a58a158a228a253
a123a175a11,a76a27a41a234a105a245a123a211a42,a25a243a252a24a253a135a9a212a154a166a123a86a7a32a13a117a123a155,
a210a23a68a139a24a199a228a253a123a218a42,a119a41Whitea218a42,Whitea52a183a123a24a199a166a9a123a166a9,
a198a41a89a199a211a42a52a198a70a44a88a169a123a24a199a136a42,a198a228a237a105a80,a198a89a199a136a42a35a143a9a212
a154a166a123a24a199a228a253a123a218a42,a183a25a243a88a21a89a89a199a136a42.
5
3 Feuchlla88a127a42,Croftona232a73a218a42a90Fary-Milnor
a189a174
a41a89a199a1a91a207a196a123a30a127,a139a74a173a7a117a159a7a255a88,a47a158a173a7a123a117a159a128a128a52a114a14
a1a91a123a117a159,a243a77a34a123a60a89a15a88a52a207a196a24a66a227a123a1a91,a183a162a25a243a178a222a193a123a89
a199a119a41a219a92a123a1a91a199a253a207a196a24a66a227a123a1a91a117a159,a89a30k,ωa209a52a140a159a123a60a106,
a126a199a162a207a196a199a123a117a159,a52a14a143a248a107a77,a88a44a205a173a7a123a52a207a196a114a30a77a34,a34a38
a139a24a30a77a34a52a85a20a123,a23a89a30a77a34a139a31a143a248a123a117a159,a89a176a139a199a58a158a7a155a123a218
a42,a233a149a89a30a85a20a123a77a34,a154a154a139a199a77a71,a79a143a85a20a234,a183a210a207a196a92a77a71,a253
a178a89a199a77a71a215a114a30a77a34a70a199a123a232a73,a149a52a210a139a196a162Feuchlla218a42
integraldisplay
c
kds ≥ 2pi,(3.10)
a210a52a128a89a199a232a73a139a199a5a142,a24a189a522pi,a89a199a134a250a89a229a117a210a52a152a42a28a24a154a77
a71a209a150a139,a89a30a77a34a210a134a34a18a5a86,a52a88a204a85a20a229a117,a28a7a134a34a18a5a86a21a234
a85a20a123a155,a15a7a178a199a107a44a117,a107a44a117a210a7a139a77a71,a0a143a152a42a21a234a107a44a117,a166a198
a117a143a48a233a222a117a123a155,a0a143a77a71a243a114a199a77a34a222a123a232a73a139a24a199a251a189a123a5a142,a89a199
a5a142a522pi,a89a210a52a196a162a123Feuchlla218a42,a183a7a121a210a89a199a218a42,a143a31a143a89a92a77a71
a7a139a24a199a5a142? a121a210a89a199a218a42a123a48a27a210a52a255a88a89a30a77a34a123a92a160a35a37a222,a253
a243a35a34a123a48a53a123a24a199a92a160a37a222,a204a7a10a39a210a52a188a126a176a132a110a243(Gau ss map).
a233a149a89a248a24a199a92a160a35a37a222,a243a251a189a154a82a203a89a199a35a37a222a52a178a113a123a0a74a92a160a37
a222,a196a49a28a139a142a245a92a160a37a222a52e1(s),a199a52a159a127a1491 a123a24a199a37a222,a196a49a28a34a38
a178a199a23a196a56a45a24a199a154,a199a210a52a243a92a160a69a222a123a24a199a154,a79a36a28a82a24a199a92a160a69,
a82e1(s)a23a196a92a160a69a222a123a24a199a154,a0a143a104a28a89a199a154a215a198a117a77a34a18a24a87a123a30a127,
e1(s)a243a92a160a69a222a196a24a30a77a34,a89a52a176a132a110a243a123a63a135,a253a196a162a123a176a132a110a243,a89
a30a77a34e1(s)a243a92a160a69a222a196a196a123a123a77a34a88a52a198a117a123a77a34,a166a234a24a171a234,a79a143a199
a123a48a199a52e1(s),a88a52x(s),e1(s) a119a22:
dσ2
ds2 = (
de1
ds,
de1
ds ) = k
2,σa52e1(s)a123a140a159,(3.11)
a92a160a69a193a152a42a139a24a199a76a203,a0a143a89a199a76a203a139a24a199a244a154,a183a119a199y,a15a210a52a128,
6
a76a203a52a6a204a203,y a52a89a199a76a203a123a244a154.a178a89a199a244a154a104a196a186a222a123a24a154,a0a143a77a34
a210a139a199a176a221,a89a199a176a221a250a82a196a52y(x),a52a199a19a232,a89a199a176a221a44a49a23a196a52a77a34
a222a123a24a199a60a106,a0a143a77a34a243a56a45a176a62a210a139a234a24a199a176a221,a183a162a89a77a34a52a85a20a123,
a104a108a139a199a33a176a154,a139a199a33a132a154,a243a33a176a154,a79a143a199a52a33a176a123,a23a108a199a123a35a34a52
a178a123,a243a33a132a154,a35a34a15a52a178a123,a196a49a210a139a220a199a248a142a154,a24a199a52a83a44a176a221a33
a176a123,a24a199a52a33a132a123,a13a166ya89a199a154a18a134a123a0a199a76a203,a24a189a203a92a160a69a222a123a77a34
a35a98,a79a143(y,dx) a89a199a19a232a520,a196a49ya203a69a193a123e1(s)a243a24a154a35a98,a79a36a243a92
a160a69a222a123a77a34a203a89a199a92a160a203a35a98,a98a52ya52a127a63a48a53,a196a49a183a162a69a222a123a176a132
a110a243a123a77a34a52a203a69a222a123a31a143a203a209a35a98,a89a52a79a143ya52a127a63a24a199a154,a149a52ya123a0
a199a6a204a203a98a183a162a89a30a77a34,a183a46a176a132a77a34a183a119a199γ,a199a203ya123a6a204a203a35a98,a79
a36a203a127a63a92a160a203a35a98,a89a52a105a7a155a123a24a199a117a159,a196a49γa89a30a77a34a24a189a7a35a104
a159,a199a243a69a222a88a52a24a30a127a91a123a77a34,a199a243a69a222a52a24a30a203a196a139a123a203a177(a76a203)a209
a35a98a123a77a34,a0a143a183a164a52a137a7a121a210a199a1392pi a89a199a5a142,a5a193a117a128a210a89a199a175a11.
a183a89a30a77a34γa52e1(s),a119a51a160a203,e1(s) a210a52a56a45a77a34a123a24a199a176a132a110a243,a199a52
a35a34a123a48a53a243a92a160a69a222a196a196a123a77a34,a13a89a30a77a34a119a41γ,a199a203a69a222a123a127a91a92
a160a203a209a35a98,a0a143a24a196a143,a89a30a77a34a35a104a159a234,a152a42a212a225,a199a150a139a89a199a117a159.
a128a199a139a24a1992pi a123a5a142,a52Croftona218a42a213
integraldisplay integraldisplay
= 4L,(3.12)
Croftona123a169a57a15a105a139a63a135,a88a52a24a199a116a42a123a106a166a169a57,a93a41a123a186a41a92
a86(Encyclopaedi a)a62a198a85a24a159a169a57,a52a7a149a1a91a150a71a123,a198a52a243a85(a1a91)a150
a71a123a169a57a30a127a178a198a123a136a42a85a243a176a62a234,a89a199a136a42a105a7a155,a166a233a155a128,a210a243a178
a193a222a89,a77a34a123a159a221a44a49a44a143a203a199a35a98a123a134a34a123a221a222,a159a221,a210a52a134a34a123a221
a222,a89a199a63a135a243a230a212a166a176,a163a83a243a26a166a209a139a97a126,a139a30a127,a28a252a14a222a123a192a220,
a7a175a199a139a245a76,a15a88a21a178a124a35a13,a52a88a80a222a123,a149a52a210a52a126a23a35a98a89a199a192a220
a123a34a42a143a221a222,a117a222a89a199a252a14a222a123a236a24a92a73a123a76a66,a89a24a92a73a106a166a24a196a119a41
a232a73a1a91,a88a52a135a73a1a91,a232a73a1a91a210a52a207a196a89a74a232a73a123a7a248a166a117a159,a232
a73a1a91a243a26a166a222a139a105a76a123a97a126,a139a105a245a229a236a102a126a102a126a89a174a88,a79a143a28a7a23
a124a123a62a14,a15a88a21a252a89a199a62a14a117a222,a210a252a34a243a199,a149a243a199a123a72a42a117a23a62a14a123
7
a76a66a249a217a199a123a117a159,a89a210a52a232a73a1a91,a0a143a183a25a243a128,(3.12)a52Croftona218a42,
a210a52a128a69a193a222a30a15a139a1a91,a52a69a193a1a91,a199a123a154a183a209a127a119,a199a123a134a34a210a52a76
a203,a196a49a25a243a183a243a69a193a222a139a30a77a34,a203a196a139a76a203a209a35a98,a0a143,a89a171a134a34a123a199
a106a139a24a199a222a221,Croftona218a42a89,a199a52a127a1494a245a199a123a222a221,Feuchlla218a42a52a69a193
a222Croftona218a42a123a24a199a136a42,a79a143Croftona218a42a128a199a123a222a221a524a245a149a199a123a143a30
a52a159,a196a49a210a188a126a89a30a77a34a203a154a24a199a76a203a209a35a98a123a117a159a210a122a116Feuchlla218a42.
a79a36a175a11a210a52a128a31a143a52a69a193a222a76a203a123a222a221,a69a193a222a123a76a203a52a69a193a222a58a78a1
a91a123a134a34,a13a233a149a134a34,a183a15a139a199a222a221(measure).a243a69a193a222a123a222a221a105a59a92,a89
a52a79a143a69a193a222a123a134a34a203a244a154a139a199a59a92a123a233a83a7a248,a79a143a28a139a24a199a76a203,a199a139
a24a199a244a154,a79a36a89a30a134a34a203a89a199a244a154a139a199a233a83a7a248,a149a52a89a199a76a203a123a222a221a210
a82a143a89a199a244a154a123a222a221,a196a49a183a25a243a76a150a89a24a89a14a143a248a121a210a69a193a222a123Crofton
a218a42,a89a199a121a210a217a34a105a59a92,a88a44a7a66a101a24a154,a210a52a46a27a6a166a166a43a41a210a113a234.
a25a243a7a48a128,γa89a30a77a34a203a24a199a76a203a35a98,a0a143a89a199a76a203a44a49a166a43a41,a152a42a166
a43a41,a132a149e1(s)a89a30a77a34a203a89a199a76a203a35a98,a13a76a203a139a199a244a154y,a196a49a7a166ya123
a43a41,a0a143a166a43a41a166a31a143a17a218a89a199a76a203a166a134a34a35a98,a13a76a203a522a145a123a56a45,a196
a49a7a139a220a199a43a41,a183a210a82a89a30a77a34a123a140a159sa42a143a24a199a43a41,a0a143,a89a199a76a203a203
a134a34a35a98a123a155,a210a139a67a222a123a89a199a60a89a213a164a98a24a199a43a41a52a31a143a17a218a76a203a139a24
a199a244a154,a89a199a244a154a52y,a23a108ya52a166e1a18a134a123,a196a49y a24a189a243e2,e3 a196a196a123a203a177
a222,a89a52a79a143e2,e3a52a203e1 a18a134a123,a79a36ya210a52a243e2,e3a123a203a177a222,a149a52ya210a44a49
a44a143
y = cosθe2(s) + sinθe3(s),(3.13)
a34a19a222,ya52e2,e3 a123a24a199a34a117a28a92,a145a79a143a199a52a199a92a160a37a222,a196a49a44a49a85a196
a89a248a123a111a231,a79a36ya89a154a139a220a199a43a41,a183a44a49a82a143s,θ,a0a143a25a243a123a175a11a52a70y
a243a89a199a193a232a195a163(element area)a123a221a222,a52a7a178y a89a154a123a221a222a23a196a52a76a203a123
a221a222,a47a158a89a88a10a70,y a52a24a199a154,a28a210a70dy,ya52a243a92a160a69a222a123a24a154,a149a52a5
a193a117a70dy,a178a89a199dya85a143a203y a18a134a123a220a199a48a53a123a60a106,a0a143a89a220a199a92a160a37a222
a248a106a123a105a232a210a52y a123a221a222,a196a49a183a25a243a89a143a41,e1,e2,e3a52a41a37,a199a162a174a199a37
8
a222a52a144a35a18a134a123a92a160a37a222,a199a162a15a209a52sa123a60a106,a196a49a183a178a199a123a218a42a85a241a117:
de1
ds = a2e2 + a3e3
de2
ds =?a2e1 + a1e3 (3.14)
de3
ds =?a3e1?a1e2
a195a88a152a91,a89a218a42a176a62a123a248a106a196a24a199a39a233a193a123a48a106,a89a210a41a5a86,a105a59a92a143,
a152a42a174a24a174dy,a210a122a116dya24a199a218a42:
dy = (?sinθe2 + cosθe3)(dθ + a3ds)?e1(a1 cosθ + a3 sinθ)ds,(3.15)
a149a52a183a162a139
a193a232a195a163= (a2 cosθ + a3 sinθ)dθ ∧ds,(3.16)
a165a215
a2 = cosτ,a3 = sinτ,(3.17)
a11
a193a232a195a163= cos(τ?θ)dθ ∧ds,(3.18)
a253a33a128cos(τ? θ)dθ ∧ ds a52a69a193a222a123a193a232a195a163,a143a234a7a70a89a199a69a193a222a123a193
a232,a70a89a199a192a220a123a173a232a73,a15a210a52a70a89a199a42a6a123a173a232a73,a0a143a183a106a89a199a154a123
a30a127,a52a31a68a113a92a253a233a138,a117a70a89a199a253a233a138a123a89a199a173a232a73,a183a166a98a34a247a139a220
a199a35a106s a203θ,a0a143a210a233a199a162a117a70a232a73,a232a73a123a30a127a11a251a189s,a82cosa123a253a233a138
a117a70a232a73,a34a38a110a221a221a24a87a123a155,|cos|a24a225a35a234a245a232a17a218cosa123a253a233a138a123a232
a73a524,a89a52a79a143a104cosa440a116pia30a52a441a116-1,a24a225a524a391,a196a49a524,a79a36a89
a248a28a210a70a116a203a77a34a35a98a123a76a203a123a221a222,a253a139a245a232a199a76a203,a89a199a221a222a52a70a89a199
a173a232a73,a178a199a174a241a117,a174a241a117a131a128,a28a144a7a70a24a39a233θa123a232a73a210a44a49a234,a16a5
a123a52a77a34a123a140a159,a196a49Cr oftona175a11a210a21a44a89a199a15a174a122a116,a28a162a168a5a117,a195
a86a46a24a46,a210a44a49a122a116,a79a36Croftona218a42a128γa89a30a77a34a150a232a139a245a159,a199a178a89
a199a77a34a203a127a91a76a203a35a98a123a117a159a44a143a24a199a221a222a123a117a159,a253a139a245a159,a132a89a199a183
a210a122a116Feuchlla189a174,a89a52a105a164a224a123a24a199a189a174,a183a162a34a38a89a199a22a111a52a85a20a123,
9
a2432a145a123a60a111,a210a52a85a20a123a77a34,a233a149a85a20a123a77a34,a199a123a15a77a71a139a24a199a24a189a123
a5a33,a31a143a30a127a89a199a5a33a21a72a116? a23a108a104a89a30a77a34a52a178a193a77a34a63a37a52a24a30
a113a247(complete)a77a34a30a52a44a49a72a116a123,a202a226a222a193a255a88a15a44a49a121a210a89a199a136a88.
a205a139a63a135a123a24a199a136a42a52a196a162a123Fray-Milnor a189a174a213a152a42a77a34Ca139a136,a11
integraldisplay
C
kds ≥ 4pi,(3.19)
a183a46Milnora52a163a74a35a117a155a41a33a128a132a123a24a199a95a192a166a27a86,a198a41a89a199a189a174a123a30
a127,a210a203a28a162a24a248,a222a49a30,a144a19a225a116a89a199a175a11,a198a201a241a24a199a30a71a31a143a30a127
a24a30a85a20a123a77a34a21a75a196a24a199a136,a34a38a139a136a123a155,a23a108a183a162a77a127a137a7a77a71a205a245
a24a74,a79a143a75a136a210a122a221a221,a88a44Fray -Milnora189a174a210a52a89a34a38a89a199a85a20a123a77
a34a139a24a199a136a123a155,a199a123a92a77a71a123a232a73a150a232a1434,a196a49a92a77a71a24a189a210a76a1494,a89
a199a121a210a105a59a92,a79a143a152a42a199a66a1494a123a155,a24a189a139a199a48a53,a243a89a199a48a53a222a123a176a221
a144a139a24a199a244a76a203a24a199a244a66a154,a196a49a183a89a30a77a34a24a189a52a203a122a89a199a244a76a154a166a244
a66a154a123a220a199a140,a0a143a165a45a210a150a139a244a76a154a90a244a66a154,a196a49a165a45a0a74a178a113a123a178a193
a203a77a34a35a98a123a155,a209a35a98a149a220a154,a0a143a126a24a30a34a227a178a89a220a154a203a229a117,a149a52a199
a89a77a34a210a140a196a24a199a75a173,a23a108a77a34a210a44a49a192a196a24a154,a196a49a210a88a52a136,a196a49a152
a42a89a92a77a71a66a1494 a123a155,a199a89a30a77a34a123a136a210a44a49a137a13,a79a36a152a42a77a34a139a199a136
a123a155,a199a123a92a77a71a24a189≥ 4,a149a52a193a47a122a241a117Fray-Milnor a189a174,a5a193a183a89a89
a166a189a41a98a123a211a42a123a7a248,a189a41a98a123a211a42a52a255a88a89a199a136,a159a113a88a144a24a30a77a34,
a44a49a139a80a1a30a77a34,a253a196a162linka123a207a196,a255a88a31a143a30a127a89a199link a21a119a44a117a77
a71a52a12a71,a89a199a175a11a212a174a166a27a105a161a108a85,a33a163,a41a234a105a245a89a48a193a123a211a42,a88
a44,a77a18Fary-Milnor a189a174a139a199a44a21a117,a210a52a255a88a35a34a123a77a71a203a12a71a233a149a189
a41a98a123a88a35a42a139a31a143a107a47,a189a41a98a52a207a196a77a34a123a233a83a51a174a123a117a159a203a135a73a42
a123a7a248,a196a49a93a116a126a243a89a199a48a193,a28a88a68a122a21a178a89a199a136a137a13,a98a52a139a44a21a104
a77a34a123a77a71a166a12a71a139a236a171a117a159a123a30a127,a198a196a189a123a89a199a88a35a42a204a127a1490,a196a49
a139a74a175a11a207a196a123a30a127a44a49a46a24a46,a152a42a139a124a38a234Harrisa115a71a123a49a123a155,a198a210
a243a89a189a41a98a123a211a42,a189a41a98a52a163a1a35a117a165a41a151a9a123a24a199a58a158a128a132a123a106a166a27.
a198a150a139a214,a88a209a166a152a144a139a31a143a7a248,a24a199a124a41a198a123a11a173,a41a58a158a1a79,a58a158a13
a99a123a211a42.
10
