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8H-27 cos
3
HtLsin
6
HtL- 27 cos
5
HtLsin
4
HtL-24 cosHtLsin
2
H2tLsin
2
HtL- 24 cos
2
HtLcosH2tLsinH2tLsinHtL+ 12 cos
3
HtLsin
2
H2tLL
ê
H
,
HH-18 sinHtLsinH2tLcos
5
HtL+ 36 cosH2tLsin
2
HtLcos
4
HtL+36 cosH2tLsin
4
HtLcos
2
HtL+ 18 sin
5
HtLsinH2tLcosHtLL^2 +
H-27 cos
3
HtLsin
6
HtL- 27 cos
5
HtLsin
4
HtL-24 cosHtLsin
2
H2tLsin
2
HtL- 24 cos
2
HtLcosH2tLsinH2tLsinHtL+
12 cos
3
HtLsin
2
H2tLL^2+H-27 sin
3
HtLcos
6
HtL- 27 sin
5
HtLcos
4
HtL-
24 sinHtLsin
2
H2tLcos
2
HtL+ 24 cosH2tLsin
2
HtLsinH2tLcosHtL+ 12 sin
3
HtLsin
2
H2tLL^2LL,
H-27 sin
3
HtLcos
6
HtL- 27 sin
5
HtLcos
4
HtL-24 sinHtLsin
2
H2tLcos
2
HtL+ 24 cosH2tLsin
2
HtLsinH2tLcosHtL+ 12 sin
3
HtLsin
2
H2tLLê
H
,
HH
-18 sinHtLsinH2tLcos
5
HtL+ 36 cosH2tLsin
2
HtLcos
4
HtL+36 cosH2tLsin
4
HtLcos
2
HtL+ 18 sin
5
HtLsinH2tLcosHtLL^2 +
H-27 cos
3
HtLsin
6
HtL- 27 cos
5
HtLsin
4
HtL-24 cosHtLsin
2
H2tLsin
2
HtL- 24 cos
2
HtLcosH2tLsinH2tLsinHtL+
12 cos
3
HtLsin
2
H2tLL^2+H-27 sin
3
HtLcos
6
HtL- 27 sin
5
HtLcos
4
HtL-
24 sinHtLsin
2
H2tLcos
2
HtL+ 24 cosH2tLsin
2
HtLsinH2tLcosHtL+ 12 sin
3
HtLsin
2
H2tLL^2LL,
H-18 sinHtLsinH2tLcos
5
HtL+ 36 cosH2tLsin
2
HtLcos
4
HtL+36 cosH2tLsin
4
HtLcos
2
HtL+ 18 sin
5
HtLsinH2tLcosHtLLê
H
,
HH
-18 sinHtLsinH2tLcos
5
HtL+ 36 cosH2tLsin
2
HtLcos
4
HtL+36 cosH2tLsin
4
HtLcos
2
HtL+ 18 sin
5
HtLsinH2tLcosHtLL^2 +
H-27 cos
3
HtLsin
6
HtL- 27 cos
5
HtLsin
4
HtL-24 cosHtLsin
2
H2tLsin
2
HtL- 24 cos
2
HtLcosH2tLsinH2tLsinHtL+
12 cos
3
HtLsin
2
H2tLL^2+H-27 sin
3
HtLcos
6
HtL- 27 sin
5
HtLcos
4
HtL-
24 sinHtLsin 2t cos
FullSimplify[%]
2
H L
2
HtL+ 24 cosH2tLsin
2
HtLsinH2tLcosHtL+ 12 sin
3
HtLsin
2
H2tLL^2LL<
:-
1
8
HL HL
csc
2
t sec
3
t
"##################
sin
6
H2 tL, -
1
8
csc
3
HtLsec
2
HtL
"##################
sin
6
H2 tL,0>
PowerExpand[%]
:-
1
8
csc
2
t sec
3
t sin
3
2 t , -
1
HL HL H L
8
csc
3
t sec
2
t sin
3
H2 tL,0>HL HL
А единичный вектор бинормали:
ubv[t]=bv[f][u]/Norm[bv[f][u]]/.u→t
9H-6sinH2 tLsin
3
HtL- 12 cosHtLcosH2tLsin
2
HtL+ 12 cos
2
HtLsinH2tLsinHtLLë
I
,
IH
-9sin
2
HtLcos
4
HtL- 9sin
4
HtLcos
2
HtLL
2
+
H6sinH2 tLcos
3
HtL- 12 cosH2 tLsinHtLcos
2
HtL- 12 sin
2
HtLsinH2tLcosHtLL^2 +
H-6sinH2tLsin
3
HtL- 12 cosHtLcosH2tLsin
2
HtL+ 12 cos
2
HtLsinH2tLsinHtLL^2MM,
H6sinH2tLcos
3
HtL- 12 cosH2 tLsinHtLcos
2
HtL- 12 sin
2
HtLsinH2tLcosHtLLë
I
,
IH
-9sin
2
HtLcos
4
HtL- 9sin
4
HtLcos
2
HtLL
2
+
H6sinH2tLcos
3
HtL- 12 cosH2 tLsinHtLcos
2
HtL- 12 sin
2
HtLsinH2tLcosHtLL^2 +
H-6sinH2tLsin
3
HtL- 12 cosHtLcosH2tLsin
2
HtL+ 12 cos
2
HtLsinH2tLsinHtLL^2MM,
H-9sin
2
HtLcos
4
HtL- 9sin
4
HtLcos
2
HtLL
ë
I
,
IH-9sin
2
HtLcos
4
HtL- 9sin
4
HtLcos
2
HtLL
2
+
H6sinH2tLcos
3
HtL- 12 cosH2 tLsinHtLcos
2
HtL- 12 sin
2
HtLsinH2tLcosHtLL^2 +
H
FullSimplify[%]
-6sinH2tLsin
3
HtL- 12 cosHtLcosH2tLsin
2
HtL+ 12 cos
2
HtLsinH2tLsinHtLL^2MM=
:
1
5
HL HL H
csc
2
t sec t
"##################
sin
4
2 tL, -
1
5
csc
HtLsec
2
HtL
"##################
sin
4
H2 tL, -
3
5
csc
2
H2 tL
"##################
sin
4
H2 tL>
PowerExpand[%]
31
:
1
5
csc
2
HtLsecHtLsin
2
H2 tL, -
1
5
csc
HtLsec
2
HtLsin
2
H2 tL, -
3
5
>
8H-27cos3HtL sin6HtL - 27cos5HtL sin4HtL - 24cosHtL sin2H2tL sin2HtL - 24cos2HtL cosH2tL sinH2tL sinHtL + 12cos3HtL sin2H2tLL ê
,
H HH-18sinHtL sinH2tL cos5HtL + 36cosH2tL sin2HtL cos4HtL + 36cosH2tL sin4HtL cos2HtL + 18sin5HtL sinH2tL cosHtLL^2 +
H-27cos3HtL sin6HtL - 27cos5HtL sin4HtL - 24cosHtL sin2H2tL sin2HtL - 24cos2HtL cosH2tL sinH2tL sinHtL +
12cos3HtL sin2H2tLL^2 + H-27sin3HtL cos6HtL - 27sin5HtL cos4HtL -
24sinHtL sin2H2tL cos2HtL + 24cosH2tL sin2HtL sinH2tL cosHtL + 12sin3HtL sin2H2tLL ^2LL,
H-27sin3HtL cos6HtL - 27sin5HtL cos4HtL - 24sinHtL sin2H2tL cos2HtL + 24cosH2tL sin2HtL sinH2tL cosHtL + 12sin3HtL sin2H2tLL ê
,
H HH-18sinHtL sinH2tL cos5HtL + 36cosH2tL sin2HtL cos4HtL + 36cosH2tL sin4HtL cos2HtL + 18sin5HtL sinH2tL cosHtLL^2 +
H-27cos3HtL sin6HtL - 27cos5HtL sin4HtL - 24cosHtL sin2H2tL sin2HtL - 24cos2HtL cosH2tL sinH2tL sinHtL +
12cos3HtL sin2H2tLL^2 + H-27sin3HtL cos6HtL - 27sin5HtL cos4HtL -
24sinHtL sin2H2tL cos2HtL + 24cosH2tL sin2HtL sinH2tL cosHtL + 12sin3HtL sin2H2tLL ^2LL,
H-18sinHtL sinH2tL cos5HtL + 36cosH2tL sin2HtL cos4HtL + 36cosH2tL sin4HtL cos2HtL + 18sin5HtL sinH2tL cosHtLLê
,
H HH-18sinHtL sinH2tL cos5HtL + 36cosH2tL sin2HtL cos4HtL + 36cosH2tL sin4HtL cos2HtL + 18sin5HtL sinH2tL cosHtLL^2 +
H-27cos3HtL sin6HtL - 27cos5HtL sin4HtL - 24cosHtL sin2H2tL sin2HtL - 24cos2HtL cosH2tL sinH2tL sinHtL +
12cos3HtL sin2H2tLL^2 + H-27sin3HtL cos6HtL - 27sin5HtL cos4HtL -
24sinHtL sin2H2tL cos2HtL + 24cosH2tL sin2HtL sinH2tL cosHtL + 12sin3HtL sin2H2tLL ^2LL<
3 "################## 2 "##################
FullSimplify[%]
:- csc H tL sec H tL sin6 H 2 tL , - csc H tL sec H tL sin6 H 2 tL , 0>
1 2 1 3
8 8
PowerExpand[%]
:- csc H tL sec H tL sin H 2 tL , - csc H tL sec H tL sin H 2 tL , 0>
1 2 3 3 1 3 2 3
8 8
А единичный вектор бинормали:
9H-6 sinH2 tL sin3HtL - 12 cosHtL cosH2 tL sin2HtL + 12 cos2HtL sinH2 tL sinHtLL ë
ubv[t]=bv[f][u]/Norm[bv[f][u]]/.u→t
I, IH-9 sin2HtL cos4HtL - 9 sin4HtL cos2HtLL2 +
H6 sinH2 tL cos3HtL - 12 cosH2 tL sinHtL cos2HtL - 12 sin2HtL sinH2 tL cosHtLL ^2 +
H-6 sinH2 tL sin3HtL - 12 cosHtL cosH2 tL sin2HtL + 12 cos2HtL sinH2 tL sinHtLL ^2MM,
H6 sinH2 tL cos3HtL - 12 cosH2 tL sinHtL cos2HtL - 12 sin2HtL sinH2 tL cosHtLLë
I, IH-9 sin2HtL cos4HtL - 9 sin4HtL cos2HtLL +
2
H6 sinH2 tL cos3HtL - 12 cosH2 tL sinHtL cos2HtL - 12 sin2HtL sinH2 tL cosHtLL ^2 +
H-6 sinH2 tL sin3HtL - 12 cosHtL cosH2 tL sin2HtL + 12 cos2HtL sinH2 tL sinHtLL ^2MM,
H-9 sin2HtL cos4HtL - 9 sin4HtL cos2HtLL ëI, IH-9 sin2HtL cos4HtL - 9 sin4HtL cos2HtLL +
2
H6 sinH2 tL cos3HtL - 12 cosH2 tL sinHtL cos2HtL - 12 sin2HtL sinH2 tL cosHtLL ^2 +
H-6 sinH2 tL sin3HtL - 12 cosHtL cosH2 tL sin2HtL + 12 cos2HtL sinH2 tL sinHtLL ^2MM=
"################## 2 "################## "##################
FullSimplify[%]
: csc H tL secH tL sin4 H 2 tL , - cscH tL sec H tL sin4H 2 tL , - csc H 2 tL sin4 H 2 tL >
1 2 1 3 2
5 5 5
PowerExpand[%]
: csc H tL secH tL sin H 2 tL , - cscH tL sec H tL sin H 2 tL , - >
1 2 2 1 2 2 3
5 5 5
31
