曲线上标架 谢锡麟 所以有 (F(t),r(t)g3, 此时 (s()=r(0)-(r(r()2;( Ir(t)23 r(t)I# P(g:()-(()/)2pro 根据内蕴正交分解,对于向量T,有 令T=(),e=m(可得 r"(s(t)) t) r(t) (t)×((t)×(t) (t)1s|(t)l8(|r()gs lr(t)I3 因此主法向量为 r(t)×(()×r(t) (t)ls|i(t)×(r(t)×(t)ls|(t)ls|(t)×(t)|R 此时曲率可以表示为 k(t)=r"s(t)kr(t)×(r(t)×r(t r(t)×r(t)lRs r(t)3 r(t)13 副法向量即为 b()=7(t)×n(t) r(t) r(tIR irOR/Fr(-( (t) (t)×(t) Ir(t)lR3/R3 r(t)IR r(t)1 挠率为 (t)= b(s(t),n(t) /db dt (t)x(s),n2(t) 1dr(t)×(t)(t)×(r(t)×(t) r(t)ldt|(t)×(t)3’|(t)lk(t)×(t)l3)g (t)×r(t) r(t)×(r(t)×F(t) r(t)lksl(t)×y(t)ls'|(t)ls|(t)×r(t) r(t)×(t), r(t) (t)(r(la-() T(t)×(t) r(t)×r(t)-r(1)+(F(t), (t) r(tIr3 (0)×F(1(()xr(,F0)=P()xO 式中倒数第二行使用了向量的内蕴正交分解 综上所述,一般形式的 Frenet标架可以表示为张量分析讲稿谢锡麟 曲线上标架 谢锡麟 所以有 d dt |r˙(t)|R3 = 1 |r˙(t)|R3 (r¨(t), r˙(t))R3 , 此时 r ′′(s(t)) = r¨(t) |r˙(t)| 2 R3 − (r¨(t), r˙(t))R3 |r˙(t)| 4 R3 r˙(t) = 1 |r˙(t)| 2 R3 [ r¨(t) − ( r¨(t), r˙(t) |r˙(t)|R3 ) R3 r˙(t) |r˙(t)|R3 ] . 根据内蕴正交分解, 对于向量 T , 有 T = (T , e)R3 e − e × (e × T ). 令 T = r¨(t), e = r˙(t) |r˙(t)|R3 可得 r ′′(s(t)) = − 1 |r˙(t)| 2 R3 r˙(t) |r˙(t)|R3 × ( r˙(t) |r˙(t)|R3 × r¨(t) ) = − r˙(t) × (r˙(t) × r¨(t)) |r˙(t)| 4 R3 . 因此主法向量为 n(t) = r ′′(s(t)) |r ′′(s(t))|R3 = − r˙(t) × (r˙(t) × r¨(t)) |r˙(t) × (r˙(t) × r¨(t))|R3 = − r˙(t) × (r˙(t) × r¨(t)) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 . 此时曲率可以表示为 κ(t) = |r ′′(s(t))|R3 = |r˙(t) × (r˙(t) × r¨(t))|R3 |r˙(t)| 4 R3 = |r˙(t) × r¨(t)|R3 |r˙(t)| 3 R3 . 副法向量即为 b(t) = τ (t) × n(t) = r˙(t) |r˙(t)|R3 × 1 |r˙(t)| 2 R3 [ r¨(t) − ( r¨(t), r˙(t) |r˙(t)|R3 ) R3 r˙(t) |r˙(t)|R3 ] = r˙(t) × r¨(t) |r˙(t)| 3 R3 . 挠率为 σ(t) = − ( d ds b(s(t)), n(t) ) R3 = − ( db dt (t) dt ds (s), n(t) ) R3 = − ( 1 |r˙(t)|R3 d dt r˙(t) × r¨(t) |r˙(t) × r¨(t)|R3 , − r˙(t) × (r˙(t) × r¨(t)) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 ) R3 = ( r˙(t) × ... r (t) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 , r˙(t) × (r˙(t) × r¨(t)) |r˙(t)|R3 |r˙(t) × r¨(t)|R3 ) R3 = 1 |r˙(t) × r¨(t)| 2 R3 ( r˙(t) × ... r (t), r˙(t) |r˙(t)|R3 × ( r˙(t) |r˙(t)|R3 × r¨(t) )) R3 = 1 |r˙(t) × r¨(t)| 2 R3 ( r˙(t) × ... r (t), −r¨(t) + ( r¨(t), r˙(t) |r˙(t)|R3 ) R3 r˙(t) |r˙(t)|R3 ) R3 = − 1 |r˙(t) × r¨(t)| 2 R3 (r˙(t) × ... r (t), r¨(t))R3 = [r˙(t), r¨(t), ... r (t)] |r˙(t) × r¨(t)| 2 R3 , 式中倒数第二行使用了向量的内蕴正交分解. 综上所述, 一般形式的 Frenet 标架可以表示为 5