求解球面的一组正交标架
球面 r ( u , v ) = ( a cos u cos v , a cos u sin v , a sin u ) \mathbf{r}(u,v)=\left(a\cos u\cos v,a\cos u\sin v,a\sin u\right) r(u,v)=(acosucosv,acosusinv,asinu),
求得
r u = ( − a sin u cos v , − a sin u sin v , a cos u ) r v = ( − a cos u sin v , a cos u cos v , 0 ) \mathbf{r}_u=(-a\sin u\cos v,-a\sin u\sin v,a\cos u)\\\mathbf{r}_v=(-a\cos u\sin v,a\cos u\cos v,0) ru=(−asinucosv,−asinusinv,acosu)rv=(−acosusinv,acosucosv,0)
计算得
E = a 2 , F = 0 , G = a 2 cos 2 u E=a^2,F=0,G=a^2\cos^2u E=a2,F=0,G=a2cos2u
于是选取
e 1 = 1 a r u = ( − sin u cos v , − sin u sin v , cos u ) \mathbf{e}_{1}=\frac{1}{a}\mathbf{r}_{u}=(-\sin u\cos v,-\sin u\sin v,\cos u) e1=a1ru=(−sinucosv,−sinusinv,cosu)
e 2 = 1 a cos u r v = ( − sin v , cos v , 0 ) \mathbf{e}_2=\frac1{a\cos u}\mathbf{r}_v=(-\sin v,\cos v,0) e2=acosu1rv=(−sinv,cosv,0)
求其外积为
e 3 = n = ( − cos u cos v , − cos u sin v , − sin u ) \mathbf{e}_3=\mathbf{n}=(-\cos u\cos v,-\cos u\sin v,-\sin u) e3=n=(−cosucosv,−cosusinv,−sinu)
这就找到了球面的一组正交活动标架。
