旋转之二 - 三维空间中的旋转:罗德里格旋转公式

先来推导三维空间中的常用旋转公式。

三维空间中的旋转

如果 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%7D
 绕着空间中的一个单位向量 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 旋转,旋转我们采用右手坐标系:

https://pic1.zhimg.com/v2-d78335b490e755055d4f053206842378_b.jpg

 

可以把 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%7D
 分解成平行于 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 的向量 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%7C%7C%7D%7D
 和 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%5Cperp%7D%7D
 :

https://pic4.zhimg.com/v2-a822a57a90754decc3a48a35628e7dc7_b.jpg

 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%7C%7C%7D%7D
 绕 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 旋转并不会造成什么改变。

当 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%7C%7C%7D%7D
 平行于旋转轴 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 时,旋转 
https://www.zhihu.com/equation?tex=%5Ctheta
 角度之后的 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%27_%7B%7C%7C%7D%7D
为:

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%27_%7B%7C%7C%7D%7D+%3D+%5Cmathbf%7Bv_%7B%7C%7C%7D%7D%5C%5C

根据正交投影公式:

https://www.zhihu.com/equation?tex=%5Cbegin%7Balign%2A%7D+++++%5Cmathbf%7Bv_%7B%7C%7C%7D%7D++%3D+%7B%7D%26+proj_u%28%5Cmathbf%7Bv%7D%29+%5C%5C++++%26%3D+%5Cfrac%7B%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%7D%7B%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bu%7D%7D+%5Cmathbf%7Bu%7D+%5C%5C+++++%26%3D+%5Cfrac%7B%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%7D%7B%5ClVert+u+%5CrVert+%5E2%7D+%5Cmathbf%7Bu%7D++%26+%5ClVert+u+%5CrVert+%5E2+%3D+1%5C%5C+++++%26%3D+%28%7B%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%7D%29+%5Cmathbf%7Bu%7D+%5Cend%7Balign%2A%7D%5C%5C+

观察 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%5Cperp%7D%7D
 :

https://pic2.zhimg.com/v2-219ee1327d6370ef242c8d903520513d_b.jpg

 

它可以表示为2D空间上的旋转,我们需要一组基来描述,可以选择 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%5Cperp%7D%7D
 和 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bw%7D
 , 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bw%7D
 可以通过叉乘构造出来,它垂直于 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 和 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%5Cperp%7D%7D
 ,注意右手准则:

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bw%7D+%3D+%5Cmathbf%7Bu%7D+%5Ctimes+%5Cmathbf%7Bv_%7B%5Cperp%7D%7D%5C%5C

因为 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 是单位向量,上式算出来的 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bw%7D
 模长和 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%5Cperp%7D%7D
 相同,所以下图是准确的:

https://pic4.zhimg.com/v2-c9aff8e1290355dbdb22339d7e283653_b.jpg

 

有了这些条件之后:

https://www.zhihu.com/equation?tex=%5Cbegin%7Balign%2A%7D++%5Cmathbf%7Bv%27_%7B%5Cperp%7D%7D+%26%3D+%5Cmathbf%7Bv%27_v%7D+%2B+%5Cmathbf%7Bv%27_w%7D++%5C%5C+++%26%3D+%5Cmathbf%7Bv%27_%7B%5Cperp%7D%7D+%5Ccos+%5Ctheta++%2B+%5Cmathbf%7Bw%7D+%5Csin+%5Ctheta+%5C%5C+++%26%3D+%5Cmathbf%7Bv%27_%7B%5Cperp%7D%7D+%5Ccos+%5Ctheta++%2B+%28%5Cmathbf%7Bu%7D+%5Ctimes++%5Cmathbf%7Bv_%7B%5Cperp%7D%7D%29+%5Csin+%5Ctheta+%5Cend%7Balign%2A%7D++%5C%5C

当 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%5Cperp%7D%7D
 正交于旋转轴 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 时,旋转 
https://www.zhihu.com/equation?tex=%5Ctheta
 角度之后的 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%27%7B%5Cperp%7D%7D
为:

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%27_%7B%5Cperp%7D%7D+%3D%5Cmathbf%7Bv_%7B%5Cperp%7D%7D+%5Ccos+%5Ctheta++%2B++%28%5Cmathbf%7Bu%7D+%5Ctimes++%5Cmathbf%7Bv_%7B%5Cperp%7D%7D%29+%5Csin+%5Ctheta%5C%5C+

所以最终旋转得到的 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%27%7D
 :

https://www.zhihu.com/equation?tex=%5Cbegin%7Balign%2A%7D++%5Cmathbf%7Bv%27%7D++%3D+%7B%7D%26+%5Cmathbf%7Bv_%7B%7C%7C%7D%7D+%2B+%5Cmathbf%7Bv%27_%7B%5Cperp%7D%7D+%5C%5C++++%26%3D+%5Cmathbf%7Bv_%7B%7C%7C%7D%7D+%2B+%5Cmathbf%7Bv_%7B%5Cperp%7D%7D+%5Ccos+%5Ctheta++%2B++%28%5Cmathbf%7Bu%7D+%5Ctimes++%5Cmathbf%7Bv_%7B%5Cperp%7D%7D%29+%5Csin+%5Ctheta+%5C%5C++%5Cend%7Balign%2A%7D%5C%5C

又:

https://www.zhihu.com/equation?tex=%5Cbegin%7Balign%2A%7D+++%5Cmathbf%7Bu%7D+%5Ctimes++%5Cmathbf%7Bv_%7B%5Cperp%7D%7D+%3D%7B%7D%26++%5Cmathbf%7Bu%7D+%5Ctimes%28+%5Cmathbf%7Bv%7D+-++%5Cmathbf%7Bv_%7B%7C%7C%7D%7D%29+%5C%5C++%26%3D+%5Cmathbf%7Bu%7D+%5Ctimes+%5Cmathbf%7Bv%7D+-+%5Cmathbf%7Bu%7D+%5Ctimes+%5Cmathbf%7Bv_%7B%7C%7C%7D%7D+%26+%28%5Cmathbf%7Bu%7D+%5Ctimes+%5Cmathbf%7Bv_%7B%7C%7C%7D%7D+%3D+0+%29+%5C%5C++%26%3D+%5Cmathbf%7Bu%7D+%5Ctimes+%5Cmathbf%7Bv%7D++%5Cend%7Balign%2A%7D%5C%5C

将上述结论, 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%7C%7C%7D%7D+%3D+%28%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%29+%5Cmathbf%7Bu%7D
 以及 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv_%7B%5Cperp%7D%7D+%3D%5Cmathbf%7Bv%7D+-++%5Cmathbf%7Bv_%7B%7C%7C%7D%7D
 代入回 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%27%7D+%3D+%5Cmathbf%7Bv_%7B%7C%7C%7D%7D+%2B+%5Cmathbf%7Bv_%7B%5Cperp%7D%7D+%5Ccos+%5Ctheta++%2B++%28%5Cmathbf%7Bu%7D+%5Ctimes++%5Cmathbf%7Bv_%7B%5Cperp%7D%7D%29+%5Csin+%5Ctheta
 :

https://www.zhihu.com/equation?tex=+%5Cbegin%7Balign%2A%7D+++%5Cmathbf%7Bv%27%7D++%3D+%7B%7D%26+%5Cmathbf%7Bv_%7B%7C%7C%7D%7D+%2B+%5Cmathbf%7Bv_%7B%5Cperp%7D%7D+%5Ccos+%5Ctheta++%2B++%28%5Cmathbf%7Bu%7D+%5Ctimes++%5Cmathbf%7Bv_%7B%5Cperp%7D%7D%29+%5Csin+%5Ctheta+%5C%5C++%26%3D+%28%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%29+%5Cmathbf%7Bu%7D+%2B+%28%5Cmathbf%7Bv%7D+-+%28%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%29%29+%5Cmathbf%7Bu%7D+%5Ccos+%5Ctheta+%2B++%28%5Cmathbf%7Bu%7D+%5Ctimes++%5Cmathbf%7Bv%7D%29+%5Csin+%5Ctheta+%5C%5C++%26+%3D+%5Cmathbf%7Bv%7D+%5Ccos+%5Ctheta+%2B++%28%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%29+%5Cmathbf%7Bu%7D+%281-+%5Ccos+%5Ctheta%29%2B+%28%5Cmathbf%7Bu%7D+%5Ctimes+%5Cmathbf%7Bv%7D%29+%5Csin+%5Ctheta+%5Cend%7Balign%2A%7D+%5C%5C

3D空间中任意一个 

https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%7D
 沿着单位向量 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bu%7D
 旋转 
https://www.zhihu.com/equation?tex=%5Ctheta+
 角度之后的 
https://www.zhihu.com/equation?tex=%5Cmathbf%7Bv%27%7D
 为:

https://www.zhihu.com/equation?tex=+%5Cmathbf%7Bv%27%7D+%3D+%5Cmathbf%7Bv%7D+%5Ccos+%5Ctheta+%2B++%28%5Cmathbf%7Bu%7D+%5Ccdot+%5Cmathbf%7Bv%7D%29+%5Cmathbf%7Bu%7D+%281-+%5Ccos+%5Ctheta%29%2B+%28%5Cmathbf%7Bu%7D+%5Ctimes+%5Cmathbf%7Bv%7D%29+%5Csin+%5Ctheta%5C%5C

 

上述公式也就是 罗德里格旋转公式(Rodrigues’ Rotation Formula) ,上述过程也就是其证明/推导过程。

 

(转载至知乎用户二圈妹

0 个评论

要回复文章请先登录注册