Cayley变换和Cayley参数化

Cayley Transformation and Cayley Parameterization

首先从李群的观点说明了反对称矩阵构成的群是特殊正交群的李代数,然后给出Cayley变换的表达式,接着证明了由Cayley变换可以生产特殊正交矩阵,并且讨论了Cayley逆变换。最后笔者比较了旋转矩阵的Cayley参数化和指数表示法,以及Rodriguez参数化之间的表达式,可以得出这三种表示法在不形式上是统一的。

In this paper,from the point of view of Lie group,it is shown that the group formed by antisymmetric matrix is lie algebra of special orthogonal group,Then the expression of Cayley transform is given.Then it is proved that special orthogonal matrix can be produced by Cayley transformation,and the inverse Cayley transform is also discussed.Finally,we compare the expressions between Cayley parameterization and exponential representation,and Rodriguez parameterization,It can be concluded that the three representations are unified in form.