es

This section is divided into a number of subsections, links to which are:

Rotations

2D Rotations

3D Rotations

Euler Rotation Theorem

3D Rotation vs Shearing

Quaternions

Compositions

 

Cayley Representation

https://rotations.berkeley.edu/other-representations-of-a-rotation/

1846, Cayley [17] introduced what is now known as the Cayley transform of a second-order skew-symmetric tensor A:

\begin{equation} \label{EqCayley.1} \mbox{Cay}({\bf A}) = \left( {\bf I} - {\bf A} \right)^{-1} \left( {\bf I} + {\bf A} \right) . \end{equation}
He showed that the Cayley transform of A = - AT is a proper-orthogonal tensor, and hence a rotation tensor1. The transform is often invertible. If C = Cay}(A = - AT), then
\begin{equation} \label{EqCayley.2} {\bf A} = \left( {\bf C} - {\bf I} \right) \left( {\bf I} + {\bf C} \right)^{-1} , \end{equation}
provided I + C is invertible. It is interesting to note that
\[ \mbox{Cay}(-{\bf A}) = \left( \mbox{Cay}({\bf A}) \right)^{\mathrm T} . \]
Thus, the inverse of a rotation is obtained by setting A ↦ - A. Given a skew-symmetric tensor Λ, we denote the representation defined by the Cayley transform of Λ as the Cayley representation of a rotation:
\begin{equation} \label{EqCayley.3} {\bf R} = {\bf R}_{Cayley} \left( \Lambda \right) = \left( {\bf I} - \Lambda \right)^{-1} \left( {\bf I} + \Lambda \right) . \end{equation}


  1. Cayley, A., Sur quelques propriétés des déterminants gauches, Journal für die reine und angewandte Mathematik 32 119-123 (1846). Reprinted in pp. 332-336 of The Collected Mathematical Papers of Arthur Cayley, Sc.D., F.R.S., Vol. 1, Cambridge University Press, Cambridge (1889).
  2. Mladenova, C. D., and Mladenov, I. M., Vector decomposition of finite rotations, Reports on Mathematical Physics 68(1) 107-117 (2011).
  3. Norris, A. N., >Euler-Rodrigues and Cayley formulae for rotation of elasticity tensors, Mathematics and Mechanics of Solids 13(6) 465-498 (2008).