Editing Cayley transform (section)

===Inverse===
Let <math>u^* = \cos \theta - r \sin \theta = u^{-1} .</math> Since
:<math>\begin{pmatrix} 1 & 1 \\ -u & u \end{pmatrix}\ \begin{pmatrix} 1 & -u^*  \\ 1 & u^* \end{pmatrix} \ = \ \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \ \sim \  \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \ ,</math>
where the equivalence is in the [[projective linear group]] over quaternions, the [[inverse function|inverse]] of <math>f(u,1)</math> is
:<math>U[p,1] \begin{pmatrix} 1 & -u^* \\ 1 & u^* \end{pmatrix} \ = \ U[p+1,\ (1-p)u^*] \sim U[u(1-p)^{-1} (p+1), \ 1] .</math>
Since homographies are [[bijection]]s, <math>f^{-1} (u,1)</math> maps the vector quaternions to the 3-sphere of versors. As versors represent rotations in 3-space, the homography <math>f^{-1}</math> produces rotations from the ball in <math>\R^3</math>.