\( % Arcus cosine. \def\acos{\cos^{-1}} % Vector projection. \def\projection#1#2{{proj_{#1}\left(#2\right)}} % Vector rejection. \def\rejection#1#2{{rej_{#1}\left(#2\right)}} % Norm. \def\norm#1{{\left\|#1\right\|}} % Cross product. \def\cross#1#2{\mathit{cross}\left(#1,#2\right)} % Dot product. \def\dot#1#2{{#1 \cdot #2}} % Magnitude. \def\mag#1{{\left|#1\right}} \def\group#1{\left(#1\right)}} \def\sbgrp#1{\left\{#1\right\}} \)

Axis angle to matrix

An axis angle \(\left(\hat{r},\alpha\right)\) represents a rotation by an angle \(\alpha\) around an axis \(\hat{r}\). The matrix representing the same transformation is given by a rotation matrix \[ R = \left[\begin{matrix} t \hat{r}_x^2 + c & t \hat{r}_x \hat{r}_y - s \hat{r}_z & t \hat{r}_x \hat{r}_z + s \hat{r}_y \\ t \hat{r}_x \hat{r}_y + s \hat{r}_z & t \hat{r}_y^2 + c & t \hat{r}_y \hat{r}_z - s \hat{r}_x \\ t \hat{r}_x \hat{r}_z - s \hat{r}_y & t \hat{r}_y \hat{r}_z + s \hat{r}_x & t \hat{r}_z^2 + c \end{matrix}\right] \] with \(c = \cos\group{\alpha}\), \(s = \sin\group{\alpha}\), and \(t = 1 - \cos\group{\alpha}\).

Proof

An axis angle \(\left(\hat{r},\alpha\right)\) represents a rotation by an angle \(\alpha\) around an axis \(\hat{r}\). Furthermore, we have an explicit matrix representation for such transformations. Substituting the \(\alpha\) and \(\hat{r}\) into that matrix gives \[ R = \left[\begin{matrix} t \hat{r}_x^2 + c & t \hat{r}_x \hat{r}_y - s \hat{r}_z & t \hat{r}_x \hat{r}_z + s \hat{r}_y \\ t \hat{r}_x \hat{r}_y + s \hat{r}_z & t \hat{r}_y^2 + c & t \hat{r}_y \hat{r}_z - s \hat{r}_x \\ t \hat{r}_x \hat{r}_z - s \hat{r}_y & t \hat{r}_y \hat{r}_z + s \hat{r}_x & t \hat{r}_z^2 + c \end{matrix}\right] \] with \(c = \cos\group{\alpha}\), \(s = \sin\group{\alpha}\), and \(t = 1 - \cos\group{\alpha}\).