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 \(t = 1 - \cos\group{\alpha}\).