Rotation: Focus Axis Angle - Exponential Map
- Matrix \(\mathrm{K}\) is such that \(\mathrm{K}\,v=n \times v\).
- \(\mathrm{K}=\left(\begin{array}{ccc}0 & -n_z & n_y \\ n_z & 0 & -n_x \\ -n_y & n_x & 0\end{array}\right)\) is skew-antisymetric. And \(K^3=-K\)
- The rotation matrix can be obtained as the matrix exponential
- \(\mathrm{R}=\mathrm{exp}(\theta\,\mathrm{K}) = \displaystyle \sum_k \frac{(\theta\,\mathrm{K})^k}{k!}\)
-
- \(\displaystyle =I + \theta \mathrm{K} + \frac{1}{2!} (\theta \mathrm{K})^2 + \frac{1}{3!} (\theta \mathrm{K})^3 + \dots\)
- \(\displaystyle = I+ \underbrace{\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-...\right)}_{\sin(\theta)}\,\mathrm{K} +
\underbrace{\left(\frac{\theta^2}{2!}-\frac{\theta^4}{4!}+\frac{\theta^6}{6!}-...\right)}_{1-\cos(\theta)}\,\mathrm{K}^2\)
- \(= I+ \sin(\theta)\,\mathrm{K} + (1-\cos(\theta))\,\mathrm{K}^2\) Rodrigues formula
- Allows for matrix interpolation with varying \(\theta\) (exponential map).