## Rodrigues’ Rotation Formula

We can compute a rotation matrix $$R \in SO(3)$$ from an angle $$\theta$$ and axis $$l$$ (unit vector)

$R = I + \sin(\theta)C + (1-\cos \theta)C^2$

where $$C$$ is the antisymmetric matrix:

$C = \begin{bmatrix} 0 & -l_z & l_y \\ l_z & 0 & -l_x \\ -l_y & l_x & 0 \end{bmatrix}$

Our proof will use the Taylor’s formula to define the exponential of a matrix $$M$$

$e^M = \sum_{k=0}^{\infty}\frac{M^k}{k!}$

where $$M^0$$ is defined to be the identity matrix $$I$$. Because $$C^3 = -C$$ and the Taylor expansions:

\begin{align}
\sin \theta &= \theta – \frac{\theta^3}{3!} + \frac{\theta^5}{5!} – \cdots + \frac{(-1)^n}{(2n+1)!}\theta^{2n+1} + \cdots \\
\cos \theta &= 1 – \frac{\theta^2}{2!} + \frac{\theta^4}{4!} – \cdots + \frac{(-1)^n}{(2n)!}\theta^{2n} + \cdots
\end{align}

then we obtain

\begin{align}
R &= e^{\theta C}\\
&= \sum_{k=0}^{\infty}\frac{(\theta C)^k}{k!} \\
&= I + \frac{1}{1!}\theta C + \frac{1}{2!}(\theta C)^2 + \frac{1}{3!}(\theta C)^3 + \frac{1}{4!}(\theta C)^4 + \frac{1}{5!}(\theta C)^5 + \frac{1}{6!}(\theta C)^6 + \cdots \\
&= I + (\theta – \frac{\theta^3}{3!} + \frac{\theta^5}{5!} – \cdots)C + (\frac{\theta^2}{2!} – \frac{\theta^4}{4!} + \frac{\theta^6}{6!} – \cdots)C^2 \\
&= I + \sin\theta C + (1-\cos \theta)C^2
\end{align}