1.4. Rodrigues' Formula

Rodrigues' Formula

Rodrigues’ formula is the closed-form expression for the matrix exponential $e^{\hat{\omega }\theta }$, converting an axis–angle pair $(\omega ,\theta )$ into a rotation matrix $R\in SO(3)$. It collapses the infinite Taylor series into a finite expression by exploiting the fact that powers of a skew-symmetric matrix cycle with period 2 through $\hat{\omega }$ and ${\hat{\omega }}^2$.

Given a unit rotation axis $\omega \in {\mathbb{R}}^3$ with $\Vert \omega \Vert =1$ and angle $\theta \in \mathbb{R}$, the rotation matrix is:

$e^{\hat{\omega }\theta }=I+\hat{\omega }\sin \theta +{\hat{\omega }}^2(1-\cos \theta )$

where $\hat{\omega }\in so(3)$ is the $3\times 3$ skew-symmetric cross-product matrix:

$\hat{\omega }=\left[\begin{array}{ccc}0& -{\omega }_z& {\omega }_y\\ {\omega }_z& 0& -{\omega }_x\\ -{\omega }_y& {\omega }_x& 0\end{array}\right]\text{such that }\omega \times v=\hat{\omega }v$

For non-unit $\omega$:

$e^{\hat{\omega }\theta }=I+\frac{\hat{\omega }}{\Vert \omega \Vert }\sin (\Vert \omega \Vert \theta )+\frac{{\hat{\omega }}^2}{\Vert \omega {\Vert }^2}(1-\cos (\Vert \omega \Vert \theta ))$

This is the 3D matrix analogue of Euler’s formula $e^{j\varphi }=\cos \varphi +j\sin \varphi$, where the role of $j$ (with $j^2=-1$) is played by $\hat{\omega }$ (with ${\hat{\omega }}^3=-\hat{\omega }$).

Intuition

The formula decomposes the rotation of a point $q$ about axis $\omega$ by angle $\theta$ into three geometric contributions:

Term	What it does
$I\cdot q$	Keeps the original point
$\hat{\omega }\sin \theta \cdot q$	Swings $q$ perpendicular to both $\omega$ and $q$ — the “sine” component of circular motion
${\hat{\omega }}^2(1-\cos \theta )\cdot q$	Pulls $q$ toward the rotation axis — the “versine” radial correction

Together, the $\sin \theta$ and $(1-\cos \theta )$ terms trace out a circle in the plane perpendicular to $\omega$, exactly as $\cos$ and $\sin$ parameterize the unit circle in 2D.

The reason this works is that higher powers of $\hat{\omega }$ don’t produce new directions — they cycle back through $\hat{\omega }$ and ${\hat{\omega }}^2$. So the infinite series naturally groups into the Taylor series for $\sin$ and $1-\cos$.

Properties

Power identities for $\hat{\omega }$ (Lemma A)

These two identities are the engine behind the proof. For any $\omega \in {\mathbb{R}}^3$:

1. ${\hat{\omega }}^2=\omega {\omega }^T-\Vert \omega {\Vert }^2{I}_3$, from the triple cross product $\omega \times (\omega \times v)=({\omega }^{T}v)\omega -\Vert \omega {\Vert }^{2}v$.

2. ${\hat{\omega }}^3=-\Vert \omega {\Vert }^2\hat{\omega }$, since $\omega \times ({\hat{\omega }}^{2}v)$ kills the $\omega \times \omega$ term, leaving $-\Vert \omega {\Vert }^2\hat{\omega }v$.

When $\Vert \omega \Vert =1$ these simplify to ${\hat{\omega }}^2=\omega {\omega }^T-I$ and ${\hat{\omega }}^3=-\hat{\omega }$, producing the cycling:

${\hat{\omega }}^3=-\hat{\omega },{\hat{\omega }}^4=-{\hat{\omega }}^2,{\hat{\omega }}^5=\hat{\omega },{\hat{\omega }}^6={\hat{\omega }}^2,\dots$

Hat conjugation identity

$\widehat{(R\omega )}=R\hat{\omega }{R}^T$

To find the skew-symmetric matrix of a rotated vector, conjugate the original by $R$. This is used to prove rotation matrices are orientation-preserving RBTs, and it recurs throughout Forward Kinematics and the Spatial Jacobian.

Special cases

Input	Output
$\theta =0$	$e^{\hat{\omega }\cdot 0}=I$ (no rotation)
$\theta =\pi$	$e^{\hat{\omega }\pi }=I+2{\hat{\omega }}^2=2\omega {\omega }^T-I$ (180° flip)
$\theta =2\pi$	$e^{\hat{\omega }\cdot 2\pi }=I$ (full revolution)
$\omega =\hat{z}$, any $\theta$	Recovers the standard 2D rotation matrix $R_z(\theta )$

Proof sketch

Expand $e^{\hat{\omega }\theta }$, substitute the cycling pattern from Lemma A, and group terms:

$e^{\hat{\omega }\theta }=I+\underset{\sin \theta }{\underbrace{\left(\theta -\frac{{\theta }^3}{3!}+\frac{{\theta }^5}{5!}-\cdots \right)}}\hat{\omega }+\underset{1-\cos \theta }{\underbrace{\left(\frac{{\theta }^2}{2!}-\frac{{\theta }^4}{4!}+\frac{{\theta }^6}{6!}-\cdots \right)}}{\hat{\omega }}^2■$

Examples

Rotation about $z$-axis by $\theta$

Let $\omega =\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$. Then $\hat{\omega }=\left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]$ and ${\hat{\omega }}^2=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right]$.

$e^{\hat{\omega }\theta }=I+\sin \theta \left[\begin{array}{ccc}0& -1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right]+(1-\cos \theta )\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right]=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$

This recovers the standard $R_z(\theta )$, confirming that elemental rotations used in Euler Angles are special cases of Rodrigues’.

180° about an arbitrary axis

With $\theta =\pi$: $\sin \pi =0$ and $1-\cos \pi =2$, so:

$e^{\hat{\omega }\pi }=I+2{\hat{\omega }}^2=I+2(\omega {\omega }^T-I)=2\omega {\omega }^T-I$

This rank-1 structure ($R+I=2\omega {\omega }^T$) is exactly what the Exponential Coordinates log map exploits in Case 3 ($\text{tr}(R)=-1$) to recover $\omega$ by normalizing any nonzero column of $R+I$.

Implementation

import numpy as np
 
def skew(w):
    return np.array([[0, -w[2], w[1]], [w[2], 0, -w[0]], [-w[1], w[0], 0]])
 
def rodrigues(omega, theta):
    """omega: unit vector (3,), theta: angle in radians -> R in SO(3)"""
    w_hat = skew(omega)
    return np.eye(3) + w_hat * np.sin(theta) + w_hat @ w_hat * (1 - np.cos(theta))

The inverse operation — recovering $(\omega ,\theta )$ from $R$ — is the log map, covered in Exponential Coordinates.