# Representations of the Rotation

• rotation matrix $$R$$
• rotation around a vector $$\omega$$ with angle $$\theta$$
• $$R\omega=1\omega$$, i.e. $$\omega$$ is an eigenvector
• one is the only real eigenvalue
• exponential form $$\exp\hat\omega\theta$$
• Rodrigues’ formula

The exponential form can be found via implicit derivation of $$R(t)R^T(t)=I$$ and solution of the resulting ODE.

• $$\dot R(t)R^T(t) =\hat\omega(t)$$ has to be skew-symmetric

# Formulæ

Definition $\exp(\hat\omega) = I + \sum_{i=1}^\infty \frac{(\hat\omega t)^i}{i!}.$

Rodrigues $\exp(\hat\omega) = I + \frac{\hat\omega}{||\omega||}\sin(||\omega||} + \frac{\hat\omega^2}{||\omega||^2}(1-\cos(||\omega||})$

# Angular Velocity

Derivation is easy from the exponential form. We have $\mathbf{X}(t) = \exp(\hat\omega t)\mathbf{X}_0$ so $\dot\mathbf{X}=\frac{\partial}{\parial t}\exp(\hat\omega t)\mathbf{X}_0 = \hat\omega\exp(\hat\omega t)\mathbf{X}_0=\hat\omega\mathbf{X}(t)$

The angular velocity is the vector $$\omega$$.