---
title: Representations of 3D Motion
categories: lectures 3D mathematics
geometry: margin=2cm
fontsize: 12pt
---

# Representation of Rotations

Consider what happens when an object rotates continuously over time,
i.e. the rotational matrix is a function $R(t)$ of time.

## The derivative

1. Rotation is represented by an orthogonal matrix $R$
   $$R(t)\cdot R^T(t)=I$$
2. Implicit derivation
   $$\dot R(t)\cdot R^T(t)+R(t)\cdot\dot R^T(t)=0$$
3. by transposing the product and moving one term across, we have
   $$\dot R(t)\cdot R^T(t) = -(\dot R(t)\cdot R^T(t))^T$$
4. This is a skew-symmetric matrix, hence
   $$\exists \vec{\omega}\in\mathbb{R}^3, \text{s.t.}
     \dot R(t)\cdot R^T(t) = \hat\omega(t)$$
5. Multiply by $R(t)$ to get
   $$\dot R(t) = \hat\omega(t)\cdot R(t)$$
6. If $R(t_0)=I$ as an initial condition, then $\dot R(t)=\hat\omega(t)$

Note $so(3)$ is the space of all skew-symmetric matrices.

<!--
7. First Order approximation
   $$ R(t_0+dt)\approx I + \hat\omega(t_0)dt$$
-->

## The differential equation

Let $x(t)$ be a point rotated over time.

Assume that $\omega$ is constant.

1. **ODE:**
   $$\dot x(t) = \hat\omega x(t), \quad x(t)\in\mathbb{R}^3$$
2. Solution:
   $$x(t) = e^{\hat\omega t} x(0)$$
3. where
   $$e^{\hat\omega t} = I + \sum_{i=1}^\infty \frac{(\hat\omega )^i}{i!}$$
4. The rotational matrix $$R(t)=e^{\hat\omega t}$$
   signifies a rotation around the axis $\omega$ by $t$ radians.

$$
\begin{align}
\exp : \mathrm{so}(3)&\to\mathrm{SO}(3)
\\
 \hat\omega& \mapsto e^{\hat\omega}
\end{align}$$

This is a map from a Lie algebra to a Lie group.

+ For any $R$, such an $\hat\omega$ can be found
    + not necessarily unique.
+ $\hat\omega$ is the axis of rotation
+ **Note** Only three degrees of freedom; since $\hat\omega$ is a 3D vector
    + a scalar factor can be applied to $t$ (change of unit) or
      to $\hat\omega$
    + useful to normalise $\hat\omega$ to unit norm
+ Rotation is obviously periodic.  
    + A rotation by $2\pi$ is back to start.

## Logarithm

*Theorem 2.8* page 27 in the textbook

$$
R = 
\begin{bmatrix}
r_{11} & r_{12} & r_{13} \\
r_{21} & r_{22} & r_{23} \\
r_{31} & r_{32} & r_{33} 
\end{bmatrix}
= \exp(\hat\omega)
$$
where
$$
\DeclareMathOperator{\tr}{trace}
||\omega|| = \cos^{-1}\big(\frac{\tr(R)-1}2\big)
$$
and

$$
\frac{\omega}{||\omega||} = \frac1{2\sin(||\omega||)}
\begin{bmatrix} r_{32}-r_{23}\\ r_{13}-r_{31}\\ r_{21}-r_{12} \end{bmatrix}
$$