## Revision fb69306a414b992332659c04926f734f6c59e436 (click the page title to view the current version)

## Changes from fb69306a414b992332659c04926f734f6c59e436 to dc7d6bfc7198414ea60130392ddb9e3dd8cbf4a0

---
title: Change of Basis
categories: 3D lecture
---
# Motion examples
1. Translation
$$\vec{x}' = \vec{x}+\vec{t}$$
1. Rotation
$$\vec{x}' = \vec{x}\cdot R$$
- But note that $R$ is not an arbitrary matrix.
- We'll return to the restrictions
# Definition: Rigid Body Motion
1. 3D Object is a set of points in $\mathbb{R}^3$
2. If the object moves, the constituent points move
3. The points have to move so that they preserve the shape of the object
## Constraints
Let $\vec{X}(t)$ and $\vec{Y}(t)$ be the coordinates of
points $\vec{x}$ and $\vec{x}$ at time $t$.
1. Preserve distance between points
+ $||\vec{X}(t)-\vec{Y}(t)||$ is constant
1. Preserve orientation
+ i.e. avoid mirroring
+ we have to preserve cross-products
+ If the right hand rule turns into a left hand rule,
we have had mirroring.
Let $u=\vec{X}-\vec{Y}$ be a vector, and
$g_*(u)=g(\vec{X})-g(\vec{Y})$ the corresponding vector after
motion.
Preserving the cross-product means
$$g_*(u)\times g_*(v) = g_*(u\times v), \forall u,v\in\mathbb{R}^3$$
# Change of Basis
## Bases