---
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