---
title: Change of Basis
categories: 3D lecture
---

**Reading** Ma 2004 Chapter 2.2-2.3

# 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

![Handdrawn illustration](Images/motion.png)
 
# 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

1.  Basis aka. frame
    + Unit vectors: $\vec{e}_1$, $\vec{e}_2$, $\vec{e}_3$
2.  The meaning of a tuple to denote a vector
    + $\vec{x}=[x_1,x_2,x_3]= x_1\cdot\vec{e}_1+x_2\cdot\vec{e}_2+x_3\cdot\vec{e}_3$
3.  Orthonormal frame: orthogonal and unit length
    $$\vec{e}_i\vec{e}_j=\delta_{ij} =
    \begin{cases}
       1 \quad\text{if } i=j\\
       0 \quad\text{if } i\neq j
    \end{cases}
    $$

## Local and Global Basis

1.  3D Scenes are built hierarchically
2.  Each object is described in a local basis
    + and then placed in the global basis.
3.  Why?
    + Save computational work
    + Local changes affect only local co-ordinates
    + Component motion independent of system motion

## Example

E.g. our co-ordinates: 62°28'19.3"N 6°14'02.6"E

Are these local or global co-ordinates?

## Rotation

Consider **common origin** first.