---
title: Change of Basis
categories: 3D lecture
geometry: margin=2cm
fontsize: 12pt
---

**Reading** Ma 2004 Chapter 2.2-2.3

# Motion examples

**Demo**

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
    $$\vec{e}_i\vec{e}_j=\delta_{ij} =
    \begin{cases}
       1 \quad\text{if } i=j\\
       0 \quad\text{if } i\neq j
    \end{cases}
    $$

## Arbitrary Choice of Basis 

+ Any set of three linearly independent vectors 
  $\vec{u}_1$, $\vec{u}_2$, $\vec{u}_3$
  can be used as a basis.
+ Usually, we prefer an orthonormal basis, i.e. 
    - the basis vectors are orthogonal
    - the basis vectors have unit length

$$ \langle\vec{u}_i, \vec{u}_j\rangle = \begin{cases} 1, \quad\text{when } i=j,\\
 0, \quad\text{when } i\neq j\end{cases}$$

+ The basis is relative to a given Origin

## 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

### Describing a Scene

+ Each object described in its own basis
    - independently of its position and orientation in the scene
    - *reusable* objects
+ Rotation and Deformation can be described locally
+ Transformation from local to global co-ordinates
    - Rotation of the basis
    - Translation of the origin
+ System of Systems
    - an object in the scene may itself be composed of multiple objects with different local 
      frames


## 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.

# Working with Different Bases

## Change of Basis

+ Point $\vec x$ represented in a basis $\vec{e}_1$, $\vec{e}_2$, $\vec{e}_3$
    - i.e. $\vec x = x_1\vec{e}_1 + x_2\vec{e}_2 + x_3\vec{e}_3$
+ Translate to a representation in
  another basis $\vec{u}_1$, $\vec{u}_2$, $\vec{u}_3$
+ Suppose we can write the old basis in terms of the new one
    - $\vec{e_i} = e_{i,1}\vec{u}_1 + e_{i,2}\vec{u}_2 + e_{i,3}\vec{u}_3$

$$\begin{split}
  p = & x_1(e_{1,1}\vec{u}_1 + e_{1,2}\vec{u}_2 + e_{1,3}\vec{u}_3) + \\
      & x_2(e_{2,1}\vec{u}_1 + e_{2,2}\vec{u}_2 + e_{2,3}\vec{u}_3) + \\
      & x_3(e_{3,1}\vec{u}_1 + e_{3,2}\vec{u}_2 + e_{3,3}\vec{u}_3) \\
    = &(x_1e_{1,1} + x_2e_{2,1} + x_3e_{2,1} )\vec{u}_1 + \\
      &(x_1e_{1,2} + x_2e_{2,2} + x_3e_{2,2} )\vec{u}_2 + \\
      &(x_1e_{1,3} + x_2e_{2,3} + x_3e_{2,3} )\vec{u}_3
\end{split}$$


Write $[x'_1, x'_2, x'_3]^\mathrm{T}$ for the coordinates in terms of the new basis.

$$ x'_i = x_1e_{1,i} + x_2e_{2,i} + x_3e_{2,i} $$

+ In matrix form we can write
  $$p = [x_1',x_2',x_3'] = [ \vec{e}_1 | \vec{e}_2 | \vec{e}_3 ]\cdot\vec{x}$$
  where $\vec{e}_i$ are written as column vectors.

## Orthonormal matrix

$$R = [ \vec{e}_1 | \vec{e}_2 | \vec{e}_3 ]$$

+ Orthonormal basis means that
  $R\cdot R^T = R^T\cdot R=I$
+ Hence $R^{-1}=R^T$
+ If $\vec{x}'=\vec{x}\cdot R$ then
  $\vec{x}=\vec{x}'\cdot R^T$ then
+ If the columns of $R$ make up the new basis,
    + then the rows make up the old basis

## Example

$$
\begin{split}
R = 
  \begin{bmatrix}
  \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0  \\
  \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0  \\
  0 & 0 & 1
  \end{bmatrix}
\end{split}
$$

## Rotation as Motion

> Why is change of basis so important?

+ It also describes rotation of rigid bodies
+ To rotate a body, rotate its local basis
+ Any orthonormal basis is a rotation of any other
    + The matrix $R$ defines the rotation
+ Any orthogonal matrix defines a rotation

## Demo 

**Use Python Here**

## Moving the Origin

+ A point is described relative to the origin

$$\mathbf x = \mathbf{0} + x_1\cdot\vec{e}_1 + x_2\cdot\vec{e}_2 + x_3\cdot\vec{e}_3$$

+ Note that I write $\mathbf{x}$ for a point and $\vec{x}$ for a vector
+ The origin is arbitrary
+ The local co-ordinate system is defined by
    1. the basis $\vec{e}_1$, $\vec{e}_2$, $\vec{e}_3$
    2. the origin $\mathbf{0}$
+ Move origin: $\mathbf{0}'=\mathbf{0}+\vec{t}$
    + for some translation vector $\vec{t}$

## Arbitrary motion

+ $\mathbf{X}\mapsto \mathbf{X}\cdot R + \vec{t}$
+ or $\mathbf{X}\mapsto (\mathbf{X}-\vec{t}_1)\cdot R + \vec{t}_2$
    + remember, what is the centre of rotation?