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

## 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.

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

## Matrix form

Still consider a point $p$ represented as

+ $[x_1, x_2, x_3]^\mathrm{T}$ in the basis $\vec{e}_1$, $\vec{e}_2$, $\vec{e}_3$
+ $[x'_1, x'_2, x'_3]^\mathrm{T}$ in the basis $\vec{u}_1$, $\vec{u}_2$, $\vec{u}_3$

We found that

$$\begin{bmatrix}x'_1 \\ x'_2 \\ x'_3\end{bmatrix} =
\begin{bmatrix}
  e_{1,1} & e_{2,1} & e_{3,1} \\
  e_{1,2} & e_{2,2} & e_{3,2} \\
  e_{1,3} & e_{2,3} & e_{3,3}
\end{bmatrix}\cdot
\begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} = R\cdot
\begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} $$

**Note** $R^\mathrm{T} = R^{-1}$ for an orthonormal basis

$$\begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix} =
\begin{bmatrix}
  e_{1,1} & e_{1,2} & e_{1,3} \\
  e_{2,1} & e_{2,2} & e_{2,3} \\
  e_{3,1} & e_{3,2} & e_{3,3}
\end{bmatrix}\cdot
\begin{bmatrix}x'_1 \\ x'_2 \\ x'_3\end{bmatrix} = R^\mathrm{T}\cdot
\begin{bmatrix}x'_1 \\ x'_2 \\ x'_3\end{bmatrix} $$

+ Rows of $R$ define the $\vec{u}_i$ basis expressed in $\vec{e}_j$
+ Columns of $R$ define the $\vec{e}_j$ basis expressed in $\vec{u}_i$

## Rotation

> Why is change of basis so important?

+ It also describes rotation of rigid bodies
+ Any orthonormal basis is a rotation of any other
    + The matrix $R$ defines the rotation

## Moving the Origin

+ A point is described relative to the origin

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

+ 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 $\vec{0}$

**TODO** Complete

# Demo 

**Use Python Here**

## Objects in 3D