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---
title: Change of Basis
categories: 3D lecture
geometry: margin=2cm
fontsize: 12pt
---
**Reading** Ma 2004 Chapter 2.2-2.3
# Motivation
+ Complex scene.
+ Lots and lots of points.
+ If an object moves, all its points changes.
+ This is unmanageable
+ Modularise
+ Describe the object in local co-ordinates.
+ Describe the location and orientation of the object once in the global co-ordinate system.
+ When the object moves, its local co-ordinate system changes relative to the global one,
but the internal description of the object does not.
# Motion examples
# 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
# Definition: Rigid Body Motion
## 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
$$
\begin{split}
R =
\begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\
0 & 0 & 1
\end{bmatrix}
\end{split}
$$
Note this is orthogonal. Compare it to
## Example in 2D
The principle is the same in 2D.
$$
R_1 =
\begin{bmatrix}
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
\end{bmatrix}
\quad
R_2 =
\begin{bmatrix}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\
\end{bmatrix}
$$
Note that
1. $\frac{\sqrt2}{2}=\sin(\pi/4)=\cos(\pi/4)$.
2. $R_1^TR_1=I$ and $R_2R_2=I$
2. The determinants $|R_1|=1$ and $|R_2|=-1$
Consider a triangle formed by the points
$(0,0)$, $(0,1)$, $(1,1)$, and consider each of them
rotated by $R_1$ and $R_2$
## Rotation around an axis
A rotation by an angle $\theta$ around the origin is given by
$$
R_\theta =
\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta \\
\end{bmatrix}
$$
In 3D, we can rotate around a given axis, by adding a column and
row to the 2D matrix above. E.g. to rotate around the $y$-axis,
we use
$$
R_\theta =
\begin{bmatrix}
\cos\theta & 0 & -\sin\theta \\
0 & 1 & 0 \\
\sin\theta & 0 & \cos\theta \\
\end{bmatrix}
$$
## Operations on rotations
+ If $R_1$ and $R_2$ are rotational matrices, then
$R=R_1\cdot R_2$ is a rotational matrix
+ There is an identity rotation $I$
+ If $R_1$ is a rotational matrices, then
there is an inverse rotation $R_1^{-1}$
+ If $R_1$, $R_2$, and $R_3$ are rotational matrices, then
$R_1(R_2R_3)= (R_1R_2)R_3$
## 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
+ [Jupyter Notebook](Python/change-of-basis.ipynb)
## 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?
