Change of Basis

Reading Ma 2004 Chapter 2.2-2.3


  • 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.




  1. Translation \[\vec{x}' = \vec{x}+\vec{t}\]
  2. Rotation \[\vec{x}' = \vec{x}\cdot R\]
    • But note that \(R\) is not an arbitrary matrix.
    • We’ll return to the restrictions
Handdrawn illustration
Handdrawn illustration

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


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
  2. 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


  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


E.g. our co-ordinates: 62°28’19.3“N 6°14’02.6”E

Are these local or global co-ordinates?


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


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

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 \(R_2R_2=I\)
  3. 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


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?