Reading Ma 2004 Chapter 2.2-2.3

Motion examples

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

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
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

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?