Combi2k2

Different Type of Geometries


Euclidean Geometry	Shape are invariant by translations, rotations & uniform scaling
Affine Geometry	Euclidean Transformations + shearing (non-uniform scaling)
Projective Geometry	Affine Transformations + projections

We always assume that all shapes lie in Euclidean space $\mathbb{R}^2$ or $\mathbb{R}^3$ .

A transformation is a map from the space to space itself $T: E^n \to E^n$

Some basic transformations are:

rotation
translation

If $T$ satisfy $d(x, y) = d(T(x), T(y)) \forall x, y\in E^n$ , then $T$ is called an isometry.

An isometry that map origin to itself is a rotation or reflection.
An isometry is always composed of a rotation/reflection and a translation
A rotation $R$ preserves inner product. i.e $<R(x), R(y)> = <x,y> \forall x, y$
A rotation $R$ is indeed just a matrix multiplication operator.

Note that:

$R$ us orthogonal ( $R^T = R^{-1}$ )
In 2D, the rotation "rotates" a vector around a point.
In 3D, the rotation is composed of:
- An axis of rotation: eigenvector of $R$ with eigenvalue $1$ .
- A rotation angle $\theta$ .

E.g:

\mathcal{R}_{z,\theta}: \begin{pmatrix}x\\ y\\ z \end{pmatrix} \mapsto \begin{pmatrix}x\cos(\theta)-y\sin(\theta)\\ x\sin(\theta)+y\cos(\theta)\\ z \end{pmatrix}

Rotation around an arbitrary axis

To rotate around an axis $l=(u,v,w)$ by angle $\theta$ , we use Rodrigues' rotation formula:

R_{(u,v,w),\theta} = I + \sin(\theta)U + (1-\cos(\theta))U^2

\text{where } U = \begin{pmatrix} 0 & -w & v \\ w & 0 & -u \\ -v & u & 0 \end{pmatrix}

Rotation - Other representations

Euler angles or Tait–Bryan angles:

Represent an arbitrary rotations by a combination of three rotation around the principle axes:

\begin{align*} R &= R_{x,\alpha}R_{y,\beta}R_{z,\gamma} &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\alpha) & -\sin(\alpha) \\ 0 & \sin(\alpha) & \cos(\alpha) \end{pmatrix} \begin{pmatrix} \cos(\beta) & 0 & \sin(\beta) \\ 0 & 1 & 0 \\ -\sin(\beta) & 0 & \cos(\beta) \end{pmatrix} \begin{pmatrix} \cos(\gamma) & -\sin(\gamma) & 0 \\ \sin(\gamma) & \cos(\gamma) & 0 \\ 0 & 0 & 1 \end{pmatrix} \end{align*}

The idea here is:

Euler angles lose degree of freedom at certain value (e.g $\beta = \frac{\pi}{2}$ )
Cannot represent all "continuous" path of rotations!

This all brings us back to the notation of quaternion:

A vector with $4$ components, with 3 imaginary ones:

q = a + bi + cj + dk, \quad\quad\quad i^2 = j^2 = k^2 = ijk = -1

Adding quaternions works simply elementwise
Multiplication is defined using the rules $i^2=j^2=k^2=ijk=-1$

Rotation - Benefit of quaternion

every rotation is represented by a unit 4D quaternion (or its opposite)
this provides a double cover of the space of rotations
- isomorphic to SU(2), closely related to spinors in physics
- no singularity to worry about
operations on rotations often simple (interpolation, composition, ...)

Other transformation

Scaling

\begin{alignat*}{3} &\begin{pmatrix} x\\ y\\ z \end{pmatrix} \mapsto \begin{pmatrix} sx\\ sy\\ sz \end{pmatrix} &&\quad \mathcal{S}_s &&= \begin{pmatrix} s & 0 & 0\\ 0 & s & 0\\ 0 & 0 & s \end{pmatrix}\\ &\begin{pmatrix} x\\ y\\ z \end{pmatrix} \mapsto \begin{pmatrix} s_xx\\ s_yy\\ s_zz \end{pmatrix} &&\quad \mathcal{S}_{s_x,s_y,s_z} &&= \begin{pmatrix} s_x & 0 & 0\\ 0 & s_y & 0\\ 0 & 0 & s_z \end{pmatrix} \end{alignat*}

Translation

\mathcal{T}_{t_x,t_y,t_z}: \begin{pmatrix} x\\ y\\ z \end{pmatrix} \mapsto \begin{pmatrix} x+t_x\\ y+t_y\\ z+t_z \end{pmatrix}

Affine Homogeneous Coordinates

Now we represent 3D coordinate $(x,y,z)$ by a a vector with another dimension:

\begin{pmatrix} x\\ y\\ z \end{pmatrix} \to \begin{pmatrix} x\\ y\\ z\\ 1 \end{pmatrix}

This is to make translation to be actually just a matrix multiplication too:

\mathcal{T}_{t_x,t_y,t_z}(x, y, z) = \begin{pmatrix} 1 & 0 & 0 & t_x\\ 0 & 1 & 0 & t_y\\ 0 & 0 & 1 & t_z\\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x\\ y\\ z\\ 1 \end{pmatrix} = \begin{pmatrix} x+t_x\\ y+t_y\\ z+t_z \end{pmatrix}

Note: Any affine transformation in 3D is a linear transformation in 4D

Projections

To project on the screen (actually just a surface) $z=0$ :

Orthogonal projection:

\begin{pmatrix}x\\ y\\ z\end{pmatrix} \mapsto \begin{pmatrix}x\\ y\\ 0\end{pmatrix}\quad \mathcal{P}_O = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}

Perspective projection:

\begin{pmatrix}x\\ y\\ z\end{pmatrix} \mapsto \begin{pmatrix}x/z\\ y/z\\ 0\end{pmatrix}\quad \mathcal{P}_O = \begin{pmatrix} ??? \end{pmatrix}

Actually, we can still make use of the homogenous coordinates:

\mathcal{P}_P(x,y,z) = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \end{pmatrix} \begin{pmatrix} x\\ y\\ z\\ 1 \end{pmatrix} = \begin{pmatrix} x\\ y\\ 0\\ z \end{pmatrix} \equiv \begin{pmatrix} x/z\\ y/z\\ 0 \end{pmatrix}

We simply used: $\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\equiv\begin{pmatrix}x/w\\ y/w\\ z/w\end{pmatrix}$

Now we know where the name homogenous comes from.

Conclusion

Geometric modeling is about shape representation & manipulation

In Euclidean world, isometric transformations are translations/rotations
Representation of rotations: rotation matrices, quaternions…
All conveniently represented w/ matrices; composition = matrix multiplication.
Homogeneous coordinates "linearize" most useful transformations

Duc Nguyen

Shape Analysis #1: Introduction