Computer Graphics - Transformation include the following topics: Linear Transformation, Homogeneous Coordinates, Affine Transformations, Inverse Transform, Composing Transform, 3D Transformations, Rodrigues' Rotation Formula.

---

CG-Modeling Transformation

Linear Transformation

Scale Matrix

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}{s}_x& 0\\ 0& s_y\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

Reflection Matrix

Horizontal reflection:

• $x^{\prime }=-x$
• $y^{\prime }=y$

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

Shear Matrix

• Horizontal shift is 0 at $y=0$
• Horizontal shift is a at $y=1$
• Vertical shift is always 0

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}1& a\\ 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$$

Rotation Matrix

$${\mathbf{R}}_{\theta }=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$$$$R_{-\theta }=R_{\theta }^{-1}=R_{\theta }^T$$

Homogeneous Coordinates

Add a third coordinate (w-coordinate):

• 2D point:$$\mathbf{p}=(x,y,1{)}^{\mathrm{T}}$$
• 2D vector:$$\mathbf{v}=(x,y,0{)}^{\mathrm{T}}$$

Valid operations if w-coordinate of result is 1 or 0:

• vector $+$ vector $=$ vector
• point $-$ point $=$ vector
• point $+$ vector $=$ point
• point $+$ point $=$ ??

$$\left[\begin{array}{c}x\\ y\\ w\end{array}\right]\text{ is the 2D point }\left[\begin{array}{c}\frac{x}{w}\\ \frac{y}{w}\\ 1\end{array}\right],w\ne 0$$

---

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ w^{\prime }\end{array}\right]=\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=\left[\begin{array}{c}x+t_x\\ y+t_y\\ 1\end{array}\right]$$

Affine Transformations

Affine map = linear map + translation:

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]+\left[\begin{array}{c}{t}_x\\ t_y\end{array}\right]$$

Using homogeneous coordinates:

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{ccc}a& b& t_x\\ c& d& t_y\\ 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

Scale:

$$\mathbf{S}(s_x,s_y)=\left[\begin{array}{ccc}{s}_x& 0& 0\\ 0& s_y& 0\\ 0& 0& 1\end{array}\right]$$

Rotation:

$$\mathbf{R}(\alpha )=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right]$$

Translation:

$$\mathbf{T}(t_x,t_y)=\left[\begin{array}{ccc}1& 0& t_x\\ 0& 1& t_y\\ 0& 0& 1\end{array}\right]$$

Inverse Transform

$${\mathbf{M}}^{-1}$$$${\mathbf{M}}^{-1}\text{ is the inverse of transform }\mathbf{M}\text{ in both a matrix and geometric sense.}$$

Composing Transform

Transform Ordering Matters!

Matrix multiplication is not commutative:

$${\mathbf{R}}_{45}\cdot {\mathbf{T}}_{(1,0)}\ne {\mathbf{T}}_{(1,0)}\cdot {\mathbf{R}}_{45}$$

Note that matrices are applied right to left:

$${\mathbf{T}}_{(1,0)}\cdot {\mathbf{R}}_{45}\cdot \left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{ccc}\cos {45}^{\circ }& -\sin {45}^{\circ }& 0\\ \sin {45}^{\circ }& \cos {45}^{\circ }& 0\\ 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

Sequence of Affine Transforms

• Compose by matrix multiplication
• Very important for performance!

$$A_n(\dots A_2(A_1(\mathbf{x})))=A_n\cdot \cdots \cdot A_2\cdot A_1\cdot \left(\begin{array}{c}x\\ y\\ 1\end{array}\right)$$

Pre-multiply $n$ matrices to obtain a single matrix representing the combined transform.

How to rotate around a given point $c$?

1. Translate center to origin
2. Rotate
3. Translate back

Matrix representation:

$$T(c)\cdot R(\alpha )\cdot T(-c)$$

3D Transformations

Use homogeneous coordinates again:

• 3D point:

  $$(x,y,z,1{)}^T$$

• 3D vector:

  $$(x,y,z,0{)}^T$$

In general, $(x,y,z,w)$ ($w\ne 0$) is the 3D point:

$$(\frac{x}{w},\frac{y}{w},\frac{z}{w})$$

Use 4×4 Matrices for Affine Transformations

$$\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\\ 1\end{array}\right)=\left(\begin{array}{cccc}a& b& c& t_x\\ d& e& f& t_y\\ g& h& i& t_z\\ 0& 0& 0& 1\end{array}\right)\cdot \left(\begin{array}{c}x\\ y\\ z\\ 1\end{array}\right)$$

Order: Linear transformations, then translation.

Scale

$$S(s_x,s_y,s_z)=\left(\begin{array}{cccc}{s}_x& 0& 0& 0\\ 0& s_y& 0& 0\\ 0& 0& s_z& 0\\ 0& 0& 0& 1\end{array}\right)$$

Translation

$$T(t_x,t_y,t_z)=\left(\begin{array}{cccc}1& 0& 0& t_x\\ 0& 1& 0& t_y\\ 0& 0& 1& t_z\\ 0& 0& 0& 1\end{array}\right)$$

Rotation

Rotation around the x-axis:

$$R_x(\alpha )=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \alpha & -\sin \alpha & 0\\ 0& \sin \alpha & \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right)$$

Rotation around the y-axis:

$$R_y(\alpha )=\left(\begin{array}{cccc}\cos \alpha & 0& \sin \alpha & 0\\ 0& 1& 0& 0\\ -\sin \alpha & 0& \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right)$$

Rotation around the z-axis:

$$R_z(\alpha )=\left(\begin{array}{cccc}\cos \alpha & -\sin \alpha & 0& 0\\ \sin \alpha & \cos \alpha & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

Compose Any 3D Rotation from $R_x$, $R_y$, $R_z$

$$R_{xyz}(\alpha ,\beta ,\gamma )=R_x(\alpha )R_y(\beta )R_z(\gamma )$$

• So-called Euler angles
• Often used in flight simulators: roll, pitch, yaw

Rodrigues' Rotation Formula

Rotation by angle $\alpha$ around axis $\mathbf{n}$

$$R(\mathbf{n},\alpha )=\cos (\alpha )\mathbf{I}+(1-\cos (\alpha ))\mathbf{n}{\mathbf{n}}^T+\sin (\alpha )\mathbf{N}$$$$\mathbf{N}=\left(\begin{array}{ccc}0& -n_z& n_y\\ n_z& 0& -n_x\\ -n_y& n_x& 0\end{array}\right)$$