Transformations

In computer graphics, transformations are used to manipulate objects in a scene. The most common transformations include translation, rotation, and scaling. These transformations can be represented using matrices, which allows for efficient computation and combination of multiple transformations.

2D

Scale

Suppose we hava a point $(x, y)$ in 2D space. To scale this point by a factor of $s$, we can use the following transformation matrix:

\[\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}\]

Shear

Suppose we have a point $(x, y)$ in 2D space. To shear this point by factors $s_x$ and $s_y$ we can use the following transformation matrix:

\[\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} 1 & s_x \\ s_y & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}\]

Rotate

To rotate a point $(x, y)$ around the origin by an angle $\theta$, we can use the following transformation matrix:

\[\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}\]

Translate

To translate a point $(x, y)$ by a vector $(t_x, t_y)$, we can use the following transformation matrix:

Add a third coordinate (w-coordinate) to represent the point in homogeneous coordinates:

• 2D point $[x, y]$ becomes $[x, y, 1]^{T}$.
• 2D vector $[x, y, 0]^{T}$

\[\begin{bmatrix} x'\\ y' \\ w' \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \\ 1 \end{bmatrix}\]

Affine

An affine transformation is a combination of linear transformations (like rotation, scaling, and shearing) and translation. It can be represented using a 3x3 matrix in homogeneous coordinates to operate a point in 2D space.

An affine transformation matrix in 2D can be represented as:

\[\begin{bmatrix} x' \\ y' \\ 1 \\ \end{bmatrix} = \begin{bmatrix} a & b & t_x \\ c & d & t_y \\ 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \\ \end{bmatrix}\]

3D

Rotate

To rotate a point $(x, y, z)$ around the origin by angles $\alpha$, $\beta$, and $\gamma$ around the x, y, and z axes respectively, we can use the following transformation matrices:

\[\begin{align*} R_x(\alpha) &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos{\alpha} & -\sin{\alpha} & 0 \\ 0 & \sin{\alpha} & \cos{\alpha} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ R_y(\beta) &= \begin{bmatrix} \cos{\beta} & 0 & \sin{\beta} & 0 \\ 0 & 1 & 0 & 0 \\ -\sin{\beta} & 0 & \cos{\beta} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ R_z(\gamma) &= \begin{bmatrix} \cos{\gamma} & -\sin{\gamma} & 0 & 0 \\ \sin{\gamma} & \cos{\gamma} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\ R_{xyz}(\alpha, \beta, \gamma) &= R_z(\gamma) R_y(\beta) R_x(\alpha) \end{align*}\]

Rodrigues’ Rotation Formula

Rotation by angle $\alpha$ around axis n

\[\bold{R}(\bold{n}, \alpha) = cos(\alpha)\bold{I} + (1 - \cos(\alpha)) \bold{nn^T} + \sin{\alpha} \bold{N}\]