GAMES 101 Introduction to Computer Graphics

Overview

The use of computers to synthesize and manipulate visual information.

Application

video game, movie, amimations, design, visualization, VR, AR, digital illustration, simulation, GUI, typography…

Course Topics

• Rasterization: Project geometry primitives (3D triangles / polygons) onto the screen.
• Curves and Mesh: Present geometry in Computer Graphs.
• Ray Tracing
• Animation / Simulation

Not for graphics API, 3D modeling, game engine.

Review of Linear Algebra

Transformations

Transformation: modeling & viewing.

2D Transformation

-> Linear Transforms (scale, reflection, shear, rotation) == Matrices

$$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$$

Homogeneous Coordinate

But linear transforms can not represent translation: $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} t_x \\ t_y \end{bmatrix}$. This problem is solved by using homogeneous coordinates, which add an extra dimension to the 2D point (x, y) to become (x, y, 1).

• 2D point: $(x, y ,1)^T$ / $(x,y,w)^T$
• 2D vector: $(x,y,0)^T$

The third coordinate naturally enforces:

• point - point = vector (1 - 1 = 0)
• point + vector = point (1 + 0 = 1)
• vector + vector = vector (0 + 0 = 0)
• vector translation invariance: multiplying a vector by a translation matrix leaves it unchanged, since $t_x \cdot 0 + t_y \cdot 0 = 0$.

Affine transformations == linear map + translation: $\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} t_x \\ t_y \end{bmatrix}$. Using homogeneous coordinate:

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

Composite Transform