Transformation Matrices in Computer Graphics

Transformation Matrices in Graphics

Computer graphics uses 4x4 transformation matrices in homogeneous coordinates to represent all common transformations uniformly.

Homogeneous Coordinates

A 3D point (x, y, z) becomes (x, y, z, 1) in homogeneous coordinates. This allows translation to be represented as matrix multiplication.

Basic Transformations

Translation by (tx, ty, tz):

T = | 1  0  0  tx |
    | 0  1  0  ty |
    | 0  0  1  tz |
    | 0  0  0   1 |

Scaling by (sx, sy, sz):

S = | sx  0   0   0 |
    | 0   sy  0   0 |
    | 0   0   sz  0 |
    | 0   0   0   1 |

Rotation around Z-axis by angle a:

Rz = | cos(a)  -sin(a)  0  0 |
     | sin(a)   cos(a)  0  0 |
     |   0        0     1  0 |
     |   0        0     0  1 |

The MVP Pipeline

In a standard graphics pipeline, vertices pass through three transformations:

1. Model matrix: positions the object in world space
2. View matrix: positions the camera (inverse of camera's transform)
3. Projection matrix: maps 3D to 2D (perspective or orthographic)

clip_position = Projection * View * Model * vertex

Combining Transforms

Multiple transformations are combined by matrix multiplication. The order is right-to-left: the rightmost matrix is applied first.

M = Translation * Rotation * Scale

This first scales, then rotates, then translates — the correct order for most object transformations.