Transformation Matrices

In 3D graphics, every object's position, orientation, and size is controlled by a 4×4 transformation matrix. This single matrix can encode any combination of translation (moving), rotation (turning), and scale (resizing).

Why 4×4 instead of 3×3?

A 3×3 matrix can handle rotation and scale, but not translation. The trick is to use homogeneous coordinates - adding a 4th coordinate (w=1 for points, w=0 for directions).

The Structure:

| Rx  Ry  Rz  Tx |     Rotation + Scale (3×3)  |  Translation
| Rx  Ry  Rz  Ty |     in the upper-left       |  in the right
| Rx  Ry  Rz  Tz |     corner                  |  column
| 0   0   0   1  |     Always [0, 0, 0, 1] for affine transforms

Identity Matrix (No Change):

| 1  0  0  0 |
| 0  1  0  0 |  × point = same point
| 0  0  1  0 |
| 0  0  0  1 |

Translation (Move):

To move by (tx, ty, tz):

| 1  0  0  tx |
| 0  1  0  ty |  × (x, y, z, 1) = (x+tx, y+ty, z+tz, 1)
| 0  0  1  tz |
| 0  0  0  1  |

Scale:

To scale by (sx, sy, sz):

| sx 0  0  0 |
| 0  sy 0  0 |  × (x, y, z, 1) = (x×sx, y×sy, z×sz, 1)
| 0  0  sz 0 |
| 0  0  0  1 |

Rotation (around Z axis by θ):

| cos(θ) -sin(θ)  0  0 |
| sin(θ)  cos(θ)  0  0 |
| 0       0       1  0 |
| 0       0       0  1 |

The Power: Composition

Multiply matrices together to combine transformations:

M_final = M_translate × M_rotate × M_scale

Order Matters!

In Skeletal Animation:

Each bone stores its local transform matrix. To find a vertex's world position:

world_position = M_root × M_spine × M_shoulder × M_arm × M_hand × local_position

This chain of multiplications is why bone hierarchies work - moving the shoulder automatically moves everything below it.