Transforms & Math

Every game engine needs 2D/3D math primitives. Even a 2D game needs:

• vec2 / vec3: positions, velocities, sizes.
• mat3 / mat4: rotation, scale, translation composed into one matrix.

Why a matrix (not a 2x2)?

A pure rotation in 2D fits in a 2x2 matrix:

R(t) = | cos(t)  -sin(t) |
       | sin(t)   cos(t) |

But translation does not fit in a 2x2 — adding a vector is not a matrix multiply. The fix is homogeneous coordinates: extend (x,y) to (x,y,1) and use a 3x3 matrix:

M = | sx*cos(t)  -sy*sin(t)  tx |
    | sx*sin(t)   sy*cos(t)  ty |
    | 0           0           1 |

Now translation, rotation, and scale all live in one matrix. Composing transforms is just matrix multiplication.

Scene graph

Entities have a Transform component with a local matrix. The world transform is computed by walking from the root and multiplying:

child.world = parent.world * child.local

Real engines cache world and only re-compute on dirty changes — entity moved, parent rotated, etc.

Coordinate conventions

• 2D: y-up (math/Bevy/Unity 2D) vs y-down (screen/SDL). Pick one and stick to it.
• 3D: left-handed (Unity, DirectX) vs right-handed (OpenGL, glTF). Differs by which way +z points relative to a screen.
• Units: meters are standard. Unreal famously uses centimeters.

The Practice problem

You'll implement a tiny stack-machine that applies TRANSLATE, ROTATE (degrees), SCALE to a single point. The same algebra scales to a full transform stack walked by a renderer.