How 4x4 transformation matrices rotate, translate, and scale 3D polygons at 120 FPS.
Points as Vectors in 3D Space
Every 3D vertex (x, y, z) in a video game mesh is expressed as a homogeneous 4D coordinate vector [x, y, z, 1]ᵀ.
Homogeneous Coordinates: [x, y, z, 1]
Transformation Matrices
Translation, scaling, and 3D rotation are represented as 4×4 numerical matrices. Combining transformations requires multiplying the respective matrices together.
Matrix Multiplication: [Combined] = [Translate] × [Rotate] × [Scale]
The Graphics Rendering Pipeline
Model Space → World Space → View (Camera) Space → Clip Space → 2D Screen Pixel Coordinates.
MVP Matrix: [Projection] × [View] × [Model] × [Vertex] = Screen_Coordinate
GPU Hardware Optimization
Modern graphics processing units (GPUs) feature thousands of tensor cores engineered to perform billions of parallel 4×4 matrix dot products per second.
Performance: Millions of vertices rendered at 144+ FPS in real time.
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