Coordinate Systems and Transformations in 3D Graphics

This collection builds the language and mathematics used to move geometry through a renderer. It begins with object and world transforms, then follows a world-space vertex through the camera, projection, clipping, normalized-device-coordinate, and viewport stages.

The Main Transformation Journey

The core sequence starts with a position already expressed in world space:

$$p_{world} \xrightarrow{V} p_{view} \xrightarrow{P} p_{clip} \xrightarrow{\text{clip}} \text{clipped primitive} \xrightarrow{/w} p_{ndc} \xrightarrow{\text{viewport}} p_{window}$$

Projection creates clip coordinates; clipping happens in that homogeneous space; and the perspective divide creates normalized device coordinates (NDC). Keeping those operations separate is the key to understanding the rest of the rasterization pipeline.

Reading Order

1. Coordinate Spaces and Hierarchies — local and world space, parent-child transforms, and the idea that coordinates are meaningful only relative to a frame.
2. Transformation Matrices, Homogeneous Coordinates, and Composition — the matrix tools behind affine transforms and transform chains.
3. Rotation Representations: Euler Angles and Quaternions — how orientations can be stored and manipulated.
4. From World Space to Screen Space — the focused camera-to-rasterizer journey:
   1. World Space → View Space: The Camera Transform
   2. View Space → Clip Space: Perspective Projection
   3. Clip Space and Homogeneous Clipping
   4. Clip Space → NDC: The Perspective Divide
   5. NDC → Window Space: The Viewport Transform
   6. Putting It Together: The Complete Vertex Journey

The world-to-screen collection deliberately treats model-to-world placement as already solved. Object transforms and scene hierarchies remain important, but they should not interrupt the more subtle camera and projection story.