MVP Pipeline

Complete Transformation Pipeline

Object coords → World coords → View coords → Clip coords → NDC → Screen coords

The MVP Matrix

$\begin{bmatrix} x_s \\ y_s \\ z_s \\ w_s \end{bmatrix} = \mathbf{P} \cdot \mathbf{V} \cdot \mathbf{M} \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix}$

Model Matrix (M)

Object coordinates → World coordinates:

• Places object in the scene
• Applies scale, rotate, translate

View Matrix (V)

World coordinates → Camera coordinates:

• Positions camera at origin
• Orients camera to look down $-z$

Projection Matrix (P)

Camera coordinates → Clip coordinates:

• Projects 3D to 2D (perspective or orthographic)
• Maps frustum to NDC cube $[-1, 1]^3$

After Projection

1. Perspective division: Divide by $w$
2. Viewport transform: Map NDC to screen pixels
3. Rasterisation: Convert to fragments

Summary

• M: Places object in world
• V: Places camera at origin
• P: Projects to 2D and maps to NDC
• Combined: $\mathbf{P} \cdot \mathbf{V} \cdot \mathbf{M}$ applied to each vertex

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Past Paper Questions

2024 Paper 3 Question 4: Explain the purpose of the model, view, and projection matrices. What coordinate system does each transform between?