Perspective-Correct Interpolation

The Problem

Linear interpolation in screen space is only correct for parallel (orthographic) projection.

Under perspective projection, 3D properties are projected non-linearly onto 2D screen. Simply interpolating in screen space causes attributes like texture coordinates to distort.

The Solution

Interpolate quantities that are linear in screen space:

• Perspective-divided attribute $Q/z$
• Reciprocal depth $1/z$

The Formula

For pixel with barycentric coordinates $(\alpha, \beta, \gamma)$ and depths $z_A, z_B, z_C$:

1. Interpolate $Q/z$: $\left(\frac{Q}{z}\right)_{\text{interp}} = \alpha \frac{Q_A}{z_A} + \beta \frac{Q_B}{z_B} + \gamma \frac{Q_C}{z_C}$

2. Interpolate $1/z$: $\left(\frac{1}{z}\right)_{\text{interp}} = \alpha \frac{1}{z_A} + \beta \frac{1}{z_B} + \gamma \frac{1}{z_C}$

3. Reconstruct: $Q(x, y) = \frac{(Q/z)_{\text{interp}}}{(1/z)_{\text{interp}}}$

Why This Works

After perspective projection, $1/z$ varies linearly in screen space. Dividing attributes by $z$ before interpolation, then dividing by $1/z$ after, gives correct results.

Summary

• Screen-space linear interpolation is wrong for perspective
• Interpolate $Q/z$ and $1/z$ in screen space
• Divide the results to recover perspective-correct $Q$

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Past Paper Questions

2022 Paper 3 Question 4: How does perspective-correct interpolation differ from linear interpolation on the screen? Why is it needed for texture mapping?