Normal Vector Transformation

The Problem

Normal vectors transform differently from positions!

For transformation matrix $\mathbf{M}$ applied to positions, normals transform using: $\mathbf{N}' = (\mathbf{M}^{-1})^\top \mathbf{N}$

Derivation

The normal $\mathbf{N}$ is perpendicular to the tangent vector $\mathbf{T}$: $\mathbf{N} \cdot \mathbf{T} = 0$

After transformation, $\mathbf{T}' = \mathbf{M}\mathbf{T}$. We need $\mathbf{N}' \cdot \mathbf{T}' = 0$: $\mathbf{N}' \cdot (\mathbf{M}\mathbf{T}) = ((\mathbf{M}^{-1})^\top \mathbf{N}) \cdot (\mathbf{M}\mathbf{T}) = \mathbf{N}^\top \mathbf{M}^{-1} \mathbf{M}\mathbf{T} = \mathbf{N} \cdot \mathbf{T} = 0$

Simplifications

Pure Rotation

If $\mathbf{M}$ is orthogonal (rotation only): $\mathbf{M}^{-1} = \mathbf{M}^\top$

Therefore: $\mathbf{N}' = \mathbf{M}\mathbf{N}$

Normals use the same rotation matrix as positions.

Uniform Scale + Rotation

The scale factor cancels (normals are renormalised anyway): $\mathbf{N}' = \mathbf{R}\mathbf{N}$

Non-Uniform Scale or Shear

Must compute and apply $(\mathbf{M}^{-1})^\top$: $\mathbf{N}' = (\mathbf{M}^{-1})^\top \mathbf{N}$

Translation

Normals are unchanged by pure translations (direction vectors).

Why It Matters

Non-uniform scaling distorts normals:

• A sphere scaled to an ellipsoid needs transformed normals
• Using $\mathbf{M}$ directly would make normals point in wrong direction

Summary

• Normals transform with $(\mathbf{M}^{-1})^\top$, not $\mathbf{M}$
• Pure rotation: normals use same matrix as positions
• Uniform scale: scale cancels after renormalisation
• Non-uniform scale/shear: must compute inverse transpose
• Translation: no effect on direction vectors

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Past Paper Questions

2025 Paper 3 Question 4(a)(ii): How would you obtain the transformation for the normal using scaling, rotation, and translation? Simplify if possible. [2 marks]