Diffuse (Lambertian) Reflection

Diffuse Surfaces

Diffuse surfaces scatter light equally in all directions. Matte surfaces like paper, chalk, and unpolished wood.

Lambert’s Cosine Law

The reflected intensity depends only on the angle between the surface normal and light direction:

$I_{\text{diffuse}} = I_l k_d \cos\theta = I_l k_d (\mathbf{N} \cdot \mathbf{L})$

where:

• $I_l$ is the light source intensity
• $k_d$ is the diffuse reflection coefficient (surface colour)
• $\mathbf{N}$ is the surface normal (unit vector)
• $\mathbf{L}$ is the direction to the light (unit vector)
• $\theta$ is the angle between $\mathbf{N}$ and $\mathbf{L}$

Physical Interpretation

$\mathbf{N} \cdot \mathbf{L} = \cos\theta$ represents:

• Projected area: A surface receives less light energy when tilted
• At glancing angles ($\theta \approx 90°$), the same energy spreads over a larger area
• At perpendicular angles ($\theta = 0°$), maximum energy per unit area

View Independence

Diffuse reflection is view-independent:

• The surface appears equally bright from any viewing angle
• Light scatters uniformly in all directions
• Only depends on $\mathbf{N}$ and $\mathbf{L}$, not on $\mathbf{V}$

When Diffuse is Zero

Diffuse is zero when:

• $\mathbf{N} \cdot \mathbf{L} \leq 0$: Light is behind or tangent to the surface
• $k_d = 0$: Surface has zero diffuse reflectance (pure mirror)

One-Sided vs Two-Sided Surfaces

One-sided surfaces: Only illuminate the side in direction of $\mathbf{N}$. If $\cos\theta < 0$, render black.

Two-sided surfaces: If $\cos\theta < 0$, use $|\cos\theta|$ (or flip $\mathbf{N}$) to illuminate the back.

Summary

• Diffuse reflection follows Lambert’s cosine law
• Intensity proportional to $\cos\theta$ = $\mathbf{N} \cdot \mathbf{L}$
• View-independent: same brightness from any angle
• Zero when light is behind the surface

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Past Paper Questions

2025 Paper 3 Question 3(d): In the Phong shading model, explain when the diffuse component becomes zero. [Part of 4 marks]