---
title: "Diffuse reflection equation"
subject: "Computer Graphics"
chapter: 8
key: "semester_5/computer_graphics:diffuse-reflection-equation"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/diffuse-reflection-equation
site: BCT Engineering
---

# Diffuse reflection equation

Computer Graphics · Chapter 8: Illumination and Surface Rendering methods

- Asked in: 2071 Shrawan, 4 marks; 2070 Asar, 8 marks; 2076 Chaitra, 5 marks (asked 3 times)
- Video: [Diffuse Lighting Equation](https://www.youtube.com/watch?v=8717ZZ-kIoo) — Jamie King
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/diffuse-reflection-equation
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-8

## Question

Explain diffuse reflection. Derive the intensity (total diffuse reflection) equation, including ambient light.

## Short answer

Diffuse reflection scatters light equally in all directions, following Lambert's cosine law: I_diff = kd·IL·( N·L ). With ambient light, total diffuse: I = ka·Ia + kd·IL·( N·L ). Example: with θ=60°, I =0.35.

## Answer

**Diffuse reflection** occurs on rough (dull) surfaces. Incident light is scattered equally in all directions, so the surface looks equally bright from every viewing position. The colour depends on the surface material and the light, not on the viewer position.

**Lambert's cosine law.** The intensity of the diffusely reflected light is proportional to the cosine of the angle θ between the unit surface normal N and the unit direction L to the light source (the surface receives less light per unit area when it is tilted):

I_diff = kd · IL · cosθ = kd · IL · (N · L), for 0 ≤ θ ≤ 90°

- kd = diffuse reflection coefficient (0 to 1)
- IL = intensity of the point light source

```text
        L  N
         \ |
          \|θ
   --------+---------- surface
```

If N·L < 0, the surface is facing away from the light and I_diff = 0.

**Ambient light.** A surface not directly lit still receives light scattered from the surroundings. This is modelled as a uniform background intensity Ia, and the reflected part is ka·Ia (ka = ambient coefficient).

**Total diffuse reflection with ambient light**

I_diff,total = ka · Ia + kd · IL · (N · L)

Often kd = ka is taken for simplicity, giving I = kd (Ia + IL·cosθ).

With distance attenuation f(d) = 1/(a0 + a1·d + a2·d²): I = ka·Ia + f(d)·kd·IL·(N·L).

**Example:** ka = kd = 0.5, Ia = 0.2, IL = 1, θ = 60°: I = 0.5·0.2 + 0.5·1·0.5 = 0.10 + 0.25 = **0.35**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How does a diffuse surface appear as the viewer moves?
   A. Equally bright from every viewing position
   B. Brighter near R
   C. Darker when closer
   D. Changes with V
   Answer: A. Light scatters equally in all directions. (Key point: Diffuse property)

2. Diffuse intensity is proportional to:
   A. cos^n of R and V
   B. cos of the angle between N and L
   C. 1/d only
   D. The viewer distance
   Answer: B. I_diff = kd IL (N . L). (Key point: Lambert's law)

3. What is the total diffuse reflection including ambient?
   A. ka Ia x kd IL
   B. kd IL / ka Ia
   C. ka Ia + kd IL (N . L)
   D. ks IL (R . V)^n
   Answer: C. Ambient and diffuse terms add. (Key point: Total with ambient)

4. ka=kd=0.5, Ia=0.2, IL=1, theta=60 deg: I = ?
   A. 0.45
   B. 0.25
   C. 0.60
   D. 0.35
   Answer: D. 0.5 x 0.2 + 0.5 x 1 x 0.5 = 0.10 + 0.25. (Key point: Example)

## Related in Computer Graphics

- [Ambient, diffuse and specular light](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aambient-diffuse-specular-expressions) — Computer Graphics, chapter 8, asked 2×
- [Define terms; Phong intensity equation](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Adefine-terms-lambert-phong-equation) — Computer Graphics, chapter 8
- [Distance consideration in intensity](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aintensity-attenuation-distance) — Computer Graphics, chapter 8, asked 2×
- [Intensity at centroid of a triangle](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Atotal-intensity-triangle-centroid) — Computer Graphics, chapter 8

## Related in other subjects

- [Total internal reflection as light conduit](https://www.bctengineer.com/question.md?key=semester_6%2Fpropagation_and_antenna%3Atotal-internal-reflection-light-conduit) — Propagation and Antenna, chapter 6
- [Derive total infiltration from Horton's equation](https://www.bctengineer.com/question.md?key=semester_5%2Fengineering_hydrology%3Ahorton-total-infiltration) — Engineering Hydrology, chapter 3
- [Light sources and specular reflection principle](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_power_distribution_and_illumination%3Alight-sources-specular-reflection) — Industrial Power Distribution and Illumination, chapter 6
- [Derive reflection coefficient for terminated LC ladder](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-coefficient-derivation) — Filter Design, chapter 5, asked 2×