---
title: "Phong illumination model expression"
subject: "Computer Graphics"
chapter: 8
key: "semester_5/computer_graphics:phong-illumination-model-derive"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/phong-illumination-model-derive
site: BCT Engineering
---

# Phong illumination model expression

Computer Graphics · Chapter 8: Illumination and Surface Rendering methods

- Asked in: 2079 Baisakh, 8 marks; 2078 Kartik, 8 marks; 2070 Chaitra, 8 marks; 2074 Chaitra, 6 marks; 2066 Magh (old course); 2076 Asoj, 8 marks; 2075 Asoj, 8 marks; 2072 Chaitra, 6 marks (asked 8 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/phong-illumination-model-derive
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-8

## Question

Derive the expression for the Phong illumination model (total light intensity at a point, with ambient, diffuse and specular components, for the different light sources and with the distance consideration), with an example.

## Short answer

The Phong illumination model sums ambient, diffuse, and specular components: I = ka·Ia + IL[kd(N·L) + ks(R·V)^n]. With distance attenuation f(d) = 1/(a0+a1·d+a2·d²) and multiple light sources, I = ka·Ia + Σ f(d_i) IL_i[kd(N·L_i) + ks(R_i·V)^n]. Example: given values, I = 0.659.

## Answer

The **Phong illumination model** (empirical) computes the intensity at a surface point as the sum of ambient, diffuse and specular components.

**1. Ambient light:** constant background light, equal in all directions.

I_amb = ka · Ia

(ka = ambient reflection coefficient, Ia = ambient intensity.)

**2. Diffuse reflection (Lambert's cosine law):** light is scattered equally in all directions; intensity depends on the angle θ between the surface normal N and light direction L.

I_diff = kd · IL · cosθ = kd · IL · (N · L)

(N and L unit vectors; use max(0, N·L).)

**3. Specular reflection:** the shiny highlight seen around the direction of perfect reflection R. With φ the angle between R and the viewing direction V:

I_spec = ks · IL · cos^n φ = ks · IL · (R · V)^n

where R = 2(N·L)N − L, ks = specular coefficient, n = shininess exponent (larger for shiny surfaces). Using H = (L + V)/|L + V|, (N·H)^n can replace (R·V)^n.

**Total for one point source**

I = ka·Ia + IL [ kd (N·L) + ks (R·V)^n ]

**With distance (attenuation):** the light intensity from a point source decreases with distance d:

f(d) = min( 1 / (a0 + a1·d + a2·d²), 1 )

I = ka·Ia + f(d) · IL [ kd (N·L) + ks (R·V)^n ]

**For several light sources** i = 1 … m:

I = ka·Ia + Σ f(d_i) · IL_i [ kd (N·L_i) + ks (R_i·V)^n ]

**Example.** ka = 0.3, Ia = 0.4, kd = 0.5, ks = 0.4, n = 10, IL = 1, f = 1; N·L = 0.8, R·V = 0.9:

I = 0.3·0.4 + 1·[0.5·0.8 + 0.4·0.9^10]
 = 0.12 + [0.40 + 0.4·0.3487]
 = 0.12 + 0.40 + 0.139 = **0.659**

## Check yourself

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

1. What is the diffuse component?
   A. kd . IL . (N . L)
   B. ks . IL . (R . V)^n
   C. ka . Ia
   D. IL / d^2
   Answer: A. Lambert's cosine law depends on N and L. (Key point: Diffuse term)

2. In ks . IL . (R . V)^n, what does larger n mean?
   A. A rougher surface
   B. A shinier surface with a smaller, sharper highlight
   C. More diffuse light
   D. Higher ambient light
   Answer: B. Large n makes the highlight fall off quickly. (Key point: Specular term)

3. Which factor models distance?
   A. f(d) = d^2
   B. f(d) = 1/n
   C. f(d) = min(1/(a0 + a1 d + a2 d^2), 1)
   D. f(d) = ka . d
   Answer: C. It multiplies the diffuse and specular terms. (Key point: Attenuation)

4. For ka=0.3, Ia=0.4, kd=0.5, ks=0.4, n=10, IL=1, f=1, N.L=0.8, R.V=0.9, what is I?
   A. About 0.52
   B. About 0.80
   C. About 1.12
   D. About 0.659
   Answer: D. 0.12 + 0.40 + 0.4 x 0.3487 = 0.659. (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×
- [Distance consideration in intensity](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aintensity-attenuation-distance) — 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
- [Specular reflection](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aspecular-reflection-expression) — Computer Graphics, chapter 8, asked 4×
- [Diffuse reflection equation](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Adiffuse-reflection-equation) — Computer Graphics, chapter 8, asked 3×
- [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

- [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