---
title: "Fast Phong shading"
subject: "Computer Graphics"
chapter: 8
key: "semester_5/computer_graphics:fast-phong-shading"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/fast-phong-shading
site: BCT Engineering
---

# Fast Phong shading

Computer Graphics · Chapter 8: Illumination and Surface Rendering methods

- Asked in: 2082 Kartik (new course), 4 marks; 2081 Chaitra (new course), 4 marks (asked 2 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/fast-phong-shading
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-8

## Question

Explain Fast Phong shading model with relevant expression. Mention the advantages of fast Phong shading over Phong shading.

## Short answer

Fast Phong approximates normal Phong by expanding the per-pixel intensity formula (with interpolated unnormalized normals) as a Taylor series up to second degree in x,y, then evaluates incrementally using forward differences. Advantages: no per-pixel square root, division or exponentiation; runs about twice as fast as Gouraud versus six-seven times for Phong with nearly identical quality.

## Answer

**Fast Phong shading** is an approximation of Phong shading that avoids the expensive per-pixel normalisation and power calculation.

In normal Phong shading, the interpolated normal N must be normalised at every pixel (a square root and a division), and (N·H)^n must be computed. Fast Phong shading reduces this work.

**Idea.** Over a triangle, the interpolated (unnormalised) normal is a linear function of the screen position:

N = Ax + By + C

where the vectors A, B, C are found from the three vertex normals. For a distant light (L constant), the diffuse term becomes

I_diff = (N·L)/(|L||N|) = (ax + by + c) / √(dx² + exy + fy² + gx + hy + i)

where the constants a, b, …, i stand for the various dot products of A, B, C and L. The method then uses:

1. **Taylor series expansion** of this expression, keeping terms up to the second degree in x and y. This removes the per-pixel square root and division:

   I_diff ≈ T5·x² + T4·xy + T3·y² + T2·x + T1·y + T0

   The specular term is approximated in the same way.
2. **Incremental (forward-difference) evaluation:** along a scan line, I(x + 1) = I(x) + ΔI and ΔI(x + 1) = ΔI(x) + Δ²I, so each pixel costs only two additions.

**Advantages over Phong shading**
- Much faster, with no per-pixel square root, division or exponentiation. Fast Phong takes about twice as long as Gouraud shading, while normal Phong takes about six to seven times as long.
- Nearly the same image quality (good highlights).
- Less computation, so it is suitable for hardware.

**Limitation:** approximation errors can appear if the polygon is large or the normals change quickly across it (especially for large n).

## Check yourself

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

1. What does Fast Phong avoid?
   A. Per-pixel normalisation and power calculation
   B. Interpolating normals at all
   C. Using vertex normals
   D. Any lighting
   Answer: A. These are the costly operations. (Key point: Purpose)

2. How are N.L and N.H updated?
   A. By recomputing from scratch
   B. By forward differences (additions)
   C. By matrix multiplication
   D. By ray casting
   Answer: B. N changes linearly along the scan line. (Key point: Incremental)

3. What is expanded in a Taylor series?
   A. The light direction
   B. The depth z
   C. The reciprocal length 1/|N|
   D. The colour
   Answer: C. This replaces the square-root normalisation. (Key point: Taylor series)

4. When are errors larger?
   A. Small polygons
   B. Flat surfaces
   C. Low n
   D. Large polygons, rapidly changing normals, large n
   Answer: D. The approximation drifts over big changes. (Key point: Limitation)

## Related in Computer Graphics

- [Phong illumination model expression](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aphong-illumination-model-derive) — Computer Graphics, chapter 8, asked 8×
- [Compare Gouraud and Phong shading](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acompare-gouraud-phong) — Computer Graphics, chapter 8, asked 7×
- [Phong shading method](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aphong-shading-method) — Computer Graphics, chapter 8, asked 9×
- [Phong shading and Mach-band effect](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aphong-mach-band-effect) — Computer Graphics, chapter 8
- [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

## Related in other subjects

- [Advantages of SSB over DSB-FC; SSB expression](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Assb-advantages-derive) — Communication Systems, chapter 2