Phong Reflection Model

The Phong Model

The complete Phong shading model combines three reflection components:

$I = I_a k_a + \sum_{\text{lights}} I_l k_d \max(0, \mathbf{N} \cdot \mathbf{L}) + \sum_{\text{lights}} I_l k_s \max(0, \mathbf{R} \cdot \mathbf{V})^n$

Components

Component	Term	Properties
Ambient	$I_a k_a$	Constant; approximates indirect illumination
Diffuse	$I_l k_d (\mathbf{N} \cdot \mathbf{L})$	View-independent; Lambertian reflection
Specular	$I_l k_s (\mathbf{R} \cdot \mathbf{V})^n$	View-dependent; creates highlights

Symbols

Symbol	Meaning
$k_a, k_d, k_s$	Reflection coefficients
$I_a, I_l$	Ambient and light intensities
$\mathbf{N}$	Surface normal (unit vector)
$\mathbf{L}$	Direction to light (unit vector)
$\mathbf{R}$	Perfect reflection direction
$\mathbf{V}$	Direction to viewer (unit vector)
$n$	Phong exponent (shininess)

Reflection Direction

$\mathbf{R} = 2(\mathbf{L} \cdot \mathbf{N})\mathbf{N} - \mathbf{L}$

Or equivalently: $\mathbf{R} = 2\cos\theta \mathbf{N} - \mathbf{L}$

Per-Channel Computation

The shading equation is computed for each RGB channel separately:

• Different $k_d$ values for each channel give surface colour
• Specular highlights often use light colour (white for plastics)

Clamping

• $\max(0, \cdot)$ prevents negative contributions (light behind surface)
• Without clamping, surfaces can become incorrectly dark

Summary

• Phong shading = ambient + diffuse + specular
• Ambient approximates global illumination
• Diffuse is view-independent (Lambertian)
• Specular is view-dependent (highlights)
• The $\max(0, \cdot)$ prevents negative values from light behind surfaces