Notes

How a surface’s final color gets computed from light hitting it. Lambert handles the matte part, Phong adds a highlight and a cheap stand-in for bounced light. Neither is physically accurate; both are still the scaffolding modern PBR is built on.

Lambert’s law

Light arriving at a surface is proportional to the cosine of the angle between the light direction and the surface normal.

Why cosine: a light beam with a fixed cross-section covers a larger patch of surface as the angle increases, so the same energy is spread thinner. The stretch factor is $1/\cos\theta$, so energy per unit area is $\cos\theta$.

Since $\text{dot}(A,B) = |A||B|\cos\theta$, two normalized vectors give the cosine directly, no trig needed.

$$I = \text{clamp}(\mathbf{N}\cdot\mathbf{L},\ 0,\ 1)$$

$\mathbf{N}$ = surface normal, $\mathbf{L}$ = direction toward the light. Diffuse means the surface scatters evenly in all directions, so the result doesn’t depend on where the viewer is. That’s also why Lambert alone has no highlight.

Phong reflection model

Three light paths reaching the eye, summed:

$$C = k_d D + k_s S + k_a A$$

Term	What it is
$D$	Diffuse — Lambert, $\text{dot}(N,L)$
$S$	Specular — the highlight
$A$	Ambient — a constant, faking bounced light

$k_d, k_s, k_a$ are the material’s reflectance for each. Addition is correct here because light is energy and superposes; multiplication is for a ratio acting on energy (texture color, the $k$ terms).

Ambient is a pure hack. Real indirect light means tracing bounces; a constant just prevents pure-black regions. It can’t depend on $N$, because “arriving equally from all directions” is precisely the assumption that makes orientation irrelevant.

Blinn-Phong specular

$$S = \max(\text{dot}(\mathbf{H},\mathbf{N}),\ 0)^{shininess} \cdot C_{spec}, \qquad \mathbf{H} = \frac{\mathbf{V}+\mathbf{L}}{2}$$

$\mathbf{H}$ is halfway between view and light direction. The highlight peaks when $\mathbf{V}$ aligns with $\mathbf{L}$ reflected about $\mathbf{N}$. Higher shininess = tighter highlight.

Unlike diffuse, this depends on $\mathbf{V}$ — move the camera and the highlight moves.

Gouraud vs Phong shading

Different axis from the reflection model: this is about where you evaluate it.

• Gouraud — compute lighting per vertex, interpolate the resulting color. Cheap. Highlights break into visible polygon facets on low-poly meshes.
• Phong — interpolate the normal, compute lighting per pixel. What OpenGL does. Smooth highlights on coarse geometry.

Code

glm::vec3 lightVec = -glm::vec3(camera.mForward);   // headlight: hoist out of the loop

// per fragment
glm::vec3 n = glm::normalize(interpNormal);         // interpolation doesn't preserve length
float lambert = glm::clamp(glm::dot(n, lightVec), 0.f, 1.f);
float intensity = glm::min(lambert + AMBIENT, 1.f);
glm::vec3 finalColor = texColor * intensity;

Gotchas

• Normalize after interpolating the normal. A weighted average of three unit vectors is shorter than 1, and the dot product silently stops being a cosine.
• Clamp the dot product. A surface facing away gives a negative value, which multiplies through into a meaningless negative color.
• Clamp the sum too. lambert + ambient reaches 1.3; qRgb takes 0–255 ints and overflows. The ceiling is the display’s, not physics’ — HDR keeps the float range and tone-maps at the end instead.
• Missing lightColor silently assumes white light. Fine until the scene has a sunset or a colored bulb.
• Light vector and normal must be in the same space. Object-space normals against a world-space light vector is a quiet, plausible-looking wrong answer. Transforming a normal uses the inverse transpose of the model matrix, not the model matrix.

Related

Dot product
Material properties
Perspective-correct interpolation
Rasterization