Concept: Dot Product

• Parent: PathTracer Learning - Phase 1 - Math for Graphics

Definition

• a · b = a.x*b.x + a.y*b.y + a.z*b.z
• Geometric interpretation: a · b = |a| * |b| * cos(θ)
• When both vectors are unit length: a · b = cos(θ)
• Result is a scalar (single number), not a vector

Key Properties

• Commutative: a · b = b · a
• Distributive: a · (b + c) = a·b + a·c
• a · a = |a|² — dot product with itself = squared length
• dot(a, b) > 0 → angle < 90° (same general direction)
• dot(a, b) = 0 → angle = 90° (perpendicular)
• dot(a, b) < 0 → angle > 90° (opposite general direction)

Uses in Path Tracing

• Lambert’s cosine law
  • irradiance = L_i * dot(N, L) where N=normal, L=light direction
  • dot(N, L) gives the cosine of the angle of incidence
  • Clamp to 0: max(0, dot(N, L)) — light from behind contributes nothing
• Checking ray direction vs surface normal
  • dot(ray.dir, N) < 0 → ray hits front face
  • dot(ray.dir, N) > 0 → ray hits back face
  • Used to flip normals for double-sided materials
• Reflection formula derivation
  • Project v onto n: proj = dot(v, n) * n
  • Reflection: reflect(v, n) = v - 2 * dot(v, n) * n
• BRDF evaluation
  • NdotL = max(0, dot(N, L)) — diffuse factor
  • NdotV = max(0, dot(N, V)) — view factor
  • NdotH = max(0, dot(N, H)) — half-vector factor (specular)
  • VdotH = max(0, dot(V, H)) — Fresnel factor

GLSL

• float d = dot(vec3 a, vec3 b);
• Works for vec2, vec3, vec4
• Hardware-accelerated — single instruction on GPU

Common Mistakes

• Forgetting to normalize before using as cosine
  • dot(a, b) is NOT cos(θ) unless both are unit vectors
  • Always normalize direction vectors: normalize(v)
• Not clamping to 0
  • dot(N, L) can be negative (light behind surface)
  • Use max(0.0, dot(N, L)) in lighting calculations