Vaibhav Rathod's Notes

PathTracer Learning - Concept - Dot Product

2 min read

  • 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

Graph View

Backlinks