Concept: Cross Product

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

Definition

• a × b = (a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y - a.y*b.x)
• Result is a vector perpendicular to both a and b
• Magnitude: |a × b| = |a| * |b| * sin(θ)
• Direction: right-hand rule (curl fingers from a to b, thumb points in result direction)

Key Properties

• Anti-commutative: a × b = -(b × a) — ORDER MATTERS
• a × a = (0, 0, 0) — cross product with itself is zero
• |a × b| = area of parallelogram formed by a and b
• If a × b = 0, vectors are parallel (or one is zero)

Uses in Path Tracing

• Building orthonormal basis (TBN matrix)
  • Given normal N, need tangent T and bitangent B
  • T = normalize(cross(N, up)) where up = (0,1,0) (or (1,0,0) if N is near up)
  • B = cross(N, T) — already unit length if N and T are unit
  • Used to transform sampled directions from tangent space to world space
• Triangle normal computation
  • Given vertices v0, v1, v2:
  • edge1 = v1 - v0
  • edge2 = v2 - v0
  • N = normalize(cross(edge1, edge2))
  • Winding order determines which side is “front”
• Möller–Trumbore intersection
  • h = cross(ray.dir, edge2)
  • det = dot(edge1, h) — used to detect parallel ray
  • q = cross(s, edge1) — used to compute barycentric v

GLSL

• vec3 c = cross(vec3 a, vec3 b);
• Only defined for vec3 (not vec2 or vec4)

Degenerate Cases

• If N is nearly parallel to up = (0,1,0):
  • cross(N, up) → near-zero vector → unstable normalization
  • Solution: choose up = (1,0,0) when |dot(N, (0,1,0))| > 0.9
  • Or use Frisvad/Duff method (see PathTracer Learning - Phase 1 - Math for Graphics)