PathTracer Learning - Concept - Ray-Triangle Intersection

• Concept: Ray-Triangle Intersection

  • Parent: PathTracer Learning - Phase 2 - CPU Ray Tracing

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• Möller–Trumbore Algorithm

  • The standard algorithm — used in virtually all ray tracers
  • Fast, numerically stable, returns barycentric coordinates
  • Paper: “Fast, Minimum Storage Ray/Triangle Intersection” (1997)

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• Derivation

  • A point on a triangle: P = (1-u-v)*v0 + u*v1 + v*v2
    • u, v ≥ 0 and u + v ≤ 1 for points inside the triangle
    • u, v are barycentric coordinates
  • Ray equation: P = origin + t * direction
  • Set equal: origin + t*dir = (1-u-v)*v0 + u*v1 + v*v2
  • Rearrange: [-dir, v1-v0, v2-v0] * [t, u, v]^T = origin - v0
  • Solve with Cramer’s rule

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• Implementation

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• Back-Face Culling

  • det < 0 means ray hits the back face of the triangle
  • For opaque geometry: if (det < 1e-8) return false; (cull back faces)
  • For double-sided materials: use abs(det)
  • In Vulkan: VK_GEOMETRY_INSTANCE_TRIANGLE_FACING_CULL_DISABLE_BIT_KHR disables culling

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• Barycentric Coordinates

  • (1-u-v, u, v) — weights for vertices v0, v1, v2
  • Interpolate any vertex attribute: attr = (1-u-v)*a0 + u*a1 + v*a2
  • Normal interpolation: N = normalize((1-u-v)*n0 + u*n1 + v*n2)
  • UV interpolation: uv = (1-u-v)*uv0 + u*uv1 + v*uv2
  • In Vulkan closest-hit shader: hitAttributeEXT vec2 baryCoords;
    • baryCoords.x = u, baryCoords.y = v
    • w = 1 - u - v

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• Watertight Intersection

  • Standard Möller–Trumbore can miss edges/vertices due to floating point
  • Watertight algorithm (Woop et al. 2013) — no gaps at shared edges
  • Important for production renderers — prevents light leaking through cracks
  • Technique: transform ray to a coordinate system where it points along +Z

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• Related

  • PathTracer Learning - Concept - Ray Definition
  • PathTracer Learning - Concept - BVH Traversal