Concept: Ray-Triangle Intersection

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

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)

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

Implementation

bool rayTriangle(Ray ray, vec3 v0, vec3 v1, vec3 v2,
                 out float t, out float u, out float v) {
    vec3 edge1 = v1 - v0;
    vec3 edge2 = v2 - v0;
    vec3 h = cross(ray.direction, edge2);
    float det = dot(edge1, h);
    
    // det ≈ 0 means ray is parallel to triangle
    if (abs(det) < 1e-8) return false;
    
    float invDet = 1.0 / det;
    vec3 s = ray.origin - v0;
    u = dot(s, h) * invDet;
    if (u < 0.0 || u > 1.0) return false;
    
    vec3 q = cross(s, edge1);
    v = dot(ray.direction, q) * invDet;
    if (v < 0.0 || u + v > 1.0) return false;
    
    t = dot(edge2, q) * invDet;
    return t > 1e-4;  // t_min to avoid self-intersection
}

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

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

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

Related

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