Ray-Shape Intersections

For each shape type, derive: does this ray hit, and where?

Ray-sphere:

Ray: P(t) = O + t*D Sphere: |P - C|² = r²

Substitute: |O + t*D - C|² = r² Expand to quadratic in t:

a = D · D = 1 (if D normalized)
b = 2 * D · (O - C)
c = (O - C) · (O - C) - r²
discriminant = b² - 4ac

Real solutions if discriminant ≥ 0:

The t parameter is the distance along the ray. We typically want closest hit > 0.

Ray-triangle (Möller-Trumbore):

Returns t (distance along ray) + barycentric coordinates.

Ray-plane:

• Plane: n · P = d.
• Substitute: n · (O + t*D) = d → t = (d - n·O) / (n·D).
• If denominator 0: parallel.

Ray-AABB:

• Slab method: 3 pairs of planes.
• t_min, t_max per axis.
• Intersection if max(t_mins) < min(t_maxes).

Used for BVH traversal. Fast.

Ray-mesh:

• Test ray against every triangle.
• Slow: O(N) per ray.
• Fix with BVH or grid: O(log N).

For complex shapes:

• Cubes, cylinders, cones: closed-form intersections exist.
• General: convert to triangles + ray-triangle.
• Implicit surfaces: ray marching (sphere tracing).