Vaibhav Rathod's Notes

PathTracer Learning - Concept - AABB

2 min read

  • Definition

    • A box whose faces are aligned with the coordinate axes
    • Defined by two points: min = (x_min, y_min, z_min) and max = (x_max, y_max, z_max)
    • Every point P inside satisfies: min.x ≤ P.x ≤ max.x (and same for y, z)
    • “Axis-aligned” means no rotation — simplifies intersection math enormously

  • Ray-AABB Intersection (Slab Method)

    • Treat the AABB as the intersection of 3 pairs of parallel planes (slabs)
    • For each axis, compute entry and exit t values
      • t_x_min = (min.x - ray.origin.x) / ray.direction.x
      • t_x_max = (max.x - ray.origin.x) / ray.direction.x
      • If ray.direction.x < 0, swap min and max
    • Combine all 3 axes
      • t_enter = max(t_x_min, t_y_min, t_z_min) — ray enters the box
      • t_exit = min(t_x_max, t_y_max, t_z_max) — ray exits the box
    • Hit condition: t_enter <= t_exit && t_exit > t_min
    • Implementation
    • Note: invDir = 1.0 / ray.direction precomputed for efficiency
    • Handle direction = 0 case: invDir = ±infinity, which works correctly with IEEE 754

  • AABB Construction

    • From a set of points: min = componentwise_min(all_points), max = componentwise_max(all_points)
    • From a triangle: min = min(v0, v1, v2), max = max(v0, v1, v2)
    • Merging two AABBs: merged.min = min(a.min, b.min), merged.max = max(a.max, b.max)

  • Surface Area

    • SA = 2 * (dx*dy + dy*dz + dz*dx) where dx = max.x - min.x etc.
    • Used in SAH (Surface Area Heuristic) for BVH construction
    • Intuition: larger surface area → more likely to be hit by a random ray

  • Why AABB and Not OBB?

    • OBB (Oriented Bounding Box) fits tighter but intersection is much more expensive
    • AABB intersection: ~6 divisions, ~6 comparisons
    • OBB intersection: requires transforming ray to box space (matrix multiply)
    • BVH with AABBs is fast enough in practice — hardware RT uses AABBs

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