Why are these BVH split heuristic formulas different?


I’ve been doing some research into changing the split heuristic I use for a work project comprised of a BVH binary tree.

The heuristic I currently use is centroid median as described here, but I seek to use the surface area heuristic (SAH) now. Several papers denote the cost function for determining the split as:

$ $ c(A, B) = t_{trav} + p_{A}\sum_{i=1}^{N_{A}}t_{isect}(a_{i}) + p_{B}\sum_{i=1}^{N_{B}}t_{isect}(b_{i})$ $

Notionally it is described in detail here, but to summarize, it is the cost of a ray intersect of primitives at this split.

However, I’ve seen an optimization technique, described as "binning", where $ K$ pre-determined bins are computed and therefore the cost function for determining the split is:

$ $ c(A, B) = A_{a}N_{a} + A_{b}N_{b}$ $

This is outlined here in Section 3.1, as well is several other documents (1, 2, 3).

What’s the rational/reasoning behind these two different cost models?