A hardware-based traversal coprocessor provides acceleration of tree traversal operations searching for intersections between primitives represented in a tree data structure and a ray. The primitives may include triangles used in generating a virtual scene. The hardware-based traversal coprocessor is configured to properly handle numerically challenging computations at or near edges and/or vertices of primitives and/or ensure that a single intersection is reported when a ray intersects a surface formed by primitives at or near edges and/or vertices of the primitives.