Correctness of primal::intersection_volume(Hexahedron, Tetrahedron) for general hexahedra

[kennyweiss]

The current implementation of primal::intersection_volume(Hexahedron, Tetrahedron) uses primal::clip(Hexahedron, Tetrahedron), which in turn uses primal::Polyhedron::from_primitive(Hexahedron).

However, since our Polyhedron class assumes planar faces, primal::Polyhedron::from_primitive(Hexahedron) is not valid for hexahedra with non-planar faces. Since our Hexahedron class is initialized with 8 unconstrained vertices, the faces of a (valid) Hexahedron will be bilinear rather than planar.

To remedy this:

• We should add a function to Hexahedron to check if its faces are (nearly) planar
• Polyhedron::from_primitive(Hexahedron) should only operate on Hexahedra with planar faces. This should either be explicitly checked at runtime, or in debug mode (via SLIC_DEBUG). At the very least, we should document this
• Similarly, primal::clip(Hexahedron, Tetrahedron) is only applicable to planar Hexahedra. We should add this to the docs, and consider adding runtime/debug checks, as appropriate
• primal::intersection_volume(Hexahedron, Tetrahedron) should be updated to split Hexahedra into 24 tets and return the sum of the volumes of their intersections. When the input Hexahedron has planar faces, we can use the current clip algorithm.
  • We'll likely want to benchmark whether the checks for planar faces + clip is more performant than the "sum of intersections" algorithm and use the more performant one
• We'll also need to update the tests for intersection_volume(Hex, Tet) to ensure we have cases with non-planar hexahedra.