Tetrahedron sampling

[mweatherley]

Objective

Add interior and boundary sampling for the Tetrahedron primitive. This is part of ongoing work to bring the primitives to parity with each other in terms of their capabilities.

Solution

Tetrahedron implements the ShapeSample trait. To support this, there is a new public method Tetrahedron::faces which gets the faces of a tetrahedron as Triangle3ds. There are more sophisticated ideas for getting the faces we might want to consider in the future (e.g. adjusting according to the orientation), but this method gives the most mathematically straightforward answer, giving the faces the orientation induced by the tetrahedron itself.

[IQuick143]

I know the compiler will probably crunch through this, esp. given the #[inline] but couldn't we just reorder the arguments?

[mweatherley]

Hmm, I guess so, but I thought this would make it clearer what's actually going on, since the relationship between vertex order and orientation is otherwise not that clear (e.g. the choice of which vertices to interchange when Triangle3d::reverse is called is essentially arbitrary).

[alice-i-cecile]

Yeah, this is clearer to me.

[alice-i-cecile]

A neat little theorem in there about linear transformations preserving probability densities correctly. Do you have a citation at hand?

[mweatherley]

A neat little theorem in there about linear transformations preserving probability densities correctly. Do you have a citation at hand?

I guess this is just the standard fact that the Lebesgue measure of a linearly transformed set is the original Lebesgue measure times the determinant of the linear transformation. I can't find a great standalone source, but I guess I could just link to something like the Wikipedia article.