Implement triangle validation and add unit tests

[therealprof]

Closes #240

Signed-off-by: Daniel Egger [email protected]

[therealprof]

Please note that for some reason I can't seem to be able to create a 2D triangle because cross product somehow only works for 3D vertices?

[hannobraun]

Thank you, @therealprof, looks good!

Please note that for some reason I can't seem to be able to create a 2D triangle because cross product somehow only works for 3D vertices?

First, I'm going to merge this besides this limitation. The math code isn't exposed in a public library (yet), and we don't seem to create 2D triangles, so it's fine for now.

Second, I think I get it:

• Mathematically, at least according to my primitive understanding, the cross product is not defined for 2D vectors. The result is a vector that is perpendicular to the input vectors, which can't work in 2D.
• So the question becomes, why does Vector::cross exist for 2D vectors? Answer: It was an accident, made possible because Matrix::cross exists for all matrices in nalgebra.
• Digging into that, nalgebra::Matrix::cross is documented to panic for non-3D vectors, and will be removed for non-dynamic/non-3D vectors in the future.

Idea for fixing this:

• Add a method that converts a Vector<D> into a Vector<3>. If it's a Vector<2>, the z component can be zero.
• Call that method before computing the triangle area, so the cross product is always done on 3D vectors.

In addition, we should define Vector::cross only for Vector<3>.

I've made notes about all of these action items. I'll either do them myself later, or open issues to track them.

[therealprof]

You're right. The cross product is a 3D version of the generalised external product. In 2D the cross product degenerates to the determinant. I think the reason why nalgebra only allows for 3-dimensional vectors is because only then the result will be again a vector. A long time ago I might have known this... 😅