C++ 3D Barycentric Coordinates: Representation and Calculation

Barycentric coordinates are a way of representing a point in a triangle using the relative weights of the triangle's vertices. In three dimensions, we can extend this concept to represent a point in a tetrahedron using the relative weights of the tetrahedron's vertices.

To find the barycentric coordinates of a point in a tetrahedron, we first need to determine which of the tetrahedron's faces the point lies on. We can do this by computing the signed volumes of the four tetrahedra formed by the point and each of the tetrahedron's faces. If the sum of the signed volumes is equal to the volume of the tetrahedron, then the point lies inside the tetrahedron. Otherwise, it lies outside.

Once we know which face the point lies on, we can compute its barycentric coordinates with respect to that face. Let the vertices of the face be A, B, and C, and let P be the point we're interested in. We can compute the barycentric coordinates of P with respect to this face by solving the following system of equations:

P = wA + xB + yC w + x + y = 1

where w, x, and y are the barycentric weights. We can solve this system using Cramer's rule, which gives us:

w = det(P, B, C) / det(A, B, C) x = det(A, P, C) / det(A, B, C) y = det(A, B, P) / det(A, B, C)

where det(u, v, w) is the determinant of the matrix formed by the columns u, v, and w.

Finally, we can extend these barycentric coordinates to the entire tetrahedron by assigning a weight of 0 to the fourth vertex of the tetrahedron, and then normalizing the weights so that they sum to 1. That is:

P = wA + xB + yC + zD w + x + y + z = 1

where z = 1 - w - x - y.