I'm working on a computational geometry algorithm.
In every iteration I solve the matrix below, where (a,b,c,d) are the vertices of a tetrahedron, and e is an arbitrary point. Solving the determinant of the matrix one can test whether point e lies inside the circumphere of the tetrahedron.
The determinant is positive if e lies inside the circumphere of tetrahedron (a,b,c,d), negative if outside, and zero if they are cospherical or coplanar.
However I would also need the volume of the tetrahedron.
Can I compute the volume of the tetrahedron from the above matrix with minimal overhead? Or should I compute it independently?