> computed the location of each Voronoi vertex by solving a 4 X 4
> linear system with a solver from LAPACK.  For the life of me, I
> can't figure what you used to solve for the Voronoi vertices

To find the location of the center of a voronoi tetrahedron, 
you have to find the center of a sphere through the four 
vertices of the tetrahedron (A, B, C, D). 

The coordinates (x,y,z) of that point P are such that: 

        AP = BP = CP = DP = radius of the sphere

or in coordinate form: 

        (a[0] - x)^2 + (a[1] - y)^2 + (a[2] - z)^2 = 
        (b[0] - x)^2 + (b[1] - y)^2 + (b[2] - z)^2 = 
        (c[0] - x)^2 + (c[1] - y)^2 + (c[2] - z)^2 = 
        (d[0] - x)^2 + (d[1] - y)^2 + (d[2] - z)^2 

These equations can be rewritten pairwise, in terms of x, y, z.
taking the first and second lines above and solving yields: 

        (2a[0]-2b[0])*x + (2a[1]-2b[1])*y + (2a[2]-2b[2])*z = 
                = a[0]^2 + a[1]^2 + a[2]^2 - b[0]^2 - b[1]^2 - b[2]^2

You get 3 such equations that you can simplify and solve using LAPACK. 

Looking at my code, the equation that i solve with LAPACK seems to be: 

        Ax = b

where A is the 4x4 matrix: 

        2a[0]      2a[1]      2a[2]      -1
        2b[0]      2b[1]      2b[2]      -1
        2c[0]      2c[1]      2c[2]      -1
        2d[0]      2d[1]      2d[2]      -1

and b is the vector: 

        a[0]^2 + a[1]^2 + a[2]^2
        b[0]^2 + b[1]^2 + b[2]^2
        c[0]^2 + c[1]^2 + c[2]^2
        d[0]^2 + d[1]^2 + d[2]^2

Please let me know if this is in the lines of what you're looking for. 

Good luck on your project :o)