```> 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 - x)^2 + (a - y)^2 + (a - z)^2 =
(b - x)^2 + (b - y)^2 + (b - z)^2 =
(c - x)^2 + (c - y)^2 + (c - z)^2 =
(d - x)^2 + (d - y)^2 + (d - 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-2b)*x + (2a-2b)*y + (2a-2b)*z =
= a^2 + a^2 + a^2 - b^2 - b^2 - b^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      2a      2a      -1
2b      2b      2b      -1
2c      2c      2c      -1
2d      2d      2d      -1

and b is the vector:

a^2 + a^2 + a^2
b^2 + b^2 + b^2
c^2 + c^2 + c^2
d^2 + d^2 + d^2

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

Good luck on your project :o)

-manolis
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