Mr. Bogomolny,

I am trying to find a source for the conversion of a tetrahedron in barycentric coordinates to a Cartesian space. Your sites have been very useful in explaining the concepts nowhere on the web can I find a conversion algorithm can you help me?

Harold Stone
Assistant Professor of Environmental Planning
East Carolina University
Greenville, NC 27858

Barycentric coordinates are defined relative to a tetrahedron whose vertices must be defined somehow else. One gets no information about their location from the barycentrics (1, 0, 0, 0), (0, 1, 0, 0), etc. A natural possibility is having the vertices in the Cartesian coordinates to start with:
(x1, y1, z1), (x2, y2, z2), etc.

Then the point with barycentrics (a, b, c, d) translates into Cartesian coordinates as

(a*x1 + b*x2 + c*x3 + d*x4, a*y1 + b*y2 + c*y3 + d*y4, a*z1 + b*z2 + c*z3 + d*z4)

Regards,
Alexander Bogomolny