How to calculate the center of mass of a polyhedron?

Hello, I am trying to calculate the center of a convex polyhedron based on its 'mass'.

I originally calculated it by simply taking the averages of the vertices in the x,y,and z directions. However, if there is a point added to an already existing face, this skews the center.

My initial thought is to break the polyhedron into tetrahedra, find their individual centers of mass and then add those up. I can't figure out how to do this. I try to use the Delaunay Triangulation, but I can't figure out the output.

Answer 1

[Amit]

http://www.mathworks.com/matlabcentral/fileexchange/8514-centroid-of-a-convex-n-dimensional-polyhedron

[Joshua]

Tried that code for a pyramid. The addition of an extra point at the center of the base (origin in my case) shifted the center.

Answer 2

[Roger Stafford]

I assume you are in three dimensions, Joshua. The centroid of any tetrahedron is always at the point which is the average value of the coordinates of its four vertices. However the centroid of a polyhedron which is decomposed into separate tetrahedra will be the weighted average of the centroids of these individual tetrahedra where this weighting is proportional to the volumes of the tetrahedra. For example, if you have a polyhedron which is composed of two tetrahedra and one tetrahedron is twice the volume of the second, then the centroid is located at 2/3*C1+1/3*C2 where C1 and C2 are centroids of the larger and smaller of the two tetrahedra, respectively. Using this method of weighted averaging, the further decomposition into more tetrahedra will not shift the overall centroid.