In barycentric coordinates - part 1 I showed a way to compute the barycentric coordinates for an interior point in a triangle, and described how the result could be used in a simple constraint example.
In part 2 I am going to show a python OpenMaya code example that could be used as the basis for some kind of data interpolation between meshes with differing topology.
Barycentric coordinates are useful when you want to transfer data between meshes that have different topology because they provide the basis for some simple interpolation. Take smooth skin binding for example. It is a commmon workflow, when painting weights, to start with a low resolution version of your mesh. When you are happy with the weighting you can use maya's copySkinWeights command to transfer those weights from the low resolution mesh to the detailed high resolution mesh, which may even be made up of several pieces. copySkinWeights interpolates the weights from the verts on the lo res mesh and copies the result to the hi res mesh. The smooth result you get with this workflow is usually something that would have been very difficult to achieve if you had tried to paint the weights directly on the hires mesh.