Noob question on smoothing / adding divisions

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I have a triangular mesh (closed surface) that I am importing. Each vertex represents a point ON the surface of the object I am modeling. I would like to increase the number of facets in a way that interpolates the data in a smooth spline-like fashion.
If I "smooth" the mesh, then I get a new surface that is inside the original surface where it is convex, and outside the original surface where it is concave. If I use "add divisions", it keeps the original flat facets, while adding more polygons inside those flat facets. I would like to be able to smooth the surface in such a way that all of the original vertices remain in their original locations, while increasing the number of faces - such that added vertices would be placed on some smoothly curving surface that goes THROUGH all of the original vertices.
Is there a way to do this in Maya?
I hope my description was clear. A simple example would be a cube. When you smooth it, you get a sphere that is smaller than the cube. I would like to get a sphere that goes through the original cube vertices. In that simple example, you could just smooth it, then scale the result, but that would not work properly for a more complicated surface that has both convex and concave regions.
Any help on how to achieve this much appreciated. Thanks.