Dr. Jörg Peters

"Shape Characterization of (subdivision) Surfaces"

In comparing shape of surfaces we are confronted with the fact that known subdivsion surfaces are everywhere twice continuously differentiable. To characterize shape in the absence of curvature we develop new metrics.

We find that - for example - the subdivision algorithms by Catmull and Clark and by Loop fail to converge, for generic input data, to the hyperbolic or elliptic limit shape suggested by the geometry of the input mesh: the limit shape s a function of the valence the control mesh rather than the mesh geometry.

We characterize the meshes for which the schemes behave as expected and indicate modifications of the shemes that prevent convergence to the wrong shape. We also introduce a type of chart that, for a specific scheme can help a designer to detect early when a mesh will lead to undesirable curvature behaviour.