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Ken Clarkson
Bell Laboratories
Approximating Surfaces with Meshes
April 12, 2006 10:00am
Abstract:
How hard is it to approximate a smooth surface M with a piecewise-linear mesh? When M is the boundary of a convex body, remarkably tight bounds are known for the smallest Hausdorff distance possible for a mesh with n simplices. In the case of more general surfaces, much less is understood. I'll show that the smallest distance, when M is a d-manifold, is O(S/n)^{2/d}, where S is the integral over M of the square root of the Gaussian curvature. (The constant factor here depends only on the dimension.) Also, under some reasonably general conditions on the surface and the mesh, this expression is also a lower bound, up to a constant factor. The upper bound construction distributes the vertices of the mesh in an "epsilon-net", in a metric based on directional curvature. The lower bound relates the volume of a simplex to its interpolation error.
It may be helpful that a version of the slides is at http://cm.bell-labs.com/who/clarkson/enet_tris/t2/t.xml, viewable with the Firefox browser.
If you have questions, or would like to meet the speaker, please contact Ponda at 4-1994 or pondabarnes@tti-c.org. For information on future TTI-C talks or events, please go to the TTI-C Events page.