T. J. Peters, University of Connecticut K. Abe, A. C. Russell, J. Bisceglio, E.. Moore, D. R. Ferguson, T. Sakkalis Topological Examples for Algorithmic Verification

Outline: Topology & Approximation Algorithms Applications

Unknot

Bad Approximation Why? Curvature? Separation?

Why Bad? Homeomorphic! Changes Knot Type Now has 4 Crossings

Good Approximation Homeomorphic vs. Ambient Isotopic (with compact support) Via Curvature (local) Separation (global)

Summary – Key Ideas Curves –Don’t be deceived by images (3D !) –Crossings versus self-intersections Local and global arguments Knot equivalence via isotopy

Initial Assumptions on a 2-manifold, M Without boundary 2 nd derivatives are continuous (curvature)

T

Proof: Similar to flow on normal field. Comment: Points need not be on surface. (noise!) Theorem: Any approximation of F in T such that each normal hits one point of W is ambient isotopic to F.

Tubular Neighborhoods Its radius defined by ½ minimum –all radii of curvature on 2-manifold –global separation distance. Estimates, but more stable than medial axis. and Ambient Isotopy

Medial Axis H. Blum, biology, classification by skeleton Closure of the set of points that have at least 2 nearest neighbors on M

X

Large Data Set ! Partitioned Stanford Bunny

Acknowledgements, NSF I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR Computational Topology for Surface Approximation, September 15, 2004,Computational Topology for Surface Approximation, September 15, 2004, #FMM