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Topological Examples for Algorithmic Verification

This paper explores topological examples for algorithmic verification, discussing topics such as topology and approximation, algorithms, and applications. It examines the reasons for bad approximation, the importance of separation and curvature, and the impact of homeomorphic changes on knot types.

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Topological Examples for Algorithmic Verification

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  1. Topological Examples for Algorithmic Verification T. J. Peters, University of Connecticut www.cse.uconn.edu/~tpeters K. Abe, A. C. Russell, J. Bisceglio, E.. Moore, D. R. Ferguson, T. Sakkalis

  2. Outline: Topology & Approximation • Algorithms • Applications

  3. Unknot

  4. Bad Approximation Why? Separation? Curvature?

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

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

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

  8. Initial Assumptionson a 2-manifold, M • Without boundary • 2nd derivatives are continuous (curvature)

  9. T

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

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

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

  13. X

  14. Large Data Set ! Partitioned Stanford Bunny

  15. Acknowledgements, NSF • I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS-0138098. • SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR - 0226504. • Computational Topology for Surface Approximation, September 15, 2004, #FMM -0429477.

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