1 / 44

Computational Topology for

Computational Topology for. Animation and Simulation. T. J. Peters, University of Connecticut www.cse.uconn.edu/~tpeters with I-TANGO Team, ++. Outline: Animation & Approximation. Animations for 3D Algorithms Applications. Animation for Understanding. ROTATING IMMORTALITY

seda
Download Presentation

Computational Topology for

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Computational Topology for Animation and Simulation T. J. Peters, University of Connecticut www.cse.uconn.edu/~tpeters with I-TANGO Team, ++

  2. Outline: Animation & Approximation • Animations for 3D • Algorithms • Applications

  3. Animation for Understanding • ROTATING IMMORTALITY • www.bangor.ac.uk/cpm/sculmath/movimm.htm • Möbius Band in the form of a Trefoil Knot • Animation makes 3D more obvious

  4. Unknot

  5. Bad Approximation Why? Separation? Curvature?

  6. Why Bad? No Intersections! Changes Knot Type Now has 4 Crossings

  7. Good Approximation All Vertices on Curve Respects Embedding Via Curvature (local) Separation (global)

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

  9. KnotPlot !

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

  11. T

  12. 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!)

  13. 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.

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

  15. X

  16. Opportunities • Bounds for animation & simulation • Surfaces move • Boundaries move • Functions to represent movement

  17. Seminal Paper, Modified Claim Surface reconstruction from unorganized points, H. Hoppe, T. DeRose, et al., 26 (2), Siggraph, `92 The output of our reconstruction method produced the correct topology in all the examples. We are trying to develop formal guarantees on the correctness of the reconstruction, given constraints on the sample and the original surface

  18. KnotPlot ! Perko Pair & Dynamic Drug Docking

  19. Animation & Simulation • Successive Frames • O(N^2) run time • risk of error versus step size • Isotopy • O(N^2) off-line • simple bound comparison at run time • formal correctness • IBM Award Nomination (Blue Gene)

  20. Mini-Literature Comparison • Similar to D. Blackmore in his sweeps also entail differential topology concepts • Different from H. Edelsbrunner emphasis on PL-approximations from Alpha-shapes, even with invocation of Morse theory. • Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Denison. • Digital topology, domain theory • Generalizations, unifications?

  21. Credits • Color image: UMass, Amherst, RasMol, web • Molecular Cartoons: T. Schlick, survey article, Modeling Superhelical DNA …, C. Opinion Struct. Biol., 1995

  22. INTERSECTIONS -- TOPOLOGY, ACCURACY, & NUMERICS FOR GEOMETRIC OBJECTS I-TANGO III NSF/DARPA

  23. 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.

  24. Scientific Collaborators • I-TANGO: D. R. Ferguson (Boeing), C. M. Hoffmann (Purdue), T. Maekawa (MIT), N. M. Patrikalakis (MIT), N. F. Stewart (U Montreal), T. Sakalis (Agr. U. Athens). • Surface Approximation: K. Abe, A. Russell, E. L. F. Moore, J. Bisceglio, C. Mow

More Related