T. J. Peters, University of Connecticut with I-TANGO Team, ++ Computational Topology for Animation and Simulation

Outline: Animation & Approximation Animations for 3D Algorithms Applications

Animation for Understanding ROTATING IMMORTALITY – –Möbius Band in the form of a Trefoil Knot Animation makes 3D more obvious

Unknot

Bad Approximation Why? Curvature? Separation?

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

Good Approximation All Vertices on Curve Respects Embedding 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

KnotPlot !

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

Opportunities Bounds for animation & simulation –Surfaces move –Boundaries move Functions to represent movement

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

KnotPlot ! Perko Pair & Dynamic Drug Docking

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)

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?

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

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

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

Scientific Collaborators I-TANGO: D. R. Ferguson (Boeing),I-TANGO: 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,Surface Approximation: E. L. F. Moore, J. Bisceglio, C. Mow