3D Graphics Projected onto 2D (Don’t be Fooled!!!!)

1. 3D Graphics Projected onto 2D(Don’t be Fooled!!!!) T. J. Peters, University of Connecticut www.cse.uconn.edu/~tpeters

2. Outline: Animation & Approximation • Animation for 3D • Approximation of 1-manifolds • Transition to molecules • Molecular dynamics and knots • Extensions to 2-manifolds • Supportive theorems • Spline intersection approximation (static)

4. Role for Animation Towards Mathematical Discovery • ROTATING IMMORTALITY • www.bangor.ac.uk/cpm/sculmath/movimm.htm • Möbius Band in the form of a Trefoil Knot • Animation makes 3D more obvious • Simple surface here • Spline surfaces joined along boundaries

5. Unknot

6. Bad Approximation Why?

7. Bad Approximation Why? Self-intersections?

8. Bad Approximation All Vertices on Curve Crossings only!

9. Why Bad? Changes Knot Type Now has 4 Crossings

10. Good Approximation All Vertices on Curve Respects Embedding

11. Good Approximation Still Unknot Closer in Curvature (local property) Respects Separation (global property)

12. Summary – Key Ideas • Curves • Don’t be deceived by images • Still inherently 3D • Crossings versus self-intersections • Local and global arguments • Applications to vizulization of molecules • Extensions to surfaces

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

19. Limitations • Tube of constant circular cross-section • Admitted closed-form engineering solution • More realistic, dynamic shape needed • Modest number of base pairs (compute bound) • Not just data-intensive snap-shots

20. Transition to Dynamics • Energy role • Embeddings • Knots encompass both

21. Interest in Tool Similar to KnotPlot • Dynamic display of knots • Energy constraints incorporated for isotopy • Expand into molecular modeling • www.cs.ubc.ca/nest/imager/contributions/scharein/

22. Topological Equivalence: Isotopy (Bounding 2-Manifold) • Need to preserve embedding • Need PL approximations for animations • Theorems for curves & surfaces

23. Opportunities • Join splines, but with care along boundaries • Establish numerical upper bounds • Maintain bounds during animation • Surfaces move • Boundaries move • Maintain bounds during simulation (FEA) • Functions to represent movement • More base pairs via higher order model

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

26. Representation: Geometric Data • Two trimmed patches. • The data is inconsistent, and inconsistent with the associated topological data. • The first requirement is to specify the set defined by these inconsistent data.

27. Rigorous Error Bounds • I-TANGO • Existing GK interface in parametric domain • Taylor’s theorem for theory • New model space error bound prototype • CAGD paper • Transfer to Boeing through GEML

28. Topology • Computational Topology for Regular Closed Sets (within the I-TANGO Project) • Invited article, Topology Atlas • Entire team authors (including student) • I-TANGO interest from theory community

29. 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?

30. 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.