T. J. Peters, University of Connecticut 3D Graphics Projected onto 2D (Don’t be Fooled!!!!)

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)

Role for Animation Towards ROTATING IMMORTALITY – –Möbius Band in the form of a Trefoil Knot Animation makes 3D more obvious Simple surface here Spline surfaces joined along boundaries Mathematical Discovery

Unknot

Bad Approximation Why?

Bad Approximation Why? Self-intersections?

Bad Approximation All Vertices on Curve Crossings only!

Why Bad? Changes Knot Type Now has 4 Crossings

Good Approximation All Vertices on Curve Respects Embedding

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

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

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

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

Transition to Dynamics Energy role Embeddings Knots encompass both

Interest in Tool Similar to KnotPlot Dynamic display of knots Energy constraints incorporated for isotopy Expand into molecular modeling

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

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

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

Intellectual Integration of Project Team New conceptual model (Stewart - UConn) Intersection improvements (Sakkalis – MIT) Polynomial evaluation (Hoffmann – Purdue) Industrial view (Ferguson – DRF Associates) Key external interactions (Peters, UConn)

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.

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

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 Topology

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?

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