1. CSE 275 F04—Graphics with OpenGL Dr. T. J. Peters, tpeters@cse.uconn.edu 486-5045 www.cse.uconn.edu/~tpeters Use of plain text files for email No attachments Dynamic syllabus on home

2. CSE 275 F04—Graphics with OpenGL Circle animation, due next week (5 pts) 3 – 4 take home labs, (60 pts) 2 tests, 9/30 & 11/04 (20 pts) Final, (15 pts) Alternate suggestions by Thurs, 9/2!!!!

3. Computational Topology and Spline Surfaces T. J. Peters, University of Connecticut Thanks: I-TANGO Team

4. Outline: Animation & Approximation Animation for 3D Spline intersection approximation (static) Transition to molecules Molecular dynamics and knots Supportive theorems

6. Role for Animation Towards ROTATING IMMORTALITY –www.bangor.ac.uk/cpm/sculmath/movimm.htmwww.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 Mathematical Discovery

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

9. 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)

10. 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.

11. 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

12. 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

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. 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

21. Transition to Dynamics Energy role Embeddings Knots encompass both

22. 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/

23. Topological Equivalence: Ambient Isotopy Need to preserve embedding –Unknot versus trefoil –Homeomorphism not enough Need PL approximations for animations Bounded perturbations to preserve topology Theorems for curves & surfaces in terms of control points

24. Finitely Many Control Points