1. T. J. Peters 2005 IBM Faculty Award www.cse.uconn.edu/~tpeters with E. L. F. Moore & J. Bisceglio Computational Topology for Scientific Visualization and Integration with Blue Gene L

3. Rotate Molecule?

4. UMass, RasMol

5. Molecular Modeling? Using Surfaces!

6. Joining Geometry

7. Static Images “A picture is worth a thousand words.” BUT, (http://commfaculty.fullerton.edu/lester/writings/ad.html) Animation is much more expensive

8. Dynamic Scientific Visualization Approximately 11M translations per hour: 100 translations per frame, at 30 frames per second (A Conservative Lower Bound)

9. Geri’s game: along boundary joins. Resolution was data-specific. Short time span was favorable DeRose, Kass and Truong, Subdivision surfaces in character animation, SIGGRAPH '98 Documented Animation Issues

10. Accumulated error versus Maya alternative. Used at BlueSky Studios (Ice Age II) Practical Animation Response

11. Mathematics for perturbing curves. Generalize to surfaces. Pragmatic Research Response

12. Approximation & Knots Approximate & compare knot types: But recognizing unknot in NP (Hass, L, P, 1998)!! Approximation as operation in geometric design Preserve original knot type (even if unknown).

13. Unknot

14. Bad Approximation! Self-intersect?

15. Good Approximation! Respects Embedding Via Curvature (local) Separation (global) (recognizing unknot in NP; Hass, L, P, 1998)

16. * Interpolation points* N r (B) B ➢ Construct the boundary of an open neighborhood N r (B) of curve B ➢ The boundary (a pipe surface) will have a radius r, with the following conditions* ➢ no local self-intersections ➢ no global self-intersections

17. Applications !

18. Subdivision for graphics Integration with sub-systems. Generation of vertices. Performance benefits. Motion driven by chemistry and physics.

19. P8P8 P7P7 P6P6 P5P5 P4P4 P3P3 P2P2 P1P1 P 10 P0P0 P9P9 ➢ Planar Degree 10 Bézier Curve ➢ Note: the control polygon is self-intersecting The Class of Unknotted Spline Curves with Knotted Control Polygons

20. Knot Projection Folk Lemma If a projection of a curve is non-self-intersecting, then the curve is unknotted.

21. Spline Projection Done by projection of control points.

22. ➢ 3D Degree 10 Bézier Curve ➢ Note: the control polygon is knotted The Class of Unknotted Spline Curves with Knotted Control Polygons P0P0 P 10 P9P9 P8P8 P7P7 P6P6 P5P5 P3P3 P2P2 P1P1

23. Algorithm for Isotopic Subdivision (cubic) Subdividing B until its control polygon is contained in Nr(B). a. Compute number of subdivisions required* b. Test to ensure there are no self-intersections N r (B) B PkPk P k+1 P k+2 q k,i lklk l k+1 l k+3 P k+2 l k+2 q k,f * Cubic: no local knotting

24. 2r Algorithm for Isotopic Subdivision 1. Computing r for B Find minimum of a. separation distance [c(s) – c(t)] c'(s) = 0 [c(s) – c(t)] c'(t) = 0 b. radius of curvature Cubic b-spline curve

25. Min distance with Newton's method

28. KnotPlot !

29. Crucial Difference Known Dynamics Versus Real-time Response (molecular simulation) (surgery)

30. Additional High Performance Issues Over 100,000 processors, with local geometry. Join across all nodes (surfaces & curves). Output to light-weight graphics clients raises bandwidth & architectural concerns. Example: Blue Gene L, Macro-Molecule Andersson-Peters-Stewart, IJCGA 00 & CAGD 98

31. Terabytes of point data. Triangulation too data intensive. Reduce by orders of magnitudes. Spline approximation, with acceptable loss. Example:Seismic Data, P. Bording, MUN, IBM Faculty Award

34. Only synthetic data. Order of magnitude reduction. Small loss. Awaiting test data. Status

35. Local constraints. Mathematically & algorithmically possible. Need domain-specific information. Options

36. Integrate Surface Approximation Provable Topological Dynamic Constraints Apply to real-time, computer-assisted cardiac surgery. Goals

37. Credits ROTATING IMMORTALITY –www.bangor.ac.uk/cpm/sculmath/movimm.htmwww.bangor.ac.uk/cpm/sculmath/movimm.htm KnotPlot –www.cs.ubc.ca/nest/imager/ contributions/scharein/KnotPlot.html

38. Acknowledgements, NSF I-TANGO,May 1, 2002, #DMS-0138098. SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR - 0226504.SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR - 0226504. Computational Topology for Surface Approximation, September 15, 2004, #FMM -0429477.Computational Topology for Surface Approximation, September 15, 2004, #FMM -0429477. IBM Faculty Award, 2005