1. T. J. Peters Kerner Graphics Topologically Encoded Animation (TEA): History & Future

3. KnotPlot: www.knotplot.comwww.knotplot.com Unknot or Trefoil? Demo A: Unknown1 & Unknown2

4. Contemporary Computational Influences Edelsbrunner: geometry & topology Sethian: Marching methods, topology changes Blackmore: differential sweeps Carlsson, Zomordian : Algebraic

5. Route to KG May discussion with Norm. NSF SBIR grant for TEA technology.

6. Little reuse or modification “Plus, we love to blow things up.” Digital Visual Effects (DVFX)

7. Challenges --- (Audacious?) Another: Inner Life of a Cell – XVIVO for Harvard

8. TEA: dimension-independent technology Provably correct temporal antialiasing Portability of animation to differing displays Efficient compression and decompression

9. Mappings and Equivalences Knots and self-intersections Piecewise Linear (PL) Approximation My Scientific Emphasis

10. Temporal Aliasing

12. Nbhd_1 about curve.

13. 1.682 Megs

16. Moore Dissertation 2006 Efficient algorithm for ambient isotopic PL approximation for Bezier curves of degree 3.

17. PL Approximation for Graphics – Animation & Visualization

18. Unknot

19. Bad Approximation! Self-intersect?

20. Good Approximation! Respects Embedding: Curvature (local) & Separation (global) Error bounds!! => Nbhd_2 about curve. But recognizing unknot in NP (Hass, L, P, 1998)!!

21. Proving 1 – 1 is central. If c is a non-self-intersecting curve and F is a homotopy of c such that each homotopic image of c is non-self-intersecting, then F is an ambient isotopy. Role of Homotopy No longer have error bounds. CONTACTCONTACT

22. Temporal Antialiasing Comparison Time to market. Produce traditionally. Produce with TEA technology.

23. Portability for Display Ipod to Big Screen by parameters. 3D TV. (Prototype shown today.)

24. Compression: TEA File (<1KB vs 1.7 Megs) Bezier degree = 3, with Control points 0.0 0.0 0.0 4.293 4.441 0.0 8.777 5.123 1.234 12.5 0.0 0.0 Perturbation vectors; constraint on each vector 1 24.1 0.0 0.0 ; 26.4 1 -12.5 0.0 5.0 ; 18.1 2 -2.1 -2.4 -3.1 ; 9.0 1 -11.6 0.0 -1.9 ; 14.0

25. Compression vs Decompression Compression, Phase I. Decompression, Phase II.

26. UMass, RasMol

27. Conclusions Time can be modeled continuously while frames remain discrete. Difference between –Perturb then approximate versus –Approximate then perturb.

28. Quotes & Interpretation “You can’t rush art.”, Woody, Toy Story 2 “Time is money”. Correct math for the most money.

29. Overview References Modeling Time and Topology for Animation and Visualization, [JMMPR], pre-print Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Special Issue of Applied General Topology, 2007 Open Problems in Topology II, 2007 NSF, Emerging Trends in Computational Topology, 1999, xxx.lanl.gov/abs/cs/9909001

30. Acknowledgements: NSF SBIR: TEA, IIP -0810023. SGER: Computational Topology for Surface Reconstruction, CCR - 0226504. Computational Topology for Surface Approximation, FMM - 0429477. Investigator’s responsibility, not NSF.

31. Acknowledgements: Images http://se.inf.ethz.ch/people/leitner/erl\_g/ www.bangor.ac.uk/cpm/sculmath/movimm.htm www.knotplot.com blog.liverpoolmuseums.org.uk/graphics/lottie_sleigh.jpg www.channel4.com/film/media/images/Channel4/film/B/be owulf_xl_01--film-A.jpg www.turbosquid.com