1. T. J. Peters University of Connecticut, Professor TEA, Knots & Molecules in Animation, Simulation & Visualization

2. T. J. Peters Topologically Encoded Animation (TEA)

4. Trefoil Knot 3D Rotation Encode: Rot_0, Rot_1, …, Rot_n

5. More Aggressive Moves Not just rigid body motion Deform shape Preserve crucial characteristics

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

7. 1.682 Megs

8. Many Frames Not just rigid body motion Deform shape Preserve crucial characteristics Role of 3D and projection

9. Homeomorphism is not enough F : X  Y, such that F is 1.continuous, 2.1 – 1 3.onto 4.and has a continuous inverse.

10. Temporal Aliasing

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

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

13. Isotopy & Animation F : X x [0,1]  Y, such that for each t in [0,1] F : X x t is a homeomorphism. We take Y to be 3D space.

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

16. DVFX vs `Blowing things up’ Modify & re-use vs destroy. But explosions are hard, for now. Provide path for integration.

17. See EagleEye

18. 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

19. Comparison XC, RFR, EC, JD 07 Singularity Solver [GE+97] Multiple objects KG folk 09 Critical points (C ) Newton, PGPU? Self-intersection 2

20. TEA Authoring Tools for DVFX Time-checker like spell-checker –runs in background; not intrusive! –very expensive if missed. Parametric re-design; similar to CAGD PTC Integrate with VFX.

22. Time and Topology Protein folding Data Volume Visualize in real time ! Geometry Slow with errors Topology Fast & correct – but scale? Versus -------- --------- K. E. Jordan (IBM), L. E. Miller (UConn), E.L.F. Moore (UConn), T. J. Peters (UConn), A. C. Russell (UConn)

23. Conclusion Time can be modeled continuously while frames remain discrete.

24. Similarity? The Need for Verifiable Visualization –Kirby and Silva, IEEE CG&A, 08 –What confidence (or error measures) can be assigned to a computer-based prediction of a complex event? –CFD: colorful faulty dynamics “First, do no harm” “Primarily, don’t introduce artifacts.”

25. Acknowledgements: NSF SBIR: TEA, IIP -0810023. SGER: Computational Topology for Surface Reconstruction, CCR - 0226504. Computational Topology for Surface Approximation, FMM - 0429477. IBM Faculty & Doctoral Awards Investigator’s responsibility, not sponsor’s.

26. Acknowledgements: Images http://se.inf.ethz.ch/people/leitner/erl_g www.knotplot.com http://domino.research.ibm.com/comm/pr.nsf/pages/rscd.bluegene-picaa.html www.bangor.ac.uk/cpm/sculmath/movimm.htm blog.liverpoolmuseums.org.uk/graphics/lottie_sleigh.jpg