T. J. Peters Kerner Graphics, Inc., CTO; University of Connecticut, Professor TEA, Knots & Molecules in Animation, Simulation & Visualization

T. J. Peters Kerner Graphics Topologically Encoded Animation (TEA)

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

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

KnotPlot: Unknot or Trefoil? Demo A: Unknown1 & Unknown2

1.682 Megs

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

Two Frames with Different Topology

Instantaneous Self-intersection

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

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

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.

Little reuse or modification “Plus, we love to blow things up.” Kerner Graphics: Digital Visual Effects (DVFX) KERNER OPTICAL

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

EagleEye

Moore Dissertation 2006 Efficient algorithm for ambient isotopic PL approximation for Bezier curves of degree 3. Now scale & accelerate.

PL Approximation for Graphics – Animation & Visualization (also for Engineeing Design)

Unknot

Bad Approximation! Self-intersect?

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

Compression: TEA File (<1KB vs 1.7 Megs) Bezier degree = 3, with Control points Perturbation vectors; constraint on each vector ; ; ; ; 14.0

Compression vs Decompression Compression, Phase I. Decompression, Phase II. Phase IB Project with Kerner Technologies??

Portability for Display Ipod to Big Screen by parameters. 3D TV. (Prototype in San Rafael.)

Dimension Independence Compute –Minimum separation distance. –Minimum radius of curvature. –Take minimum. Tubular neighborhood: –Constant radius = limit. –Adaptive options?

Stadium Curve Curvature & MSD

Tubular Neighborhood for Stadium Curve

Computing Curvature – calculus problem Minimum Separation Distance: –Candidate line segments. –Nearly normal at both ends. –Newton’s Method to converge.

Infinitely many good seeds

Symmetry & Performance Important for animation. Not used in initial test cases. Role for PGPU’s (updates!!) Pre-print 09 –

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

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.

Visualization for Simulations Animation `on-the-fly’. No human in the loop. Recall update issue (fast!!).

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)

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.”

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

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

Overview References Modeling Time and Topology for Animation and Visualization …., [JMMPR], TCS08 Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Special Issue of Applied General Topology, 2007 Open Problems in Topology II, 2007 [BP] NSF, Emerging Trends in Computational Topology, 1999, xxx.lanl.gov/abs/cs/

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

Acknowledgements: Images blog.liverpoolmuseums.org.uk/graphics/lottie_sleigh.jpg

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

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

Nbhd_1 about curve.