Topology Animation Science Technology T J Peters UConn

Topology & Animation : Science & Technology T. J. Peters, UConn Computer Science & Engineering Research & Education

http: //en. wikipedia. org/wiki/Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Not to be confused with topography.

Topology --- Mobius Strip

http: //en. wikipedia. org/wiki/Topography (from Greek τόπος topo-, "place", and γράφω graphia, "writing") is the study of Earth's surface shape and features or those of planets, moons, and asteroids. It is also the description of such surface shapes and features (especially their depiction in maps).

Topography – Contour lines hiking maps versus Google maps

Stowe Google Maps Different static viewing options

Topology --- Change

Topography – Contour lines Static

Knot. Plot: www. knotplot. com Unknot or Trefoil? Demo A: Unknown 1 & Unknown 2

Projection of Knot For a closed curve, c, if there exists some projection such that there are no self-intersections, then c is the unknot. Proof: ?

TEA & To. AST Knots & Molecules in Animation, Simulation & Visualization T. J. Peters, Kerner Graphics & UConn

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

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

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

1. 682 Megs

Homeomorphism is not enough F : X Y, such that F is 1. 2. 3. 4. continuous, 1– 1 onto 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

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

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 3 D space.

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

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

Eagle. Eye

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 (<1 KB vs 1. 7 Megs) Bezier degree = 3, with Control points 0. 0 4. 293 4. 441 0. 0 8. 777 5. 123 1. 234 12. 5 0. 0 Perturbation vectors; constraint on each vector 1 24. 1 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

Compression: TEA File (<1 KB vs 1. 7 Megs)

High Resolution

Low Resolution

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

Portability for Display • Ipod to Big Screen by parameters. • 3 D 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 – www. cse. uconn. edu/~tpeters

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

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

Artifacts? • 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. ”

Graphics Supercomputers? • PGPU: – Programmable Graphics Processing Unit – GPU • System level • Put pixels on screen – Generic programming interface • Nvidia – Quadro FX 5800 , 4 GB on board memory – Use of CUDA and possible interns

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], TCS 08 • 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/9909001

Acknowledgements: $$$ • SBIR: TEA, IIP -0810023. • SGER: Computational Topology for Surface Reconstruction, CCR - 0226504. • Computational Topology for Surface Approximation, FMM - 0429477. • IBM Faculty & Doctoral Awards • Nvidia: boards, more pending • UCRF: machines • Investigator’s responsibility, not sponsors’.

Acknowledgements: Images • http: //se. inf. ethz. ch/people/leitner/erl_g • http: //www. koshermealstogo. com/images/french-toast. jpg • 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

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.