Topology of Geometry in Motion (Blowing Things Up)

Topology of Geometry in Motion (Blowing Things Up) T. J. Peters University of Connecticut Kerner Graphics

Temporal Aliasing

Unknot

Bad Approximation! Self-intersect?

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

Proof Outline [Maekawa, Patrikalakis, Sakkalis, Yu, 1998] Nr(B) B lk(j) = consecutive line segments k(1) lk(1) k(2) k(0) lk(2) lk(0) Dk(1) Dk(2) Dk(0)

Nr(B) B r Proof Outline • Subdivide B until the entire control polygon is: 1) non-self-intersecting ([Neagu, Calcoen, Lacolle, 2000]) 2) contained in Nr(B)([MPSY, 1998]).

Minimal Number for Nontrivial Stick Knot Any nontrivial stick knot must have at least six sticks. C. Adam, Williams College

Nr(B) B curve degree = 3 control points = [0, 0, 0] [2.0, 0, 0] [0, 0.5, 0] [5, 1.25, 0] knot vector = { 0, 0, 0, 0, 1, 1, 1, 1 } curve degree = 3 control points = [0, 0, 0] [-2.0, 0, 0] [0, 0.25, 0] [-5, 1.25, 0] knot vector = { 0, 0, 0, 0, 1, 1, 1, 1 }

Technical Lemma Let w be a point where a regular cubic Be´zier curve c is subdivided. Let P be the plane normal to c at w. There exists a subdivision of c such that the control polygon of the single segment Be´zier sub-curve ending at w and the control polygon of the single segment Be´zier sub-curve beginning at w intersect P only in the single point w. DeRose result, planar & convex. (Asked G. Farin.) Computing article, from Dag05,with Moore & Roulier.

Moore Dissertation 2006 Efficient algorithm for ambient isotopic PL approximation for Bezier curves of degree 3.Degree 4? (locally unknotted.)

1.682 Megs 1.682 Megs

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

Route to KG May discussion with N. Noble. NSF SBIR grant for TEA technology.

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

Challenges --- (Audacious?)

TEA: dimension-independent technology • Provably correct temporal antialiasing • Portability to differing displays (3D TV) • Efficient compression and decompression

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

Richer Approximations Manifolds with continuous 2nd derivatives. Bound distance & curvature. Isotopy along arclength?

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

Modeling Time and Topology for Animation and Visualization, [JMMPR], TCS0? • 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 Overview References

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.

Acknowledgements: Images • http://se.inf.ethz.ch/people/leitner/erl\_g/ • blog.liverpoolmuseums.org.uk/graphics/lottie_sleigh.jpg • www.channel4.com/film/media/images/Channel4/film/B/beowulf_xl_01--film-A.jpg

Topology of Geometry in Motion (Blowing Things Up)