T. J. Peters, University of Connecticut

www.cse.uconn.edu/~tpeters

with I-TANGO Team, ++

Computational Topology for

Animation and Simulation

Outline: Animation & Approximation

• Animations for 3D

• Algorithms

• Applications

Animation for Understanding

• ROTATING IMMORTALITY – www.bangor.ac.uk/cpm/sculmath/movimm.htm – Möbius Band in the form of a Trefoil Knot

• Animation makes 3D more obvious

Unknot

BadApproximation

Why?

Curvature?

Separation?

Why Bad?

No Intersections!

Changes Knot Type

Now has 4Crossings

Good Approximation

All Vertices on Curve

Respects Embedding

Via

Curvature (local)

Separation (global)

Summary – Key Ideas

• Curves– Don’t be deceived by images (3D !)– Crossings versus self-intersections

• Local and global arguments

• Knot equivalence via isotopy

KnotPlot !

Initial Assumptionson a 2-manifold, M

• Without boundary

• 2nd derivatives are continuous (curvature)

T

Proof: Similar to flow on normal field.

Comment: Points need not be on surface. (noise!)

Theorem: Any approximation of F in T such that each normal hits one point of

W is ambient isotopic to F.

Tubular Neighborhoods

• Its radius defined by ½ minimum

– all radii of curvature on 2-manifold

– global separation distance.

• Estimates, but more stable than medial axis.

and Ambient Isotopy

Medial Axis

• H. Blum, biology, classification by skeleton

• Closure of the set of points that have at least 2 nearest neighbors on M

X

Opportunities

• Bounds for animation & simulation

– Surfaces move

– Boundaries move

• Functions to represent movement

Seminal Paper, Modified Claim

Surface reconstruction from unorganized points, H. Hoppe, T. DeRose, et al., 26 (2), Siggraph, `92

The output of our reconstruction method produced the correct topology in all the examples.

We are trying to develop formal guaranteeson the correctness of the reconstruction, given constraints on the sample and the original surface

KnotPlot !

Perko Pair & Dynamic Drug Docking

Animation & Simulation

• Successive Frames

• O(N^2) run time

• risk of error versus step size

• Isotopy

• O(N^2) off-line

• simple bound comparison at run time

• formal correctness

• IBM Award Nomination (Blue Gene)

Mini-Literature Comparison• Similar to D. Blackmore in his sweeps also

entail differential topology concepts

• Different from H. Edelsbrunner emphasis on PL-approximations from Alpha-shapes, even with invocation of Morse theory.

• Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Denison.– Digital topology, domain theory– Generalizations, unifications?

Credits

• Color image: UMass, Amherst, RasMol, web

• Molecular Cartoons: T. Schlick, survey article, Modeling Superhelical DNA …, C. Opinion Struct. Biol., 1995  

INTERSECTIONS -- TOPOLOGY,

ACCURACY, &

NUMERICS FOR

GEOMETRIC

OBJECTS

I-TANGO III

NSF/DARPA

Acknowledgements, NSF

• I-TANGO: Intersections --- Topology, Accuracy and Numerics for Geometric Objects (in Computer Aided Design), May 1, 2002, #DMS-0138098.

• SGER: Computational Topology for Surface Reconstruction, NSF, October 1, 2002, #CCR - 0226504.

• Computational Topology for Surface Approximation, September 15, 2004,

#FMM -0429477.

Scientific Collaborators

• I-TANGO: D. R. Ferguson (Boeing),

C. M. Hoffmann (Purdue),

T. Maekawa (MIT),

N. M. Patrikalakis (MIT),

N. F. Stewart (U Montreal),

T. Sakalis (Agr. U. Athens).

• Surface Approximation: K. Abe, A. Russell,

E. L. F. Moore,

J. Bisceglio, C. Mow