T. J. Peters, University of Connecticut

www.cse.uconn.edu/~tpeters

3D Graphics Projected onto 2D(Don’t be Fooled!!!!)

Outline: Animation & Approximation

• Animation for 3D

• Approximation of 1-manifolds

• Transition to molecules

• Molecular dynamics and knots

• Extensions to 2-manifolds

• Supportive theorems

• Spline intersection approximation (static)

Role for Animation Towards

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

• Animation makes 3D more obvious

• Simple surface here

• Spline surfaces joined along boundaries

Mathematical Discovery

Unknot

BadApproximation

Why?

BadApproximation

Why?

Self-intersections?

BadApproximation

All Vertices onCurve

Crossings only!

Why Bad?

Changes Knot Type

Now has 4Crossings

GoodApproximation

All Vertices onCurve

RespectsEmbedding

Good Approximation

Still Unknot

Closer in Curvature(local property)

RespectsSeparation(global property)

Summary – Key Ideas

• Curves– Don’t be deceived by images– Still inherently 3D– Crossings versus self-intersections

• Local and global arguments

• Applications to vizulization of molecules

• Extensions to surfaces

Credits

• Color image: UMass, Amherst, RasMol, web

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

Limitations

• Tube of constant circular cross-section

• Admitted closed-form engineering solution

• More realistic, dynamic shape needed

• Modest number of base pairs (compute bound)

• Not just data-intensive snap-shots

Transition to Dynamics

•Energy role

•Embeddings

•Knots encompass both

Interest in Tool Similar to KnotPlot

• Dynamic display of knots

• Energy constraints incorporated for isotopy

• Expand into molecular modeling

• www.cs.ubc.ca/nest/imager/contributions/scharein/

Topological Equivalence: Isotopy

• Need to preserve embedding

• Need PL approximations for animations

• Theorems for curves & surfaces

(Bounding 2-Manifold)

Opportunities

• Join splines, but with care along boundaries

• Establish numerical upper bounds

• Maintain bounds during animation– Surfaces move– Boundaries move

• Maintain bounds during simulation (FEA)

• Functions to represent movement

• More base pairs via higher order model

INTERSECTIONS -- TOPOLOGY,

ACCURACY, &

NUMERICS FOR

GEOMETRIC

OBJECTS

I-TANGO III

NSF/DARPA

Representation: Geometric Data

• Two trimmed patches.• The data is

inconsistent, and inconsistent with the associated topological data.

• The first requirement is to specify the set defined by these inconsistent data.

Rigorous Error Bounds

• I-TANGO – Existing GK interface in parametric domain– Taylor’s theorem for theory – New model space error bound prototype

• CAGD paper

• Transfer to Boeing through GEML

• Computational Topology for Regular Closed Sets (within the I-TANGO Project)

– Invited article, Topology Atlas

– Entire team authors (including student)

– I-TANGO interest from theory community

Topology

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?

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.