3D Graphics Projected onto 2D (Don’t be Fooled!!!!)
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Transcript of 3D Graphics Projected onto 2D (Don’t be Fooled!!!!)
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T. J. Peters, University of Connecticut
www.cse.uconn.edu/~tpeters
3D Graphics Projected onto 2D(Don’t be Fooled!!!!)
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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)
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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
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Unknot
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BadApproximation
Why?
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BadApproximation
Why?
Self-intersections?
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BadApproximation
All Vertices onCurve
Crossings only!
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Why Bad?
Changes Knot Type
Now has 4Crossings
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GoodApproximation
All Vertices onCurve
RespectsEmbedding
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Good Approximation
Still Unknot
Closer in Curvature(local property)
RespectsSeparation(global property)
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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
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Credits
• Color image: UMass, Amherst, RasMol, web
• Molecular Cartoons: T. Schlick, survey article, Modeling Superhelical DNA …, C. Opinion Struct. Biol., 1995
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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
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Transition to Dynamics
•Energy role
•Embeddings
•Knots encompass both
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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/
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Topological Equivalence: Isotopy
• Need to preserve embedding
• Need PL approximations for animations
• Theorems for curves & surfaces
(Bounding 2-Manifold)
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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
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INTERSECTIONS -- TOPOLOGY,
ACCURACY, &
NUMERICS FOR
GEOMETRIC
OBJECTS
I-TANGO III
NSF/DARPA
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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.
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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
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• 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
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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?
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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.