3D Graphics Projected onto 2D (Don

1
3D Graphics Projected onto 2D(Dont be
Fooled!!!!)
T. J. Peters, University of Connecticut www.cse.uc
onn.edu/tpeters
2
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)

3
(No Transcript)
4
Role for Animation Towards
Mathematical Discovery

• 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

5
Unknot
6
Bad Approximation Why?
7
Bad Approximation Why? Self-intersections?
8
Bad Approximation All Vertices on Curve
Crossings only!
9
Why Bad? Changes Knot Type Now has 4 Crossings
10
Good Approximation All Vertices
on Curve Respects Embedding
11
Good Approximation Still Unknot Closer in
Curvature (local property) Respects Separation (
global property)
12
Summary Key Ideas

• Curves
• Dont be deceived by images
• Still inherently 3D
• Crossings versus self-intersections
• Local and global arguments
• Applications to vizulization of molecules
• Extensions to surfaces

13
(No Transcript)
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17

18
Credits

• Color image UMass, Amherst, RasMol, web
• Molecular Cartoons T. Schlick, survey article,
  Modeling Superhelical DNA , C. Opinion Struct.
  Biol., 1995  

19
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

20

• Transition to Dynamics
• Energy role
• Embeddings
• Knots encompass both

21
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/

22
Topological Equivalence Isotopy
(Bounding 2-Manifold)

• Need to preserve embedding
• Need PL approximations for animations
• Theorems for curves surfaces

23
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

24
(No Transcript)
25

INTERSECTIONS -- TOPOLOGY, ACCURACY,
NUMERICS FOR GEOMETRIC OBJECTS
I-TANGO III
NSF/DARPA
26
Intellectual Integration of Project Team

• New conceptual model (Stewart - UConn)
• Intersection improvements (Sakkalis MIT)
• Polynomial evaluation (Hoffmann Purdue)
• Industrial view (Ferguson DRF Associates)
• Key external interactions (Peters, UConn)

27
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.

28
Rigorous Error Bounds

• I-TANGO
• Existing GK interface in parametric domain
• Taylors theorem for theory
• New model space error bound prototype
• CAGD paper
• Transfer to Boeing through GEML

29
Topology

• 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

30
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?

31
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.

32
(No Transcript)