t. j. peters tpeters computational topology : a personal overview
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T. J. Peters
www.cse.uconn.edu/~tpeters
Computational Topology :A Personal Overview
My Topological Emphasis:
General Topology (Point-Set Topology)
Mappings and Equivalences
Vertex, Edge, Face: Connectivity
Euler Operations
Thesis: M. Mantyla; “Computational Topology …”, 1983.
Contemporary Influences
• Grimm: Manifolds, charts, blending functions
• Blackmore: differential sweeps
• Kopperman, Herman: Digital topology
• Edelsbrunner, Zomordian, Carlsson : Algebraic
KnotPlot !
Comparing Knots
• Reduced two to simplest forms
• Need for equivalence
• Approximation as operation in geometric design
Unknot
BadApproximation!
Self-intersect?
Why Bad?
No Intersections!
Changes Knot Type
Now has 4Crossings
Good Approximation!
Respects Embedding
Via
Curvature (local)
Separation (global)
But recognizing unknot in NP (Hass, L, P, 1998)!!
NSF Workshop 1999 for Design
• Organized by D. R. Ferguson & R. Farouki
• SIAM News: Danger of self-intersections
• Crossings not detected by algorithms
• Would appear as intersections in projections
• Strong criterion for ‘lights-out’ manufacturing
Summary – Key Ideas
• Space Curves: intersection versus crossing
• Local and global arguments
• Knot equivalence via isotopy
• Extensions to surfaces
UMass, RasMol
Proof: 1. Local argument with curvature.
2. Global argument for separation.
(Similar to flow on normal field.)
Theorem: If an approximation of F has a unique intersection with each
normalof F, then it is ambient isotopic to F.
Good Approximation!
Respects Embedding
Via
Curvature (local)
Separation (global)
But recognizing unknot in NP (Hass, L, P, 1998)!!
Global separation
Mathematical Generalizations
• Equivalence classes: – Knot theory: isotopies & knots– General topology: homeomorphisms & spaces– Algebra: homorphisms & groups
• Manifolds (without boundary or with boundary)
Overview References
• Computation Topology Workshop, Summer Topology Conference, July 14, ‘05, Denison,
planning with Applied General Topology
• NSF, Emerging Trends in Computational Topology, 1999, xxx.lanl.gov/abs/cs/9909001
• Open Problems in Topology 2 (problems!!)
• I-TANGO,Regular Closed Sets (Top Atlas)
Credits• ROTATING IMMORTALITY
– www.bangor.ac.uk/cpm/sculmath/movimm.htm
• KnotPlot– www.knotplot.com
Credits• IBM Molecule
– http://domino.research.ibm.com/comm/pr.nsf/pages/rscd.bluegene-picaa.html
• Protein – Enzyme Complex– http://160.114.99.91/astrojan/protein/pictures/
parvalb.jpg
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.