Computational topology a personal overview
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Computational Topology : A Personal Overview. T. J. Peters www.cse.uconn.edu/~tpeters. My Topological Emphasis:. General Topology (Point-Set Topology) Mappings and Equivalences. Vertex, Edge, Face: Connectivity. Euler Operations.

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Computational Topology : A Personal Overview

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Computational Topology :A Personal Overview

T. J. Peters

www.cse.uconn.edu/~tpeters


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


Bad

Approximation!

Self-intersect?


Why Bad?

No

Intersections!

Changes

Knot Type

Now has 4

Crossings


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


Theorem: If an approximation of F has a unique intersection with each normalof F, then it is ambient isotopic to F.

Proof:

1. Local argument with curvature.

2. Global argument for separation.

(Similar to flow on normal field.)


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.


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