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

Computational Topology :A Personal Overview

T. J. Peters

www.cse.uconn.edu/~tpeters


My topological emphasis

My Topological Emphasis:

General Topology (Point-Set Topology)

Mappings and Equivalences


Computational topology a personal overview

Vertex, Edge, Face:

Connectivity

Euler Operations

Thesis: M. Mantyla; “Computational Topology …”, 1983.


Contemporary influences

Contemporary Influences

  • Grimm: Manifolds, charts, blending functions

  • Blackmore: differential sweeps

  • Kopperman, Herman: Digital topology

  • Edelsbrunner, Zomordian, Carlsson : Algebraic


Knotplot

KnotPlot !


Comparing knots

Comparing Knots

  • Reduced two to simplest forms

  • Need for equivalence

  • Approximation as operation in geometric design


Computational topology a personal overview

Unknot


Computational topology a personal overview

Bad

Approximation!

Self-intersect?


Computational topology a personal overview

Why Bad?

No

Intersections!

Changes

Knot Type

Now has 4

Crossings


Computational topology a personal overview

Good

Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

But recognizing unknot in NP (Hass, L, P, 1998)!!


Nsf workshop 1999 for design

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

Summary – Key Ideas

  • Space Curves: intersection versus crossing

  • Local and global arguments

  • Knot equivalence via isotopy

  • Extensions to surfaces


Computational topology a personal overview

UMass, RasMol


Computational topology a personal overview

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.)


Computational topology a personal overview

Good

Approximation!

Respects Embedding

Via

Curvature (local)

Separation (global)

But recognizing unknot in NP (Hass, L, P, 1998)!!


Computational topology a personal overview

Global separation


Mathematical generalizations

Mathematical Generalizations

  • Equivalence classes:

    • Knot theory: isotopies & knots

    • General topology: homeomorphisms & spaces

    • Algebra: homorphisms & groups

  • Manifolds (without boundary or with boundary)


Overview references

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

Credits

  • ROTATING IMMORTALITY

    • www.bangor.ac.uk/cpm/sculmath/movimm.htm

  • KnotPlot

    • www.knotplot.com


Credits1

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

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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