Computational Topology Counterexamples with 3D Visualization of Bézier Curves

Ji Li

United States

University of Connecticut

Department of Mathematics

T. J. Peters

United States

University of Connecticut

Department of Computer Science and Engineering

D. Marsh

United States

Pratt and Whitney

K. E. Jordan

United States

IBM T.J. Watson Research ; Cambridge Research Center

Submitted: 2013-07-26

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Accepted: 2013-07-26

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DOI: https://doi.org/10.4995/agt.2012.1624
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Supporting agencies:

NSF grants CCF 0429477

CMMI 1053077 and CNS 0923158

IBM Faculty Award and IBM Doctoral Fellowships

Abstract:

For applications in computing, Bézier curves are pervasive and are defined by a piecewise linear curve L which is embedded in R3 and yields a smooth polynomial curve C embedded in R3. It is of interest to understand when L and C have the same embeddings. One class ofc ounterexamples is shown for L being unknotted, while C is knotted. Another class of counterexamples is created where L is equilateral and simple, while C is self-intersecting. These counterexamples were discovered using curve visualizing software and numerical algorithms that produce general procedures to create more examples.
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