Unknots with highly knotted control polygons

Authors:
J. Bisceglio;T. J. Peters;J. A. Roulier;C. H. Séquin
Affiliations:
BlueSky Studios, Greenwich, CT, United States;Department of Computer Science and Engineering, University of Connecticut, Storrs, CT, United States and Kerner Graphics, Inc., 90 Windward Way, San Rafael, CA, United States;Department of Computer Science and Engineering, University of Connecticut, Storrs, CT, United States;Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA, United States
Venue:
Computer Aided Geometric Design
Year:
2011

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Abstract

An example is presented of a cubic Bezier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bezier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its Bezier curve.