Computational Topology Counterexamples with 3D Visualization of Bézier Curves


  • Ji Li University of Connecticut
  • T. J. Peters University of Connecticut
  • D. Marsh Pratt and Whitney
  • K. E. Jordan IBM T.J. Watson Research ; Cambridge Research Center



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.


Download data is not yet available.

Author Biographies

Ji Li, University of Connecticut

Department of Mathematics

T. J. Peters, University of Connecticut

Department of Computer Science and Engineering


C .C. Adams, The Knot Book: An Elementary Introduction To The Mathematical Theory Of Knots, American Mathematical Society, 2004.

J. W. Alexander and G. B. Briggs, On types of knotted curves Annals of Mathematics 28 (1926-1927), 562-586.

L. E. Andersson, T. J. Peters and N. F. Stewart, Selfintersection of composite curves and surfaces, CAGD 15 (1998), 507-527.

M. A. Armstrong, Basic Topology, Springer, New York, 1983.

R. H. Bing, The Geometric Topology of 3-Manifolds, American Mathematical Society, Providence, RI, 1983.

J. Bisceglio, T. J. Peters, J. A. Roulier and C. H. Sequin, Unknots with highly knotted control polygons, CAGD 28, no. 3 (2011), 212-214.

G. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, San Diego, CA, 1990.

J. Hass, J. C. Lagarias and N. Pippenger, The computational complexity of knot and link problems, Journal of the ACM, 46, no, 2 (1999), 185-221.

K. E. Jordan, R. M. Kirby, C. Silva and T. J. Peters, Through a new looking glass*: Mathematically precise visualization, SIAM News 43, no. 5 (2010), 1-3.

K. E. Jordan, L. E. Miller, E. L. F. Moore, T. J. Peters and A. C. Russell, Modeling time and topology for animation and visualization with examples on parametric geometry, Theoretical Computer Science 405 (2008), 41-49.

K. E. Jordan, L. E. Miller, T. J. Peters and A. C. Russell, Geometric topology and visualizing 1-manifolds, In V. Pascucci, X. Tricoche, H. Hagen, and J. Tierny, editors, Topology-based Methods in Visualization, pages 1 - 12, New York, 2011. Springer.

C. Livingston, Knot Theory, volume 24 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1993.

T. J. Peters and D. Marsh, Personal home page of T. J. Peters.

L. Piegl and W. Tiller, The NURBS Book, Springer, New York, 2nd edition, 1997. R. Scharein, The knotplot site,

C. H. Sequin, Spline knots and their control polygons with differing knottedness,


How to Cite

J. Li, T. J. Peters, D. Marsh, and K. E. Jordan, “Computational Topology Counterexamples with 3D Visualization of Bézier Curves”, Appl. Gen. Topol., vol. 13, no. 2, pp. 115–134, Oct. 2012.