Computational Topology Counterexamples with 3D Visualization of Bézier Curves

J. Li, T.J. Peters, D. Marsh, K.E. Jordan
2013 Applied General Topology  
For applications in computing, Bézier curves are pervasive and are defined by a piecewise linear curve L which is embedded in R 3 and yields a smooth polynomial curve C embedded in R 3 . It is of interest to understand when L and C have the same embeddings. One class of counterexamples 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
more » ... vered using curve visualizing software and numerical algorithms that produce general procedures to create more examples. (a) Unknotted L with knotted C (b) Zoomed-in view of C
doi:10.4995/agt.2012.1624 fatcat:7r7k227f4fbb5gcrzjct5nbh34