Computational topology counterexamples with 3D visualization of B’ezier curves. Applied General Topology

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 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 dis-covered 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