Computational Topology Counterexamples with 3D Visualization of Bezier Curves

TL;DR

This paper presents counterexamples in computational topology where Bezier curves and their linear control polygons differ in embedding properties, using visualization software and algorithms to generate and analyze such cases.

Contribution

It introduces new counterexamples showing discrepancies between control polygons and Bezier curves in 3D, with methods for generating more examples.

Findings

Counterexamples of unknotted control polygons with knotted Bezier curves

Counterexamples of simple control polygons with self-intersecting Bezier curves

Algorithms for creating diverse topological discrepancies in 3D curves

Abstract

For applications in computing, Bezier 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 visualizing software and numerical algorithms that produce general procedures to create more examples.