Exact Computation for Existence of a Knot Counterexample

TL;DR

This paper provides a formal existence proof for a self-intersecting Bezier curve with an unknot control polygon, advancing the understanding of geometric properties in scientific visualization.

Contribution

It introduces an exact computation method that formally proves the existence of a self-intersecting Bezier curve with an unknot control polygon, filling a gap left by previous numerical evidence.

Findings

Established a formal existence proof for the self-intersecting curve

Demonstrated the limitations of numerical methods in proving geometric properties

Contributed to the theoretical understanding of Bezier curve intersections

Abstract

Previously, numerical evidence was presented of a self-intersecting Bezier curve having the unknot for its control polygon. This numerical demonstration resolved open questions in scientific visualization, but did not provide a formal proof of self-intersection. An example with a formal existence proof is given, even while the exact self-intersection point remains undetermined.