Isotopic Equivalence from Bezier Curve Subdivision

We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035

Exact computation for existence of a knot counterexample

[EN] 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 scientic 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.The authors acknowledge, with appreciation, the contributions of •D. Marsh, for software that generated experimental visualizations and related computations, •the reviewers, for singularly comprehensive and constructive comments,and •the editors, for their keen insight and informed perspective in selecting those reviewers.Marinelli, K.; Peters, TJ. (2019). Exact computation for existence of a knot counterexample. 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