Isotopic Equivalence from Bezier Curve Subdivision
J. Li, T. J .Peters, J. A. Roulier
TL;DR
This paper proves that subdividing a Bezier curve's control polygon makes it topologically equivalent and isotopic to the original curve, providing formulas for the number of subdivisions needed.
Contribution
It establishes the topological equivalence of Bezier curves and their control polygons through subdivision, with explicit formulas for subdivision iterations.
Findings
Control polygons become homeomorphic to Bezier curves after subdivision.
Exterior angles of control polygons decrease exponentially with subdivision.
Formulas for the number of subdivisions needed to ensure topological equivalence.
Abstract
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.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Computational Geometry and Mesh Generation · Polynomial and algebraic computation