A general framework for the optimal approximation of circular arcs by parametric polynomial curves

We propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves. The approach is based on a constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximatin...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2019-01, Vol.345, p.146-158
Main Authors: Vavpetič, Aleš, Žagar, Emil
Format: Article
Language:English
Subjects:
Bézier curve
Circular arc
Geometric interpolation
Optimal approximation
Parametric polynomial
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Summary:We propose a general framework for a geometric approximation of circular arcs by parametric polynomial curves. The approach is based on a constrained uniform approximation of an error function by scalar polynomials. The system of nonlinear equations for the unknown control points of the approximating polynomial given in the Bézier form is derived and a detailed analysis provided for some low degree cases which were not studied yet. At least for these cases the solutions can be, in principal, written in a closed form, and provide the best known approximants according to the simplified radial distance. A general conjecture on the optimality of the solution is stated and several numerical examples conforming theoretical results are given.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2018.06.020