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

Vavpetič, Aleš (author), Žagar, Emil (author)

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.

• Language: English
• Keywords: geometric interpolation, circular arc, parametric polynomial, Bézier curve, optimal approximation
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• Number of pages: Str. 146-158
• Numbering: Vol. 345
• PID: 20.500.12556/RUL-116750
• UDC: 519.651
• ISSN on article: 0377-0427
• DOI: 10.1016/j.cam.2018.06.020
• COBISS.SI-ID: 18388057
• Publication date in RUL: 08.06.2020

Record is a part of a journal

• Title: Journal of computational and applied mathematics
• Shortened title: J. comput. appl. math.
• Publisher: Elsevier
• ISSN: 0377-0427
• COBISS.SI-ID: 27496960

Secondary language

• Language: Slovenian
• Title: Splošen pristop k optimalni aproksimaciji krožnih lokov s parametričnimi polinomskimi krivuljami

Abstract

V članku opišemo splošen pristop h geometrijski aproksimaciji krožnih lokov s parametričnimi polinomskimi krivuljami. Pristop temelji na enakomerni aproksimaciji funkcije napake z omejitvami. Izpeljan je ustrezen sistem algebraičnih enačb za neznane kontrolne točke aproksimacijske krivulje v Bézierjevi obliki in podana natančna analiza rešljivosti za nekaj primerov nizkih stopenj, ki še niso bili obravnavani. Vsaj za te primere lahko rešitev zapišemo v zaključeni obliki in dokažemo optimalnost rešitve glede na poenostavljeno radialno napako. Podana je splošna domneva o optimalnosti rešitve ter nekaj numeričnih primerov, ki potrjujejo teoretične rezultate.

• Keywords: geometrijska interpolacija, krožni lok, parametrični polinom, Bézierjeva krivulja, optimalna aproksimacija