The best \(G^{1}\) cubic and \(G^{2}\) quartic Bézier approximations of circular arcs
type: Item
instance of: scholarly article
title: The best \(G^{1}\) cubic and \(G^{2}\) quartic Bézier approximations of circular arcs
zbMATH Open document ID: 1236.65014
author: Seok Hur
author: Tae-wan Kim
published in: Journal of Computational and Applied Mathematics
publication date: 2011-12-21T00:00:00Z
review text: For a cubic Bézier representation of a curve approximating a circular arc, four control points are needed. Two of them coincide with the end-points of the arc (\(G^0\) condition) and two other are placed at a distance \(h\) along the tangents in these end-points (\(G^1\) condition). This gives a family of approximating curves depending on the parameter \(h\). Conditions are given to select the best one in this class in a Hausdorff sense. For quartic curves, there is one more parameter but an extra \(G^2\) condition. Again necessary and sufficient conditions are formulated to find the best approximant in Hausdorff sense. The best solutions are unique. Because of the special character of a circular arc, the Hausdorff distance can be replaced by a (signed) function that is easily computable.
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 5991189
zbMATH Keywords: cubic Bézier curve
zbMATH Keywords: quartic Bézier curve
zbMATH Keywords: \(G^{1}\) and \(G^{2}\) end-point interpolation
zbMATH Keywords: best approximation curve
zbMATH Keywords: circular arc
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: control points
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
full work available at URL: j.cam.2011.08.002
OpenAlex ID: W169091707
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DOI: 10.1016/J.CAM.2011.08.002
description: scientific article; zbMATH DE number 5991189