\(q\)-Bernstein polynomials and Bézier curves
type: Item
instance of: scholarly article
title: \(q\)-Bernstein polynomials and Bézier curves
zbMATH Open document ID: 1014.65015
DOI: 10.1016/S0377-0427(02)00733-1
author: Halil Oruç
author: George M. Phillips
published in: Journal of Computational and Applied Mathematics
publication date: 2003-03-16T00:00:00Z
review text: Bézier curve techniques are extended by using a generalization of the Bernstein basis, called the \(q\)-Bernstein basis. A one-parameter family of generalized Bernstein polynomial is defined. It is proved that the approximation to a convex function by its \(q\)-Bernstein polynomials is one sided. It is shown also that the difference of two consecutive \(q\)-Bernstein polynomials has a representation involving second-order divided differences. The connection with the power basis, the Bernstein basis and the \(q\)-Bernstein basis is revealed by deriving their transformation matrices. Parametric curves are constructed using the \(q\)-Bernstein basis; their shape properties are discussed using the total positivity. Moreover, a degree elevation algorithm for \(q\)-Bernstein parametric curves is presented.
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 1882237
zbMATH Keywords: generalized Bernstein polynomials
zbMATH Keywords: Bézier curves
zbMATH Keywords: \(q\)-Bernstein polynomials
zbMATH Keywords: degree elevation algorithm
zbMATH Keywords: \(q\)-Bernstein parametric curves
reviewed by: Dana Petcu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1882237