Bézier curves of positive curvature
type: Item
instance of: scholarly article
title: Bézier curves of positive curvature
zbMATH Open document ID: 0648.65004
DOI: 10.1016/0167-8396(88)90021-0
author: John A. Roulier
published in: Computer Aided Geometric Design
publication date: 1988-01-01T00:00:00Z
review text: Let \(n+1\) distinct points \(P_ i\), \(i=0,1,...,n\), be given and let \(v_ i\), \(i=0,1,...,n\), be the vector from \(P_ i\) to \(P_{i+1}\). Let \(\vartheta_{k,j}\), \(k=0,1,...,n-1\), \(j=0,1,...,n-1\), be the counter- clockwise angle from \(v_ k\) to \(v_ j\). The data \(P_ i\) have positive discrete curvature if \(0<\vartheta_{i-1,i}<\pi\) for \(i=1,2,...,n-1\). Let \(B_ n(t)=(\dot x(t),y(t))=\sum^{n}_{k=0}P_ k\binom{n}{k}t^ k(1-t)^{n-k}\) be the nth degree Bézier curve for the control points \(P_ i=(x_ i,y_ i)\). The positive discrete curvature of the data points \(P_ i\) is not sufficient to insure the positivity of the curvature of the resulting curve \(B_ n(t)\). The author obtains sufficient conditions on the control points in the form of easy to check algorithms for positive curvature of \(B_ n(t)\). Many interesting examples are also given.
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 4058718
zbMATH Keywords: shape control
zbMATH Keywords: positive discrete curvature
zbMATH Keywords: Bézier curve
reviewed by: Luigi Gatteschi
describes a project that uses: pchip
MaRDI profile type: Publication
full work available at URL: 0167-8396(88)90021-0
OpenAlex ID: W1989351125
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description: scientific article; zbMATH DE number 4058718