Some remarks on Bézier curves
type: Item
instance of: scholarly article
title: Some remarks on Bézier curves
zbMATH Open document ID: 0649.41004
DOI: 10.1016/0167-8396(88)90007-6
published in: Computer Aided Geometric Design
publication date: 1988-01-01T00:00:00Z
review text: Parametric polynomial curves in Bézier-Bernstein representation are considered as projections of rational norm curves of degree n in n-space; from this point of view the singularities of a planar Bézier cubic are determined and expressed by its affine invariants. Secondly, for an arbitrary pair of adjacent parametric curves in homogeneous coordinates, the general conditions for geometric continuity of any order k, \(G^ k\), are established. This result generalizes the corresponding conditions in the non-homogeneous (affine) case, recently obtained by \textit{T. N. T. Goodman} [J. Approximation Theory 44, 132-153 (1984; Zbl 0569.41010)]. Some applications are given for Bézier curves. In particular, for \(\gamma\)-splines \textit{W. Boehm} [Comput. Aided Geom. Des. 4, 67-77 (1987; Zbl 0632.65005)] the existence of a global rational parameter that makes it to a \(C^ 2\) parametric curve is shown. Furthermore, for two adjacent rational Bézier curves the complete conditions for \(G^ 3\) are stated using the projective properties of the control points only.
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 4060017
zbMATH Keywords: Parametric polynomial curves
zbMATH Keywords: Bézier-Bernstein representation
zbMATH Keywords: Bézier curves
author: Wendelin L. F. Degen
MaRDI profile type: Publication
full work available at URL: 0167-8396(88)90007-6
OpenAlex ID: W182550205
cites work: A survey of curve and surface methods in CAGD
cites work: Curvature continuous curves and surfaces
cites work: Rational geometric splines
cites work: Q3284338
cites work: Q4745933
cites work: Some remarks on \(V^ 2\)-splines
cites work: Über berührende Kegelschnitte einer ebenen Kurve
cites work: Properties of \(\beta\)-splines
cites work: Q3660449
cites work: Q3241626
cites work: Q3925060
cites work: Continuity conditions for spline curves
cites work: Q3220682
cites work: A locally controllable spline with tension for interactive curve design
cites work: Q3665803
description: scientific article; zbMATH DE number 4060017