Bézier representation for cubic surface patches
type: Item
instance of: scholarly article
title: Bézier representation for cubic surface patches
zbMATH Open document ID: 0766.65014
DOI: 10.1016/0010-4485(92)90019-7
published in: CAD. Computer-Aided Design
publication date: 1993-05-16T00:00:00Z
review text: It would be nice to have an easy method to model cubic surfaces in computer aided design. The authors offer a method of parametrization that yields a biquadratic Bézier representation of a surface patch given by 4 vertices of the surface \(F_ 3\) that lie pairwise on a straight line contained in the surface. The parametrization is based on a nice idea and deserves some study. \{Unfortunately, the method does not work in general. Of the 27 straight lines on a nondegenerate real cubic surface over \(C^ 3\), only 3 have to be real and these may form a triangle. In addition, there are 18 classes of singular cubics and two classes of ruled cubic surfaces. Also, if one already has two straight lines on the surface, one has all the tools to write down its equation in Cayley-Salmon normal form and the Clebsch birational representation of the surface onto the projective plane and with it a parametric representation of the implicitly given surface. It would be worthwhile to find an easy algorithmic implementation of the Clebsch representation or other reductions to normal forms. All mathematics necessary to make cubic surfaces useful in CAD is assembled in article IIIC10a by \textit{W. Fr. Meyer} of the old Encyklopädie der mathematischen Wissenschaften, 3. Band, 2. Teil, 2. Hälfte, p. 1437- 1531 (1928)\}.
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 14Q10
zbMATH DE Number: 166651
zbMATH Keywords: cubic surfaces
zbMATH Keywords: computer aided design
zbMATH Keywords: biquadratic Bézier representation
author: Suresh Kumar Lodha
author: J. Warren
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
full work available at URL: 0010-4485(92)90019-7
OpenAlex ID: W1967526463
description: scientific article; zbMATH DE number 166651