Degree, multiplicity, and inversion formulas for rational surfaces using \(u\)-resultants
type: Item
instance of: scholarly article
title: Degree, multiplicity, and inversion formulas for rational surfaces using \(u\)-resultants
zbMATH Open document ID: 0756.65011
DOI: 10.1016/0167-8396(92)90009-E
author: S. Singh
published in: Computer Aided Geometric Design
publication date: 1992-10-13T00:00:00Z
review text: Given an algebraic surface, the basic problems of identification of base points, minimal degree, generic representation, etc. are solved in algebraic geometry by intersecting the surface with a linear pencil. The authors note that if the surface is given parametrically in projective (3-)space by \(x_ i=f_ i(s_ k)\) then the intersection of the surface with a line given as intersection of two planes gives two equations \(\sum a^{(j)}_ if(s_ 1,s_ 2,s_ 3)=0\), (\(j=1,2\)). The adjunction of a third \(\sum u_ ks_ k=0\) with indeterminates \(u_ k\) allows one to compute the resultant which can be used to solve problems one is interested in \{unless one runs into a surface with infinitely close singularities, the problem that is unsolvable with methods of enumerative algebraic geometry\}. However, in the untractable cases the resultant will vanish and therefore it is a very safe indicator of trouble and that is really the most important aspect for computational geometry.
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 14Q10
zbMATH DE Number: 68931
zbMATH Keywords: degree
zbMATH Keywords: multiplicity
zbMATH Keywords: inversion formulas
zbMATH Keywords: rational surfaces
zbMATH Keywords: algebraic surface
zbMATH Keywords: algebraic geometry
zbMATH Keywords: intersection
zbMATH Keywords: resultant
zbMATH Keywords: computational geometry
reviewed by: Heinrich Guggenheimer
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: A survey of curve and surface methods in CAGD
cites work: Q4002478
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full work available at URL: 0167-8396(92)90009-e
OpenAlex ID: W2006512450
Wikidata QID: Q127972163
description: scientific article; zbMATH DE number 68931