Rationality of the offsets to algebraic curves and surfaces
type: Item
instance of: scholarly article
title: Rationality of the offsets to algebraic curves and surfaces
zbMATH Open document ID: 0814.14048
DOI: 10.1007/BF02663776
published in: Applied Mathematics. Series B (English Edition)
publication date: 1995-06-20T00:00:00Z
review text: The parallel curves and surfaces of algebraic curves and surfaces are algebraic as envelopes of algebraic systems of circles and spheres. However, the parallel curves and surfaces of rational curves and surfaces are not in general rational (i.e., of genus 0). The condition of rationality is clearly the existence of a rational parametrization of the system of circles or spheres, i.e., of the unit normal. In addition to the literature quoted by the author one may add the papers of \textit{E. Laguerre} [Oeuvres, Vol. 2: Géométrie. pp. 592 ff.]. Rational parallel curves have by definition a rational representation in Laguerre geometry [cf. also \textit{H. Wieleitner}, ``Spezielle ebene Kurven'' (Leipzig 1908); \S16]. Using essentially Laguerre's criterion, the author proves that the parallel surfaces to paraboloids, spheres, and hyperboloids of one sheet are rational.
Mathematics Subject Classification ID: 14M20
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 713529
zbMATH Keywords: algebraic surfaces
zbMATH Keywords: rational surfaces
zbMATH Keywords: parallel curves
zbMATH Keywords: rational curves
zbMATH Keywords: Laguerre geometry
zbMATH Keywords: parallel surfaces
author: Wei Lü
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
cites work: Automatic parameterization of rational curves and surfaces. I: Conics and conicoids
cites work: Q4041056
cites work: Exact offset procedures for simple solids
cites work: Q4026160
cites work: Algebraic properties of plane offset curves
cites work: Q4178157
cites work: Rational curves and surfaces with rational offsets
cites work: Algebraically rectifiable parametric curves
cites work: Improperly parametrized rational curves
cites work: Implicit representation of parametric curves and surfaces
cites work: Q3287734
description: scientific article; zbMATH DE number 713529