Quadric-surface intersection curves: Shape and structure
type: Item
instance of: scholarly article
title: Quadric-surface intersection curves: Shape and structure
zbMATH Open document ID: 0794.65102
DOI: 10.1016/0010-4485(93)90018-J
author: Itzhak Wilf
author: Yehuda Manor
published in: CAD. Computer-Aided Design
publication date: 1994-04-18T00:00:00Z
review text: The authors modify \textit{J. Levin's} algorithm [Commun. ACM 19, 555-563 (1976; Zbl 0334.68050)] for the tracing of the curve of intersection of two quadrics to take advantage of the knowledge about the curve of intersection obtained from the elementary divisors and their exponents of the pencil defined by the two quadrics. [The problem was treated comprehensively by \textit{W. Killing} in his thesis [Das Flächenbüschel 2. Ordnung, Berlin (1872)]; a modern representation of Killing's classification is, e.g., \textit{O. Mayer} [Geometrie Proiectivă, Ed. Academiei, Bucureşti (1970) \S 130]. Ray tracing is really unnecessary since the curve of intersection has a parametric representation by elliptic \(\sigma\)-functions (Killing) or \(\theta\)-functions, cf. \textit{G. Loria} [Rend. Accad. Lincei, mat.-fis. cl. Rome (4)6, 179 (1890)] that is immediately given once the relative eigenvalues and eigenvectors of the pencil are computed; and the authors' method requires even knowledge of the Weierstrass decomposition of the matrix of the pencil. If the curve of intersection has a singularity, it is of genus 0 and has a rational representation, given by Killing for all his classes. A modern reworking of Killing's book would be a great service to CAD].
Mathematics Subject Classification ID: 65D18
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 51N35
Mathematics Subject Classification ID: 51N05
zbMATH DE Number: 516702
zbMATH Keywords: curve of intersection
zbMATH Keywords: quadrics
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
full work available at URL: 0010-4485(93)90018-j
OpenAlex ID: W2051651621
Recommended article: Computing quadric surface intersections based on an analysis of plane cubic curves
Recommended article: Tools for analyzing the intersection curve between two quadrics through projection and lifting
Recommended article: Automatic parsing of degenerate quadric-surface intersections
Recommended article: Enhancing Levin's method for computing quadric-surface intersections
Recommended article: Using signature sequences to classify intersection curves of two quadrics
Recommended article: Curve implicitization using moving lines
Recommended article: Near-optimal parameterization of the intersection of quadrics
Recommended article: Near-optimal parameterization of the intersection of quadrics. III. Parameterizing singular intersections
description: scientific article; zbMATH DE number 516702