Tools for analyzing the intersection curve between two quadrics through projection and lifting
type: Item
instance of: scholarly article
title: Tools for analyzing the intersection curve between two quadrics through projection and lifting
zbMATH Open document ID: 1465.14032
author: Laureano Gonzalez-Vega
author: Alexandre Trocado
published in: Journal of Computational and Applied Mathematics
publication date: 2021-04-23T00:00:00Z
review text: Determining the topology of and parametric equations for the intersection curve of two quadrics are two well-studied topics in computational geometry. Nonetheless, details of these tasks can be quite involved, in particular if an accurate and robust treatment of special cases is required. This article proposes a concept based on 1) projecting the intersection curve onto a plane, 2) analyzing this projection and 3) lifting back correct curve parts onto the quadrics. The projection step via resultants is straightforward but yields an algebraic curve while the actual projection of real points of the intersection curve is bounded by the two quadrics' silhouettes, thus producing a semi-algebraic set, consisting of a quartic curve that is bounded by two conics (the ``cutcurve''). The authors exploit special properties of the cutcurve for analyzing its topology and for gathering information on diverse singularities that allow to conveniently lift back portions of the cutcurve onto the quadrics. All of this results in an (as expected) rather involved algorithm to determine the intersection curve's topology and square root parametrizations. The authors argue that it is of moderate algebraic complexity and compares favourable with more general curve plotting methods. They also illustrate it at hand of some examples. While the computations work best with rational input, they can be adapted to floating point numbers. The authors give some hints on that.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 68U05
Mathematics Subject Classification ID: 13P15
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 7337489
zbMATH Keywords: quadric
zbMATH Keywords: intersection curve
zbMATH Keywords: cutcurve
zbMATH Keywords: lifting
MaRDI profile type: Publication
full work available at URL: j.cam.2021.113522
OpenAlex ID: W3133547174
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DOI: 10.1016/J.CAM.2021.113522
description: scientific article; zbMATH DE number 7337489