Solving the interference problem for ellipses and ellipsoids: new formulae
type: Item
instance of: scholarly article
title: Solving the interference problem for ellipses and ellipsoids: new formulae
zbMATH Open document ID: 1491.65018
author: Jorge Caravantes
author: Gema Maria Diaz Toca
author: Mario Fioravanti
author: Laureano Gonzalez-Vega
published in: Journal of Computational and Applied Mathematics
publication date: 2022-02-16T00:00:00Z
review text: Ellipses and ellipsoids are essential entities for modelling (and/or enclosing) the shape of the objects under consideration. So, in this sense, the problem of detecting when two moving ellipses or ellipsoids overlap is of interest to robotics, CAD/CAM, computer animation, etc. In this paper, the authors consider the characteristic polynomial of the pencil defined by two ellipses/ellipsoids \(A\) and \(B\) given by \(X^T AX = 0\) and \(X^T BX = 0\) and they analyze symbolically the sign of the real roots of the characteristic polynomial. From this, new formulae are derived when \(A\) and \(B\) overlap, are separate, or touch each other externally. This characterization is defined by a minimal set of polynomial inequalities depending only on the entries of \(A\) and \(B\). Thus, one of the fascinating result of this paper is that one needs only to compute the characteristic polynomial of the pencil defined by \(A\) and \(B\), det\((TA + B)\), and not the intersection points between them. The authors compare the results with the best available approach dealing with this problem, and they show that the new formulae involve a smaller set of polynomials and less sign conditions. Finally, as an application, it is shown that, since the analysis of the univariate polynomials (depending on the time) in the formulae provides the collision events between them, the obtained characterization provides a new approach for exact collision detection of two moving ellipses or ellipsoids.
reviewed by: Sonia Pérez Díaz
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 7474412
zbMATH Keywords: interference problem
zbMATH Keywords: ellipses
zbMATH Keywords: ellipsoids
zbMATH Keywords: events detection
MaRDI profile type: Publication
full work available at URL: j.cam.2021.114072
OpenAlex ID: W4205111140
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DOI: 10.1016/J.CAM.2021.114072
description: scientific article; zbMATH DE number 7474412