Vol. 11, No. 2, 2018

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On computable classes of equidistant sets: finite focal sets

Csaba Vincze, Adrienn Varga, Márk Oláh, László Fórián and Sándor Lőrinc

Vol. 11 (2018), No. 2, 271–282
Abstract

The equidistant set of two nonempty subsets K and L in the Euclidean plane is the set of all points that have the same distance from K and L. Since the classical conics can be also given in this way, equidistant sets can be considered as one of their generalizations: K and L are called the focal sets. The points of an equidistant set are difficult to determine in general because there are no simple formulas to compute the distance between a point and a set. As a simplification of the general problem, we are going to investigate equidistant sets with finite focal sets. The main result is the characterization of the equidistant points in terms of computable constants and parametrization. The process is presented by a Maple algorithm. Its motivation is a kind of continuity property of equidistant sets. Therefore we can approximate the equidistant points of K and L with the equidistant points of finite subsets Kn and Ln. Such an approximation can be applied to the computer simulation, as some examples show in the last section.

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Keywords
generalized conics, equidistant sets
Mathematical Subject Classification 2010
Primary: 51M04
Milestones
Received: 21 August 2016
Revised: 26 January 2017
Accepted: 4 February 2017
Published: 17 September 2017

Communicated by Michael Dorff
Authors
Csaba Vincze
Institute of Mathematics
University of Debrecen
Hungary
Adrienn Varga
Faculty of Engineering
University of Debrecen
Hungary
Márk Oláh
BSc Mathematics
University of Debrecen
Hungary
László Fórián
BSc Mathematics
University of Debrecen
Hungary
Sándor Lőrinc
BSc Electric Engineering
University of Debrecen
Hungary