On computable classes of equidistant sets: finite focal sets

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

doi:10.2140/involve.2018.11.271

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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

DOI: 10.2140/involve.2018.11.271

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 $K_{n}$ and $L_{n}$ . 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

Vol. 11, No. 2, 2018