## Involve: A Journal of Mathematics

- Involve
- Volume 11, Number 2 (2018), 271-282.

### On computable classes of equidistant sets: finite focal sets

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

#### 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.

#### Article information

**Source**

Involve, Volume 11, Number 2 (2018), 271-282.

**Dates**

Received: 21 August 2016

Revised: 26 January 2017

Accepted: 4 February 2017

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513775062

**Digital Object Identifier**

doi:10.2140/involve.2018.11.271

**Mathematical Reviews number (MathSciNet)**

MR3733957

**Zentralblatt MATH identifier**

06817020

**Subjects**

Primary: 51M04: Elementary problems in Euclidean geometries

**Keywords**

generalized conics equidistant sets

#### Citation

Vincze, Csaba; Varga, Adrienn; Oláh, Márk; Fórián, László; Lőrinc, Sándor. On computable classes of equidistant sets: finite focal sets. Involve 11 (2018), no. 2, 271--282. doi:10.2140/involve.2018.11.271. https://projecteuclid.org/euclid.involve/1513775062