Involve: A Journal of Mathematics
- Volume 11, Number 2 (2018), 271-282.
On computable classes of equidistant sets: finite focal sets
The equidistant set of two nonempty subsets and in the Euclidean plane is the set of all points that have the same distance from and . Since the classical conics can be also given in this way, equidistant sets can be considered as one of their generalizations: and 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 and with the equidistant points of finite subsets and . Such an approximation can be applied to the computer simulation, as some examples show in the last section.
Involve, Volume 11, Number 2 (2018), 271-282.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 51M04: Elementary problems in Euclidean geometries
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