On computable classes of equidistant sets: finite focal sets

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.