On Equidistant Sets and Generalized Conics: The Old and the New

• Published in: The American mathematical monthly Vol. 121; no. 1; pp. 18 - 32
• Main Authors: Mario Ponce, Patricio Santibáñez
• Format: Journal Article
• Language: English
• Published: Washington Mathematical Association of America 01-01-2014; Taylor & Francis Ltd
• Subjects: Approximation; Circles; Computer simulation; Conic sections; Ellipses; Euclidean space; Geometric planes; Geometry; Hyperbolas; Mathematical functions; Mathematical theorems; Parabolas; Pixels; Set theory; Topological manifolds; Topological theorems
• ISSN: 0002-9890; 1930-0972
• DOI: 10.4169/amer.math.monthly.121.01.018

This article is devoted to the study of classical and new results concerning equidistant sets, both from the topological and metric point of view. We include a review of the most interesting known facts about these sets in Euclidean space and we prove two new results. First, we show that equidistant sets vary continuously with their focal sets. We also prove an error estimate result about approximative versions of equidistant sets that should be of interest for computer simulations. Moreover, we offer a viewpoint in which equidistant sets can be thought of as a natural generalization for conics. Along these lines, we show that the main geometric features of classical conics can be retrieved from more general equidistant sets.