A generalization of the Minkowski distance and new definitions of the central conics

Abstract

In this paper, we give a generalization of the well-known Minkowski distance family in the n-dimensionalCartesian coordinate space. Then we consider three special cases of this family, which are also generalizations of thetaxicab, Euclidean, and maximum metrics, respectively, and we determine some circle properties of them in the realplane. While we determine some properties of circles of these generalized distances, we discover a new definition ofellipses, and then we also determine a similar definition of hyperbolas, which will be new members among differentmetrical definitions of central conics in the Euclidean plane.