On surface of Apollonius of two ellipsoids

Abstract

Apollonius defined the circle as the set of points that have a given ratio \(\mu \) of distances from two given points, where the ratio is not equal to one. In a more general sense, consider two 0-symmetric, bounded, convex bodies K and \(K'\), which define two norms. Their unit balls are K and \(K'\). The surface of Apollonius is defined as the set of points equidistant from the centres of bodies K and \(K'\) with respect to the aforementioned norms. In this paper we demonstrate that the surface of Apollonius of two ellipsoids is a quadratic surface. We also examine the circumstances under which this surface becomes a sphere.