Algebraic study of the Apollonius circle of three ellipses

We study the external tritangent Apollonius (or Voronoi) circle to three ellipses. This problem arises when one wishes to compute the Apollonius (or Voronoi) diagram of a set of ellipses, but is also of independent interest in enumerative geometry. This paper is restricted to non-intersecting ellipses, but the extension to arbitrary ellipses is possible. We propose an efficient representation of the distance between a point and an ellipse by considering a parametric circle tangent to an ellipse. The distance of its center to the ellipse is expressed by requiring that their characteristic polynomial have at least one multiple real root. We study the complexity of the tritangent Apollonius circle problem, using the above representation for the distance, as well as sparse (or toric) elimination. We offer the first nontrivial upper bound on the number of tritangent circles, namely 184.