Descartes Circle Theorem, Steiner Porism, and Spherical Designs

Abstract

A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be constructed starting with any initial circle, tangent to the parent circles. What do the circles in these 1-parameter family of Steiner chains of length k have in common? We prove that the first k-1 moments of their curvatures remain constant within a 1-parameter family. For k=3, this follows from the Descartes Circle Theorem. We extend our result to Steiner chains in the spherical and hyperbolic geometries and present a related more general theorem involving spherical designs.