3D extension of Steiner chains problem

Abstract

A natural 3D extension of the Steiner chains problem, original to the authors of this article, where circles are substituted by spheres, is presented. Given three spheres such that either two of them are contained in (or intersect) the third one, chains of spheres, each one externally tangent to its two neighbors in the chain and to the first and second given spheres, and internally tangent to the third given sphere, are considered. A condition for these chains to be closed has been stated and the Steiner alternative or Steiner porism has been extended to 3D. Remarkably, the process is of symbolic-numeric nature. Using a computer algebra system is almost a must, because a theorem in the constructive theory in the background requires using the explicit general solution of a non-linear algebraic system. However, obtaining a particular solution requires computing concatenated processes involving trigonometric expressions. In this case, it is recommended to use approximated calculations to avoid obtaining huge expressions.