Geometric Determination of the Spheres which Are Tangent to Four Given Ones

摘要

Apollonius' problem (find the tangent circles to three given ones) has attracted many mathematicians and has been solved using different methods along more than 22 centuries. Nowadays computers allow to mechanize the solving process and to treat its generalization to higher dimension using algebraic methods. Starting from the classical Vieta-Steiner solution for dimension 2, we have developed a method valid for dimension n, that, thanks to the use of an original coding, allows to choose in advance the relative position of the solution sphere w.r.t. the given ones (i.e., if each tangency is exterior or interior). Moreover, the possible degeneracy of some of the solution (hyper-)spheres in (hyper-) planes and the existence of configurations with an infinity number of solutions are considered.