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

  • E. Roanes-Macías
  • E. Roanes-Lozano
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2330)

Abstract

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.

Keywords

Tangent Plane Computer Algebra Polynomial System Algebraic Method Original Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • E. Roanes-Macías
    • 1
  • E. Roanes-Lozano
    • 1
  1. 1.Dept. AlgebraUniversidad Complutense de MadridMadridSpain

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