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)


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.


Tangent Plane Computer Algebra Polynomial System Algebraic Method Original Code 
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  1. 1.
    Berger M.: Geometry I, Springer-Verlag, Berlin-Heidelberg, 1987.zbMATHGoogle Scholar
  2. 2.
    Coxeter H.S.M.: The Problem of Apollonius, Am. Math. Monthly, Vol. 75 (1968) 5–15.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hadamard J.: Lecons de Géométrie Elementaire, A. Colin, Paris, 1947-49.Google Scholar
  4. 4.
    Lemoine E.: Application de d’une méthode d’évaluation de la simplicité des constructions a la comparaison de quelques solutions du probléme d’Apollonius, Nouvelles Ann. Math. (1892) 453–474.Google Scholar
  5. 5.
    Lewis R. H.: Apollonius Meets Computer Algebra. In: Proceedings of AGA’2001,
  6. 6.
    Pedoe D.: On a theorem in Geometry, Am. Math. Monthly, Vol. 74 (1967) 627–640.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Pedoe D.: Geometry, Dover Pub., New York, 1988.zbMATHGoogle Scholar
  8. 8.
    Ogilvy C.S.: Excursions in Geometry, Dover Pub., New York, 1990.zbMATHGoogle Scholar
  9. 9.
    Roanes Lozano E.: El Problema de Apolonio, Bol. Soc. Puig Adam, Vol. 14 (1987) 13–41.Google Scholar
  10. 10.
    Roanes Macías E., Roanes Lozano E.: Nuevas tecnologías en Geometría, Editorial Complutense, Madrid, 1994.Google Scholar
  11. 11.
    Soddy F.: The Kiss Precise, Nature, 137 (1936) 1021.CrossRefGoogle Scholar
  12. 12.
    Vieta F.: Varia Responsa. IX: Apollonius Gallus, Real Academia de Ciencias, Madrid, not dated edition (Reprint of the original dated 1600).Google Scholar

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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