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.
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References
Berger M.: Geometry I, Springer-Verlag, Berlin-Heidelberg, 1987.
Coxeter H.S.M.: The Problem of Apollonius, Am. Math. Monthly, Vol. 75 (1968) 5–15.
Hadamard J.: Lecons de Géométrie Elementaire, A. Colin, Paris, 1947-49.
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.
Lewis R. H.: Apollonius Meets Computer Algebra. In: Proceedings of AGA’2001, http://math.warn.edu/ACA/2001/Proceedings/NonStd/
Pedoe D.: On a theorem in Geometry, Am. Math. Monthly, Vol. 74 (1967) 627–640.
Pedoe D.: Geometry, Dover Pub., New York, 1988.
Ogilvy C.S.: Excursions in Geometry, Dover Pub., New York, 1990.
Roanes Lozano E.: El Problema de Apolonio, Bol. Soc. Puig Adam, Vol. 14 (1987) 13–41.
Roanes Macías E., Roanes Lozano E.: Nuevas tecnologías en Geometría, Editorial Complutense, Madrid, 1994.
Soddy F.: The Kiss Precise, Nature, 137 (1936) 1021.
Vieta F.: Varia Responsa. IX: Apollonius Gallus, Real Academia de Ciencias, Madrid, not dated edition (Reprint of the original dated 1600).
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Roanes-Macías, E., Roanes-Lozano, E. (2002). Geometric Determination of the Spheres which Are Tangent to Four Given Ones. In: Sloot, P.M.A., Hoekstra, A.G., Tan, C.J.K., Dongarra, J.J. (eds) Computational Science — ICCS 2002. ICCS 2002. Lecture Notes in Computer Science, vol 2330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46080-2_6
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DOI: https://doi.org/10.1007/3-540-46080-2_6
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