Forma, Vol. 21 (No. 3), pp. 197-225, 2006
Original Paper

Packing and Minkowski Covering of Congruent Spherical Caps on a Sphere for N = 2, ..., 9

Teruhisa Sugimoto1* and Masaharu Tanemura1,2

1The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan
2Department of Statistical Science, The Graduate University for Advanced Studies, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan
*E-mail address:

(Received October 31, 2005; Accepted May 26, 2006)

Keywords: Spherical Cap, Sphere, Packing, Covering, Minkowski Set

Abstract. Let Ci (i = 1, ..., N) be the i-th open spherical cap of angular radius r and let Mi be its center under the condition that none of the spherical caps contains the center of another one in its interior. We consider the upper bound, rN (not the lower bound!) of r of the case in which the whole spherical surface of a unit sphere is completely covered with N congruent open spherical caps under the condition, sequentially for i = 2, ..., N - 1, that Mi is set on the perimeter of Ci-1, and that each area of the set ( ∪ Cv) ∩ Ci becomes maximum. In this study, for N = 2, ..., 9, we found out that the solutions of the above covering and the solutions of Tammes problem were strictly correspondent.

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