No. 901

Title:

Packing and Minkowski covering of congruent spherical caps on a sphere

Author(s):

Sugimoto, Teruhisa (Institute of Statistical Mathematics)
Tanemura, Masaharu (Institute of Statistical Mathematics)

Key words:

Sphere; Minkowski Set; Packing; Covering; Tammes Problem.

Abstract:  

Let $C_{i}$ ($\,i=1,\ldots ,N\,$) be the $i$-th open spherical cap of angular radius $r$ and let $M_{i}$ 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, $r_{N} $, (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 $i=2,\ldots ,N-1\,$, that $M_{i}$ is set on the perimeter of $C_{i-1}$, and that each area of the set $(\cup _{\nu =1}^{i-1}C_{\nu })\cap C_{i}$ becomes maximum. In this study, for $N=2,\,\ldots ,12$, we found out that the solutions of above covering and the solutions of Tammes problem were strictly correspondent. Especially, we succeeded to obtain the exact value $r_{10}$ for $N = 10$.