Visualization of the Dirichlet-Voronoi cells in S²×R space

Pallagi, János and Szirmai, Jenő (2012) Visualization of the Dirichlet-Voronoi cells in S²×R space. Pollack Periodica, 7 (Supple). pp. 95-104. ISSN 1788-1994

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Abstract

The S²×R geometry can be derived by the direct product of the spherical plane S² and the real line R. In [1] J. Z. Farkas has classified and given the complete list its space groups. In [6] the second author has studied the geodesic balls and their volumes in S²×R space, moreover he has introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of S²×R.The aim of this paper to develop a method to study and visualize the Dirichlet-Voronoi cells belonging to a given ball packing. We apply our former results on the equidistant surfaces of the S²×R geometry (see [5]) to determine the D-V cells to locally optimal ball packings belonging to S²×R space groups generated by glide reflections.E. Molnár has shown in [3], that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere, in our work we will use this projective model of S²×R geometry. We will use the Wolfram Mathematica software for visualization of the arrangement of a locally optimal geodesic ball packing and its Dirichlet-Voronoi cell of a given glide reflection space group.

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