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Visualization of Geodesic Curves, Spheres and Equidistant Surfaces in S^2×R Space

János Pallagi ; Institute of Mathematics, Department of Geometry, Budapest University of Technology and Economics, Budapest, Hungary
Benedek Schultz ; Institute of Mathematics, Department of Geometry, Budapest University of Technology and Economics, Budapest, Hungary
Jenő Szirmai orcid id orcid.org/0000-0001-9610-7993 ; Institute of Mathematics, Department of Geometry, Budapest University of Technology and Economics, Budapest, Hungary


Puni tekst: engleski pdf 387 Kb

str. 35-40

preuzimanja: 932

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Sažetak

The S^2×R geometry is derived by direct product of the
spherical plane S^2 and the real line R. In [9] the third author has determined the geodesic curves, geodesic balls of S^2×R space, computed their volume and defined the notion of the geodesic ball packing and its density. Moreover, he has developed a procedure to determine the density of the geodesic ball packing for generalized Coxeter space groups of S^2×R and applied this algorithm to them.
E. MOLNÁR showed in [3], that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere
PS^3(V^4,V_ 4,R). In our work we shall use this projective
model of S^2×R geometry and in this manner the geodesic
lines, geodesic spheres can be visualized on the Euclidean
screen of computer.
Furthermore, we shall define the notion of the equidistant surface to two points, determine its equation and visualize it in some cases. We shall also show a possible way of making the computation simpler and obtain the equation of an equidistant surface with more possible geometric meaning. The pictures were made by the Wolfram Mathematica software.

Ključne riječi

non-Euclidean geometries; projective geometry; geodesic sphere; equidistant surface

Hrčak ID:

62863

URI

https://hrcak.srce.hr/62863

Datum izdavanja:

29.12.2010.

Podaci na drugim jezicima: hrvatski

Posjeta: 1.914 *