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Russia Orel
Year
2017
Volume
27
Issue
1
Pages
60-68
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Section Mathematics
Title Methods of conformal mappings of polyhedra in $\mathbb{R}^3$
Author(-s) Radygin V.M.a, Polyanskii I.S.a
Affiliations The Academy of Federal Security Guard Service of the Russian Federationa
Abstract Methods necessary to solve problems of conformal mapping of polyhedra in $\mathbb{R}^3$ are developed. The results are obtained with the use of quaternion algebra and geometric representations. The direct and inverse conformal mappings are defined: those of the upper half-space onto the unit ball, those of a ball crescent onto the dihedral angle and those of dihedral and polyhedral angles onto the upper half-space. Solutions to the direct and inverse problems of conformal mapping of the polyhedrons onto the upper half-space are found using the results obtained. The solution to the direct problem of conformal mapping is based on the results of the Christoffel-Schwarz theorem. The solution of the inverse problem is obtained by the method of successive conformal mappings. In general, the one-to-one mappings obtained are based on the fact that, by the Liouville theorem, all conformal diffeomorphisms of any area in the space are the Möbius transformations.
Keywords conformal mapping, polyhedron, dihedral angle, polyhedral angle, upper half-space
UDC 517.54
MSC 30C20
DOI 10.20537/vm170106
Received 27 October 2016
Language Russian
Citation Radygin V.M., Polyanskii I.S. Methods of conformal mappings of polyhedra in $\mathbb{R}^3$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 60-68.
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