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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 Federation^{a} 
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 halfspace onto the unit ball, those of a ball crescent onto the dihedral angle and those of dihedral and polyhedral angles onto the upper halfspace. Solutions to the direct and inverse problems of conformal mapping of the polyhedrons onto the upper halfspace are found using the results obtained. The solution to the direct problem of conformal mapping is based on the results of the ChristoffelSchwarz theorem. The solution of the inverse problem is obtained by the method of successive conformal mappings. In general, the onetoone 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 halfspace 
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. 6068. 
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