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## Archive of Issues

Russia Orel
Year
2017
Volume
27
Issue
1
Pages
60-68
 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. References Arkhipov N.S., Polyanskij I.S., Stepanov D.E. The barycentric method for the field analysis in a regular waveguide with arbitrary cross-section, Antenny, 2015, no. 1 (212), pp. 32-40 (in Russian). Polyansky I.S. Poisson barycentric coordinates for multivariate approximation of scalar potential within an arbitrary area (Part 1), Vestn. Sarat. Gos. Tekh. Univ., 2015, vol. 1, no. 1 (78), pp. 30-36 (in Russian). Polyansky I.S. Poisson barycentric coordinates for multivariate approximation of scalar potential within an arbitrary area (Part 2), Vestn. Sarat. Gos. Tekh. Univ., 2015, vol. 1, no. 1 (78), pp. 36-42 (in Russian). Fil'chakov P.F. Priblizhennye metody konformnykh otobrazhenii. Spravochnoe rukovodstvo (Approximate methods of conformal mappings. Handbook), Kiev: Naukova Dumka, 1964, 536 p. Radygin V.M., Polansky I.S. Modified method of successive conformal mappings of polygonal domains, Vestn. Tomsk. Gos. Univ., Mat. Mekh., 2016, no. 1 (39), pp. 25-35 (in Russian). DOI: 10.17223/19988621/39/3 Sudbery A. Quaternionic analysis, Mathematical Proceedings of the Cambridge Philosophical Society, 1979, vol. 85, issue 2, pp. 199-225. DOI: 10.1017/S0305004100055638 Ahlfors L. Möbius transformations in several dimensions, University of Minnesota, 1981. Translated under the title Preobrazovanie Mebiusa v mnogomernom prostranstve, Moscow: Mir, 1986, 112 p. Norden A.P. Teoriya poverkhnostei (Theory of surfaces), Moscow: Moscow: Gos. Izd. Tekh. Teor. Lit., 1956, 260 p. Lavrent'ev M.A., Shabat B.V. Metody teorii funktsii kompleksnogo peremennogo (Methods of the theory of functions of a complex variable), Moscow: Nauka, 1973, 736 p. Full text