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
|
|