1 Kavrayskiy A.V.

State research navigation-hydrographic Institute, PJSC

The experience of mathematical modeling of the 3D-sphere in the 4D-space and projecting it by mathematical cartography methods in the 3D-Euclidian space is presented. The problem is solved by introduction of spherical coordinates for the 3D-sphere and their transformation into the rectangular coordinates, using the mathematical cartography methods. The mathematical relationship for calculating the length distortion mp(s) of the ds linear element when projecting the 3D-sphere from the 4-dimensional Euclidian space into three-dimensional Euclidian space is derived. Numerical examples, containing the modeling of the ds small linear element by spherical coordinates of 3D-sphere, projecting this sphere into the 3D-Euclidian space and length of ds calculating by means of its projection dL and size of distortion mp(s) are solved. Based on the model of the Universe known in cosmology as the 3D-sphere, the hypothesis of connection between distortion mp(s) and the known observed effects Redshift and Microwave Background Radiation is considered.
1.   Babenko I.K., Bogatyj S.A. K otobrazheniyu sfery v evklidovo prostranstvo [The mapping of a sphere into a Euclidean space]. Matematicheskie zametki, M.: Nauka, 1989, Vol. 46, 3. pp. 3–8.
2.   Graur A.V. Matematicheskaya kartografiya [Mathematical Cartography]. L.: Izd-vo Leningradskogo un-ta, 1956, 372 p.
3.   Zasov A.V., Postnov K.A. Obshchaya astrofizika: Galaktiki i skopleniya galaktik [General astrophysics: Galaxies and clusters of galaxies]. Fryazino, 2006, 496 p.
4.   Klimishin I.A. Relyativistskaya astronomiya [Relativistic astronomy]. M.: Nauka, 1989, 288 p.
5.   Layzer D. Sozdavaya kartinu Vselennoj [Constructing the Universe]. M.: Mir, 1988, 324 p.
6.   Makeev V.V. Zadacha Knastera o nepreryvnyh otobrazheniyah sfery v evklidovo prostranstvo [The Knaster problem on continuous mappings of a sphere into a Euclidean space]. Issledovaniya po topologii. 6: Zap. nauch. seminarov LOMI, L.: Nauka, 1988, Vol. 167, pp. 169–178.
7.   Pauli V. Teoriya otnositel'nosti [Theory of relativity]. M.: Nauka, 1983, 336 p.
8.   Sazhin M.V. Sovremennaya kosmologiya v populyarnom izlozhenii [Modern cosmology in the popular presentation]. M.: Editorial URSS, 2002, 240 p.
9.   Chernij A.N. Relyativistskaya fizika kosmosa [Theory of relativity]. M.: Nauchnyj mir, 2010, 480 p.
10.   Yang C.T. (1957) On maps from spheres to Euclidean spaces. American Journ. of Mathematics, Volume 79, no. 4, pp. 725–732.
Kavrayskiy A.V., 
(2017) The experience of employment of the mathematical cartography methods in cosmology. Geodesy and cartography = Geodezia i Kartografia, 923(5), pp. 7-16. (In Russian). DOI: 10.22389/0016-7126-2017-923-5-7-16
Publication History
Received: 07.07.2016
Accepted: 19.12.2016
Published: 19.06.2017


2017 May DOI:

QR-code page

QR-код страницы