Cite this article as:

Romakina L. N. Parabolic parallelograms of the plane Ĥ. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 1, pp. 20-28. DOI: https://doi.org/10.18500/1816-9791-2014-14-1-20-28

Language: 
Russian
Heading: 
Mathematics
UDC: 
514.133

Parabolic parallelograms of the plane Ĥ

Authors: 
Romakina Lyudmila Nikolaevna, Saratov State University, Russia, Saratov, Astrakhanskaya 83 , romakinaln@mail.ru
Abstract: 

Parabolic parallelograms on a Hyperbolic Plane Hˆ with the positive curvature in the Cayley–Klein model are investigated. We conducted their classification, obtained the metric correlations between the measure of angles and the expressions of lengths of the edges through a measure of included angles. 

Key words: 
hyperbolic plane Hˆ with positive curvature, de Sitter plane, parallelogram, parabolic parallelogram.
DOI: 
https://doi.org/10.18500/1816-9791-2014-14-1-20-28
References
1. Romakina L. N. Simple partitions of a hyperbolic plane of positive curvature. Sbornik : Mathematics, 2012, vol. 203, no. 9, pp. 1310–1341. Available at : http://dx.doi.org/10.1070/SM2012v203n09ABEH004266.
2. Romakina L. N. Veernye trianguliatsii giperbolicheskoi ploskosti polozhitel’noi krivizny [Fan triangulations of hyperbolic plane positive curvature]. Matematicheskie trudy, 2013, vol. 16, no. 2, pp. 142–168 (in Russian).
3. Romakina L. N. Geometriia giperbolicheskoi ploskosti polozhitel’noi krivizny : v 4 ch. Ch. 2 : Preobrazovaniia i prostye razbieniia [Geometry of the hyperbolic plane of positive curvature : in 4 pt. Pt. 2 : Transformations and simple splittings]. Saratov, Saratov Univ. Press, 274 p. (in Russian).
4. De Sitter W. On the Relativity of Inertia. Remarks Concerning Einstein’s Latest Hypothesis. Proc. Royal Acad. Amsterdam, 1917, vol. 19, iss. 2, pp. 1217–1225.
5. Akutagawa K. On space-like hypersurfaces with constant mean curvature in the de Sitter space. Math. Z., 1987, vol. 196, pp. 13–19.
6. Montiel S. An integral inequality for compact spacelike hypersurfaces in a de Sitter space and application to the case of constant mean curvature. Indiana Univ. Math. J., 1988, vol. 37, pp. 909–917.
7. Cho Yun. Trigonometry in extended hyperbolic space and extended de Sitter space. Bull. Korean Math. Soc., 2009, vol. 46, no. 6, pp. 1099–1133. DOI : 10.4134/BKMS.2009.46.6.1099.
8. Asmus Im. Duality between hyperbolic and de Sitter geometry. J. of Geometry, 2009, vol. 96, iss. 1–2, pp. 11–40.
9. Romakina L. N. Hyperbolic parallelograms of the plane b H. Izv. Sarat. Univ. (N. S.), Ser. Math. Mech. Inform., 2013, vol. 13, iss. 3, pp. 45–52 (in Russian).
10. Romakina L. N. Analogs of a formula of Lobachevsky for angle of parallelism on the hyperbolic plane of positive curvature. Siberian Electronic Mathematical Reports, 2013, vol. 10, pp. 393–407 (in Russian). Available at: http ://semr.math.nsc.ru
11. Romakina L. N. Geometriia giperbolicheskoi ploskosti polozhitel’noi krivizny : v 4 ch. Ch. 1 : Trigonometriia [Geometry of the hyperbolic plane of positive curvature : in 4 pt. Pt. 1 : Trigonometry]. Saratov, Saratov Univ. Press., 244 p. (in ussian).
12. Efimov N. V. Vysshaia geometriia [The highest geometry]. Moscow, Nauka, 1971, 576 p. (in Russian).
Short text (in English): 
isu482.pdf
50
Full text:
download
75