Research Article
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Year 2023, , 272 - 282, 30.04.2023
https://doi.org/10.36890/iejg.1270550

Abstract

References

  • Busemann, H., Kelly P.: Projective Geometry and Projective Metrics. Academic Press Inc. New York (1953).
  • Efimov, N.: Higher Geometry. MAIK. Nauka/Interperiodika. FIZMATLIT (2004).
  • Glaeser G., Stachel H., Odehnal B.: The Universe of quadrics, Springer Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-61053-4
  • Hodge, W., Pedoe, D.: Methods of Algebraic Geometry. Vol. I (Book II). Cambridge University Press (1994) [1947]. https://doi.org/10.1017/CBO9780511623875
  • Romakina, L.: Coordinates of the midpoints of non-parabolic segments of the hyperbolic plane of positive curvature in the canonical frame of the first type. Mathematics. Mechanics, \textbf{20}, 70-72 (2018).
  • Romakina, L.: Geometries of co-Euclidean and co-pseudo-Euclidean planes. Scientific book. Saratov (2008).
  • Romakina, L.: Geometry of the hyperbolic plane of positive curvature. P. 1: Trigonometry. Publishing house of the Saratov university. Saratov (2013).
  • Romakina, L., Kharchenko, A., Kharchenko, N.: \emph{Geometric constructions on the ideal domain of the Lobachevskii plane}. In: Materials of the International Forum on Mathematical Education dedicated to the 225th anniversary of N. I. Lobachevskii (MATHEDU\,--\,2017), Oct. 18\,--\,22/2017, Kazan, RUSSIA. Kazan (Volga Region) Federal University, 114-118 (2017).
  • Romakina, L.: On the area of a simple $4$-contour of a hyperbolic plane of positive curvature. In: Lomonosov readings in Altai: Collection of scientific articles of the International conference, Nov. 11-14/2014, Barnaul, RUSSIA. Altai State University, 346-353 (2014).
  • Romakina, L., Ushakov, I.: The Chaos game in the hyperbolic plane of positive curvature. In: Abstracts of Satellite International Conference on Nonlinear Dinamics $\&$ Integrability and Scientific School “Nonlinear days” (NDI\,--\,2022), June 27\,---\,July 1/2022, Yaroslavl, RUSSIA. YarSU, 86-88 (2022).
  • Rosenfel’d, B.: Noneuclidean spaces. Nauka. Moscow (1969).
  • Rosenfel’d, B.: Geometry of Lie Groups. Springer New York, NY (1997). https://doi.org/10.1007/978-1-4757-5325-7
  • Rosenfel’d, B., Zamakhovskii M.: Geometry of Lie groups. Symmetric, parabolic and periodic spaces. Moscow center for continuous mathematical education. Moscow (2003).
  • Shafarevich, I., Remizov, A.: Linear Algebra and Geometry. Springer. Berlin Heidelberg (2012). https://doi.org/10.1007/978-3-642-30994-6
  • pyv // github.com URL: https://github.com/MrReDoX/pyv (reference date: 24.03.2023).

Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space

Year 2023, , 272 - 282, 30.04.2023
https://doi.org/10.36890/iejg.1270550

Abstract

In this article, we find an analytical characteristic of the type of a line and derive the formulae for calculating the coordinates of the midpoints and quasi-midpoints of elliptic, hyperbolic, and parabolic segments in an extended hyperbolic space $H^3$ in the frame of the first type. The space $H^3$ we consider in the Cayley\,--\,Klein projective model as a projective three-dimensional space with an oval quadric $\gamma$ fixed in it.

References

  • Busemann, H., Kelly P.: Projective Geometry and Projective Metrics. Academic Press Inc. New York (1953).
  • Efimov, N.: Higher Geometry. MAIK. Nauka/Interperiodika. FIZMATLIT (2004).
  • Glaeser G., Stachel H., Odehnal B.: The Universe of quadrics, Springer Berlin, Heidelberg (2020). https://doi.org/10.1007/978-3-662-61053-4
  • Hodge, W., Pedoe, D.: Methods of Algebraic Geometry. Vol. I (Book II). Cambridge University Press (1994) [1947]. https://doi.org/10.1017/CBO9780511623875
  • Romakina, L.: Coordinates of the midpoints of non-parabolic segments of the hyperbolic plane of positive curvature in the canonical frame of the first type. Mathematics. Mechanics, \textbf{20}, 70-72 (2018).
  • Romakina, L.: Geometries of co-Euclidean and co-pseudo-Euclidean planes. Scientific book. Saratov (2008).
  • Romakina, L.: Geometry of the hyperbolic plane of positive curvature. P. 1: Trigonometry. Publishing house of the Saratov university. Saratov (2013).
  • Romakina, L., Kharchenko, A., Kharchenko, N.: \emph{Geometric constructions on the ideal domain of the Lobachevskii plane}. In: Materials of the International Forum on Mathematical Education dedicated to the 225th anniversary of N. I. Lobachevskii (MATHEDU\,--\,2017), Oct. 18\,--\,22/2017, Kazan, RUSSIA. Kazan (Volga Region) Federal University, 114-118 (2017).
  • Romakina, L.: On the area of a simple $4$-contour of a hyperbolic plane of positive curvature. In: Lomonosov readings in Altai: Collection of scientific articles of the International conference, Nov. 11-14/2014, Barnaul, RUSSIA. Altai State University, 346-353 (2014).
  • Romakina, L., Ushakov, I.: The Chaos game in the hyperbolic plane of positive curvature. In: Abstracts of Satellite International Conference on Nonlinear Dinamics $\&$ Integrability and Scientific School “Nonlinear days” (NDI\,--\,2022), June 27\,---\,July 1/2022, Yaroslavl, RUSSIA. YarSU, 86-88 (2022).
  • Rosenfel’d, B.: Noneuclidean spaces. Nauka. Moscow (1969).
  • Rosenfel’d, B.: Geometry of Lie Groups. Springer New York, NY (1997). https://doi.org/10.1007/978-1-4757-5325-7
  • Rosenfel’d, B., Zamakhovskii M.: Geometry of Lie groups. Symmetric, parabolic and periodic spaces. Moscow center for continuous mathematical education. Moscow (2003).
  • Shafarevich, I., Remizov, A.: Linear Algebra and Geometry. Springer. Berlin Heidelberg (2012). https://doi.org/10.1007/978-3-642-30994-6
  • pyv // github.com URL: https://github.com/MrReDoX/pyv (reference date: 24.03.2023).
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Lyudmila N. Romakina 0000-0002-3695-2076

Publication Date April 30, 2023
Acceptance Date April 19, 2023
Published in Issue Year 2023

Cite

APA Romakina, L. N. (2023). Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space. International Electronic Journal of Geometry, 16(1), 272-282. https://doi.org/10.36890/iejg.1270550
AMA Romakina LN. Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space. Int. Electron. J. Geom. April 2023;16(1):272-282. doi:10.36890/iejg.1270550
Chicago Romakina, Lyudmila N. “Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space”. International Electronic Journal of Geometry 16, no. 1 (April 2023): 272-82. https://doi.org/10.36890/iejg.1270550.
EndNote Romakina LN (April 1, 2023) Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space. International Electronic Journal of Geometry 16 1 272–282.
IEEE L. N. Romakina, “Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space”, Int. Electron. J. Geom., vol. 16, no. 1, pp. 272–282, 2023, doi: 10.36890/iejg.1270550.
ISNAD Romakina, Lyudmila N. “Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space”. International Electronic Journal of Geometry 16/1 (April 2023), 272-282. https://doi.org/10.36890/iejg.1270550.
JAMA Romakina LN. Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space. Int. Electron. J. Geom. 2023;16:272–282.
MLA Romakina, Lyudmila N. “Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space”. International Electronic Journal of Geometry, vol. 16, no. 1, 2023, pp. 272-8, doi:10.36890/iejg.1270550.
Vancouver Romakina LN. Coordinates of the Midpoint of a Segment in an Extended Hyperbolic Space. Int. Electron. J. Geom. 2023;16(1):272-8.