On the Generalized Poincare Distance

Yıl 2023, Cilt: 5 Sayı: 2, 1 - 5, 30.12.2023

Ayşe Bayar Sezin Cirdi Şaan

Öz

In this work, the concept of the generalized Poincaré distance is given and the distance between two points on vertical lines, horizontal lines and semi-ellipses in the upper half-plane are examined. It is also shown that translations parallel to the x-axis and reflections in the vertical lines preserve the generalized Poincaré distance in the upper half-plane.

Anahtar Kelimeler

Poincare distance, the upper half-plane, hyperbolic plane

Kaynakça

• Stahl, S. (1993). The Poincare half-plane: A Gateway to modern geometry. Jones & Bartlett Learning.
• Anderson, J. W. (2005). Hyperbolic geometry. Springer Undergraduate Mathematics Series, Springer-Verlag. https://doi.org/10.1007/1-84628-220-9
• Greenberg, M. J. (1993). Euclidean and non-Euclidean geometries: Development and history. Macmillan.
• Kaya, R. (2022). Generalized Poincare half-planes. arXiv preprint, arXiv:1904.01899. https://doi.org/10.48550/arXiv.1904.01899.
• Bayar, A., Ekmekçi, S., & Akça, Z. (2008). On the plane geometry with generalized absolute value metric. Mathematical Problems in Engineering, Article ID 673275, 1-8.
• Çolakoglu, H. B., & Kaya, R. (2011). A generalization of some well-known distances and related isometries. Mathematical Communications, 16(1), 21-35.
• Ekmekçi, S., Bayar, A., & Altıntaş, A. K. (2015). On the group of isometries of the generalized taxicab plane. International Journal of Contemporary Mathematical Sciences, 10(4), 159-166.
• Ekmekçi, S., Akça, Z., & Altıntaş, K. (2015). On trigonometric functions and norm in the generalized taxicab metric Mathematical Sciences and Applications E-Notes, 3(2), 27-33.
• Kaya, R., Gelişgen, Ö., Ekmekçi, S., & Bayar, A. (2009). On the group of isometries of the plane with generalized absolute value metric. The Rocky Mountain Journal of Mathematics, 39(2), 591-603.