The Alpha Distance Formulae

Year 2024, , 60 - 71, 31.07.2024

Harun Barış Çolakoğlu

https://doi.org/10.54974/fcmathsci.1312895

Abstract

In this study, we define the concept of tangent in two and three-dimensional alpha spaces concerning the alpha circles and the alpha spheres. Then using this concept, we derive the alpha distance formulae between points, a point and a line, between two lines and a point and a plane of the alpha spaces. Finally, we give simple area and volume formulas in the three dimensional space in terms of the alpha distances.

Keywords

Alpha metric, distance formula, metric geometry

References

• Akça Z., Kaya R., On the distance formulae in three dimensional taxicab space, Hadronic Journal, 27, 521-532, 2004.
• Chen G., Lines and Circles in Taxicab Geometry, M. Sc., Central Missouri State University, Missouri, USA, 1992.
• Çolakoğlu H.B., Concerning the alpha distance, Algebras, Groups, and Geometries, 8, 1-14, 2011.
• Çolakoğlu H.B., On the distance formulae in the generalized taxicab geometry, Turkish Journal of Mathematics, 43, 1578-1594, 2019.
• Çolakoğlu H.B., A generalization of the Minkowski distance and new definitions of the central conics, Turkish Journal of Mathematics, 44, 319-330, 2020.
• Çolakoğlu H.B., Gelişgen Ö., Kaya R., Area formulas for a triangle in the alpha plane, Mathematical Communications, 18, 123-132, 2013.
• Çolakoğlu H.B., Gelişgen Ö., Kaya R., Pythagorean theorems in the alpha plane, Mathematical Communications, 14, 211-221, 2009.
• Deza M.M., Deza E., Encyclopedia of Distances, Springer, 2009.
• Gelişgen Ö., Ermiş T., Inversions and fractal patterns in alpha plane, International Electronic Journal of Geometry, 16(1), 398-411, 2023.
• Gelişgen Ö., Kaya R., Özcan M., Distance formulae in the Chinese checker space, International Journal of Pure and Applied Mathematics, 26, 35-44, 2006.
• Gelişgen Ö., Kaya R., On α-distance in three dimensional space, APPS Applied Sciences, 8, 65-69, 2006.
• Gelişgen Ö., Kaya R., Generalization of α-distance to n-dimensional space, Croatian Society for Geometry and Graphics, 10, 33-35, 2006.
• Krause E.F., Taxicab Geometry, Dover, 1986.
• Menger K., You Will Like Geometry, A Guide Book for the Illinois Institute of Technology Geometry Exhibition, Museum of Science and Industry, 1961.
• Tian S., Alpha-distance - A generalization of Chinese checker distance and taxicab distance, Missouri Journal of Mathematical Sciences, 17, 35-40, 2005.
• Wallen L.J., Kepler, the taxicab metric, and beyond: An isoperimetric primer, The College Mathematics Journal, 26, 178-190, 1995.