Őren, İdrıs; Anil Coban, H. Some invariant properties of curves in the taxicab geometry. (English) Zbl 1311.51010 Missouri J. Math. Sci. 26, No. 2, 107-114 (2014). Summary: Let \(E^{2}_{T}\) be the group of all isometries of the \(2\)-dimensional taxicab space \(R^{2}_{T}\). For the taxicab group \(E^{2}_{T}\), the taxicab type of curves is introduced. All possible taxicab types are found. For every taxicab type, an invariant parametrization of a curve is described. The \(E^{2}_{T}\)-equivalence of curves is reduced to the problem of the \(E^{2}_{T}\)-equivalence of paths. Cited in 1 Document MSC: 51K05 General theory of distance geometry 51K99 Distance geometry 51N30 Geometry of classical groups 51F20 Congruence and orthogonality in metric geometry 53A55 Differential invariants (local theory), geometric objects 53A35 Non-Euclidean differential geometry Keywords:curve; taxicab geometry; invariant parametrization × Cite Format Result Cite Review PDF Full Text: Euclid References: [1] Z. Akca and R. Kaya, On the distance formulae in three dimensional taxicab space , Hadronic Journal, 27.5 (2004), 521-532. · Zbl 1078.51017 [2] Z. Akca and R. Kaya, On the norm in higher dimensional taxicab spaces , Hadronic J. Suppl., 19 (2004), 491-501. · Zbl 1093.51501 [3] R. G. Aripov and D. Khadjiev, The complete system of differential and integral invariants of a curve in Euclidean geometry , Russian Mathematics, 51.7 (2007), 1-14. · Zbl 1148.53302 · doi:10.3103/S1066369X07070018 [4] D. Caballero, Taxicab geometry: some problems and solutions for square grid-based fire spread simulation , V. International Conference on Forest Fire Research, 2006. [5] S. S. Chern, Curves and surfaces in Euclidean space , Global Diff. Geom., 27 (1989), 99-139. [6] Ő. Gelisken and R. Kaya, The taxicab space group , Acta Math.Hungar., 122.1-2 (2009), 187-200. · Zbl 1199.51002 · doi:10.1007/s10474-008-8006-9 [7] D. Khadjiev, An Application of Invariant Theory to Differential Geometry of Curves , Fan Publ., Tashkent, 1988. [Russian] [8] D. Khadjiev and Ö. Pekşen, The complete system of global integral and differential invariants for equi-Affine curves , Differ. Geom. Appl., (2004), no. 20, 167-175. · Zbl 1050.53016 · doi:10.1016/j.difgeo.2003.10.005 [9] E. F. Krause, Taxicab Geometry , Addison-Wesley, Menlo Park, 1975. [10] K. Menger, You Will Like Geometry, Guidebook for Illinois Institute of Technology Geometry Exhibit , Museum of Science and Industry, Chicago, III., 1952. [11] M. Őzcan, S. Ekmekci, and A. Bayar, A note on the variation of taxicab length under rotations , Pi Mu Epsilon Journal, 11.7 (Fall 2002), 381-384. [12] Y. Sagiroglu and Ö. Pekşen, The equivalence of centro-equi-affine curves , Turk. J. Math, 34 (2010), 95-104. [13] D. J. Schattschneider, The taxicab group , Amer. Math. Monthly, 97.7 (1984), 423-428. · Zbl 0564.51005 · doi:10.2307/2322995 [14] M. W. Sohn, Distance and cosine measures of niche overlap , Soc. Networks, 23 (2001), 141-165. [15] R. A. Struble, Non-linear Differential Equations , McGraw-Hill Comp., New York, 1962. · Zbl 0124.04904 [16] K. P. Thompson, The nature of length, area, and volume in taxicab geometry , Int. Electron. J. Geom., 4.2 (2011), 193-207. · Zbl 1308.51014 [17] D. L. Warren, R. E. Glor, and M. Turelli, Environmental niche equivalency versus conservatism: quantitative approaches to niche evolution , Evolution, 62.11 (2008), 2868-2883. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.