SHAPE CURVATURES OF THE LORENTZIAN PLANE CURVES
Year 2017,
Volume: 66 Issue: 2, 276 - 288, 01.08.2017
Hakan Sımsek
Mustafa Özdemır
Abstract
In this paper, we examine the Lorentzian similar plane curvesusing the hyperbolic structure and spherical arc length parameter. We classifyall self-similar Lorentzian plane curves and give formulas for pseudo shapecurvatures of evolute, involute and parallel curves of a nonnull plane curve
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Year 2017,
Volume: 66 Issue: 2, 276 - 288, 01.08.2017
Hakan Sımsek
Mustafa Özdemır
References
- A. Gray, Modern Diğ erential Geometry of Curves and Surfaces, CRC Press, Boca Raton, A. Saloom and F. Tari, Curves in the Minkowski plane and their contact with pseudo-circles, Geometriae Dedicata (2012), 159:109-124.
- A. Schwenk-Schellscmidt, U. Simon, M Wiehe, Eigenvalue equations in curve theory Part I: characterization of conic sections, Results in Mathematics, 40, 273-285 (2001).
- B. B. Mandelbrot, The Fractal Geometry of Nature, New York: W. H. Freeman, 1983.
- B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., London, 1983.
- D. Hestenes, New Foundations for Classical Mechanics, Kluwer Academic Publisher, Second Edition, 1999.
- D. Hestenes, G. Sobczyk, Cliğ ord Algebra to Geometric Calculus: A Uni…ed Language for Mathematics and Physics, Kluwer Academic Publishing, Dordrecht, 1987.
- F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, P. Zampetti, Geometry of Minkowski Space- time, Springer Briefs in Physics, ISBN: 978-3-642-17977-8 (2011).
- H.B. Öztekin, M. Ergüt, Eigenvalue equations for Nonnull curve in Minkowski plane, Int. J. Open Probl. Compt. Math. 3, 467–480 (2010).
- H. Simsek, M. Özdemir, On Conformal Curves in 2-Dimensional de Sitter Space, Adv. Appl. Cliğord Algebras 26, 757–770 (2016).
- H. Simsek, M. Özdemir, Similar and Self-Similar Curves in Minkowski n-Space, Bull. Korean Math. Soc., 52 , No. 6, pp. 2071-2093 (2015).
- I. R. Porteous, Cliğ ord Algebras and Classical Groups, Cambridge: Cambridge University Press, ISBN 978-0-521-55177-3 (1995).
- J. E. Hutchinson, Fractals and Self-Similarity, Indiana University Mathematics Journal, Vol. , N:5, (1981).
- J. G. Alcázar, C. Hermosoa, G. Muntinghb, Detecting similarity of rational plane curves, Journal of Computational and Applied Mathematics, 269, 1–13 (2014).
- K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Second Edition, John Wiley & Sons, Ltd., 2003.
- KS. Chou, C. Qu, Integrable equations arising from motions of plane curves, Pysica D, 162 (2002), 9-33.
- KS. Chou, C. Qu, Motions of curves in similarity geometries and Burgers-mKdv hierarchies, Chaos, Solitons & Fractals 19 (2004), 47-53.
- M. Berger: Geometry I. Springer, New York 1998.
- M. K. Karacan, B. Bükcü, Parallel (Oğ set) Curves in Lorentzian Plane, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 24 (1-2), 334- 345 (2008).
- R. Encheva and G. Georgiev, Curves on the Shape Sphere, Results in Mathematics, 44 (2003), 288.
- S. Müller, A.Schwenk-Schellscmidt, U. Simon, Eigenvalue equations in curve theory Part II: Evolutes and Involutes, Results in Mathematics, 50, 109-124 (2007).