Year 2019, Volume 7, Issue 1, Pages 228 - 232 2019-04-15

A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$

Mücahit Akbıyık [1] , Nurten Gürses [2] , Salim Yüce [3]

31 64

In this study, we define a base curve, a rolling curve and a roulette on generalized complex number plane ($\mathfrak{p}$-complex plane) $\mathbb{C}_{J}$. We examine the third one of these curves under the condition that two others given. We also re-obtain

the Euler Savary's formula in $\mathbb{C}_{J}$ as a generalization of the Euler Savary's formula for complex plane $\mathbb{C}$, hyperbolic plane $\mathbb{H}$ and dual plane $\mathbb{D}$.



base curve, rolling curve, roulette, associated curve, Generalized complex number plane
  • [1] Fjelstad, P., Extending special relativity via the perplex numbers. Amer. J. Phys. 54 no.5 (1986), 416–422.
  • [2] Alfsmann, D., On families of 2N dimensional Hypercomplex Algebras suitable for digital signal Processing. Proc. EURASIP 14th European Signal Processing Conference(EUSIPCO 2006), Florence, Italy, 2006.
  • [3] Yaglom, I.M., Complex numbers in geometry, Academic Press, New York, 1968.
  • [4] Y¨uce S., and N. Kuruo˘glu, One-Parameter Plane Hyperbolic Motions, Adv. appl. Clifford alg. 18 (2008), 279-285.
  • [5] P. Fjelstad and S.G. Gal, n-dimensional hyperbolic complex numbers, Adv. Appl. Clifford Algebr. 8 no.1 (1998), 47-68.
  • [6] P. Fjelstad and S.G. Gal, Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers, Adv. Appl. Clifford Algebra 11 no. 1 (2001) 81–107.
  • [7] D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, Anal. Univ. Oradea, fasc. math. 11 (2004), 71–110.
  • [8] F. Catoni, R. Cannata, V. Catoni and P. Zampetti, Hyperbolic Trigonometry in two-dimensional space-time geometry, N. Cim. B 118 B (2003).
  • [9] E. Study, Geometrie der Dynamen, Verlag Teubner, Leipzig, 1903.
  • [10] A. A. Harkin and J. B. Harkin, Geometry of Generalized Complex Numbers, Mathematics Magazine, 77 no. 2 (2014).
  • [11] F. Catoni, D. Boccaletti, , R. Cannata, V. Catoni, E. Nichelatti and P. Zampetti, The mathematics of Minkowski space-time and an introduction to commutative hypercomplex numbers, Birkh auser Verlag, Basel, 2008.
  • [12] I. M. Yaglom, A simple non-Euclidean geometry and its Physical Basis, Springer-Verlag, New York, 1979.
  • [13] E. Pennestri and R. Stefanelli, Linear Algebra and Numerical Algorithms Using Dual Numbers, Multibody System Dynamics, 18 no. 3 (2007), 323–344.
  • [14] G. Sobczyk, The Hyperbolic Number Plane, The College Math. J. 26 no. 4 (1995), 268–280.
  • [15] F. Klein, Uber die sogenante nicht-Euklidische Geometrie, Gesammelte Mathematische Abhandlungen, (1921), 254–305.
  • [16] F. Klein, Vorlesungen ¨uber nicht-Euklidische Geometrie, Springer, Berlin, 1928.
  • [17] H. Es, Motions and Nine Different Geometry, PhD Thesis, Ankara University Graduate School of Natural and Applied Sciences, 2003.
  • [18] G. Helzer,Special Relativity with Acceleration, The American Mathematical Monthly, 107 no. 3 (2000), 219-237.
  • [19] R. Salgado, Space-Time Trigonometry, AAPT Topical Conference: Teaching General Relativity to Undergraduates, AAPT Summer Meeting, Syrauce University, NY, July 20-21,22-26 2006.
  • [20] M. A. F. Sanjuan, Group Contraction and Nine Cayley-Klein Geometries, International Journal of Theoretical Physics, 23(1) (1984).
  • [21] V. V. Kisil, Geometry of M¨obius Transformations:Eliptic, Parabolic and Hyperbolic Actions of SL2 (R), Imperial College Press, London, 2012.
  • [22] N. A. Gromov and V. V. Kuratov, Possible quantum kinematics, J. Math. Phys. 47 no. 1 (2006).
  • [23] J.C. Alexander , Maddocks J.H., On the Maneuvering of Vehicles, SIAM J. Appl. Math. 48(1): 38-52,1988.
  • [24] R. Buckley, E.V. Whitfield, The Euler-Savary Formula, The Mathematical Gazette 33(306): 297-299, 1949.
  • [25] D.B. Dooner, M.W. Griffis, On the Spatial Euler-Savary Equations for Envelopes, J. Mech. Design 129(8): 865-875, 2007.
  • [26] Ito N., Takahashi K., Extension of the Euler-Savary Equation to Hypoid Gears, JSME Int.Journal. Ser C. Mech Systems 42(1): 218-224, 1999.
  • [27] G.R. Pennock, N.N. Raje, Curvature Theory for the Double Flier Eight-Bar Linkage, Mech. Theory 39: 665-679, 2004.
  • [28] S. Ersoy, M. Akyigit, One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula, Adv. Appl. Clifford Algebras, Vol. 21 ,pp. 297 - 313 , 2011 .
  • [29] D. L. Wang, J. Liu, D. Xaio, A unified Curvature Theory in kinematic geometry of mechanism, Science in China (Series E). 41(2): 196-202, 1998.
  • [30] W. Blaschke, H.R. M¨uller, Ebene Kinematik, Verlag Oldenbourg, M¨unchen, 1959.
  • [31] O. R¨oschel, Zur Kinematik der isotropen Ebene., Journal of Geometry, 21, 146–156,1983.
  • [32] I. Aytun, Euler-Savary formula for one-parameter Lorentzian plane motion and its Lorentzian geometrical interpretation, M.Sc. Thesis, Celal Bayar University, 2002.
  • [33] T. Ikawa, Euler-Savary’s formula on Minkowski geometry, Balkan Journal of Geometry and Its Applications, 8 (2), 31-36, 2003.
  • [34] M. Akbıyık, S. Y¨uce, The Moving Coordinate System And Euler-Savary’s Formula For The One Parameter Motions On Galilean (Isotropic) Plane, International Journal of Mathematical Combinatorics, vol.2015, pp.88-105, 2015.
  • [35] M. Akbıyık, S. Y¨uce, A Note On Euler Savary’s Formula On Galilean Plane, IWBCSM-2013 1st International Western Balkans Conference of Mathematical Sciences, Tiran, Albania, May 30-June 1, 2013, pp.173-174
  • [36] M. Masal, M. Tosun, A. Z. Pirdal, Euler Savary formula for the one parameter motions in the complex plane C; International Journal of Physical Sciences, 5 (1), 006-010, 2010.
  • [37] M. Akbıyık, S. Y¨uce, Euler-Savary’s Formula on Complex Plane C, Applied Mathematics E-Notes, vol.2016, pp.65-71, 2016.
  • [38] M. Akbıyık, S. Y¨uce, Euler-Savary’s Formula on The Dual Plane D, 13th Algebraic Hyperstructures and its Applications Conference (AHA2017), Istanbul, Turkey, 24-27 July 2017, pp.119.
  • [39] N. Gurses, M. Akbıyık, S. Y¨uce, One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane CJ , Adv. Appl. Clifford Algebras, 26(2016), 115-136.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Author: Mücahit Akbıyık
Country: Turkey


Author: Nurten Gürses
Country: Turkey


Author: Salim Yüce
Country: Turkey


Dates

Publication Date: April 15, 2019

Bibtex @research article { konuralpjournalmath347776, journal = {Konuralp Journal of Mathematics (KJM)}, issn = {}, eissn = {2147-625X}, address = {Mehmet Zeki SARIKAYA}, year = {2019}, volume = {7}, pages = {228 - 232}, doi = {}, title = {A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane \$\\mathbb\{C\}\_\{J\}\$}, key = {cite}, author = {Akbıyık, Mücahit and Gürses, Nurten and Yüce, Salim} }
APA Akbıyık, M , Gürses, N , Yüce, S . (2019). A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$. Konuralp Journal of Mathematics (KJM), 7 (1), 228-232. Retrieved from http://dergipark.org.tr/konuralpjournalmath/issue/31492/347776
MLA Akbıyık, M , Gürses, N , Yüce, S . "A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$". Konuralp Journal of Mathematics (KJM) 7 (2019): 228-232 <http://dergipark.org.tr/konuralpjournalmath/issue/31492/347776>
Chicago Akbıyık, M , Gürses, N , Yüce, S . "A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$". Konuralp Journal of Mathematics (KJM) 7 (2019): 228-232
RIS TY - JOUR T1 - A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$ AU - Mücahit Akbıyık , Nurten Gürses , Salim Yüce Y1 - 2019 PY - 2019 N1 - DO - T2 - Konuralp Journal of Mathematics (KJM) JF - Journal JO - JOR SP - 228 EP - 232 VL - 7 IS - 1 SN - -2147-625X M3 - UR - Y2 - 2019 ER -
EndNote %0 Konuralp Journal of Mathematics (KJM) A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$ %A Mücahit Akbıyık , Nurten Gürses , Salim Yüce %T A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$ %D 2019 %J Konuralp Journal of Mathematics (KJM) %P -2147-625X %V 7 %N 1 %R %U
ISNAD Akbıyık, Mücahit , Gürses, Nurten , Yüce, Salim . "A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$". Konuralp Journal of Mathematics (KJM) 7 / 1 (April 2019): 228-232.
AMA Akbıyık M , Gürses N , Yüce S . A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$. Konuralp J. Math.. 2019; 7(1): 228-232.
Vancouver Akbıyık M , Gürses N , Yüce S . A Survey on the Base, Rolling and Roulette Curves on Generalized Complex Number Plane $\mathbb{C}_{J}$. Konuralp Journal of Mathematics (KJM). 2019; 7(1): 232-228.