On the Construction of Generalized Bobillier Formula

Year 2019, Volume: 68 Issue: 1, 111 - 124, 01.02.2019

Tülay Erisir , Mehmet Ali Güngör , Soley Ersoy

https://doi.org/10.31801/cfsuasmas.443657

Abstract

In this study, we consider the generalized complex number system C_{p}={x+iy:x,y∈R,i²=p∈R} corresponding to elliptical complex number, parabolic complex number and hyperbolic complex number systems for the special cases of p<0,[kern]<LaTeX>\kern</LaTeX>1pt[kern]<LaTeX>\kern</LaTeX>1ptp=0,[kern]<LaTeX>\kern</LaTeX>1pt[kern]<LaTeX>\kern</LaTeX>1ptp>0, respectively. This system is used to derive Bobillier Formula in the generalized complex plane. In accordance with this purpose we obtain this formula by two different methods for one-parameter planar motion in C_{p}; the first method depends on using the geometrical interpretation of the generalized Euler-Savary formula and the second one uses the usual relations of the velocities and accelerations.

Keywords

Bobillier Formula, Euler-Savary Formula, Generalized Complex Numbers

References

• Akbıyık, M. and Yüce, S., Euler-Savary's Formula on Galilean Plane, The Algerian-Turkish International days on Mathematics, 1(2012), no. 1, 133.
• Akbıyık, M. and Yüce, S., Euler Savary's Formula on Complex Plane C^{*}, Applied Mathematics E-Notes, 16(2016), 65--71.
• Blaschke, W. and Müller, H. R., Ebene Kinematik, Verlag Oldenbourg, München, 1956.
• Bottema, O. and Roth, B., Theoretical Kinematics North-Holland Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam, 1979.
• Buckley, R. and Whitfield, E. V., The Euler Savary Formula, The Mathematical Gazette 33(1949), no. 306, 297--299.
• Dündar, F. S., Ersoy, S. and Sá Pereira, N. T., Bobillier Formula for the Elliptical Harmonic Motion, An. St. Univ. Ovidius Constanta, accepted.
• Ersoy, S. and Akyigit, M., One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula, Adv. Appl. Clifford Algebr., 21 (2011), 297--313.
• Erişir, T., Güngör, M. A. and Tosun, M., A Generalised Method for Centres of Trajectories in Kinematics, Journal of Advanced Research in Applied Mathematics, DOI: 10.5373/jaram (2016), no: 8, 1--18.
• Ersoy, S. and Bayrak, N., Bobillier Formula for One Parameter Motions in the Complex Plane, J. Mechanisms Robotics, 4(2012), no. 2, 024501-1-024501-4.
• Ersoy, S. and Bayrak, N., Lorentzian Bobillier Formula, Appl. Math. E-Notes, 13(2013), 25--35.
• Fayet, M., Une Nouvelle Formule Relative aux Courbures Dans un Mouvement Plan Nous Proposons de I'appeler: Formule De Bobillier, Mech. Mach. Theory, 23(1988), no. 2, 135--139.
• Fayet, M., Bobillier Formula as a Fundamental Law in Planar Motion, Z. Angew. Math. Mech., 82(2002), no. 3, 207--210.
• Gürses, N., Akbıyık, M. and Yüce, S., Galilean Bobillier Formula for One-Parameter Motions, International Journal of Mathematical Combinatorics, 4 (2015), 74--83.
• Gürses, N. and Yüce, S., Euler-Savary Formula for One-Parameter Motions in Affine Cayley-Klein Planes, 13. Geometry Symposium, Istanbul, Turkey, 27-30 July 2015, pp.20.
• Gürses, N. and Yüce, S., One-Parameter Planar Motions in Generalized Complex Number Plane C_{J}, Adv. Appl. Clifford Algebr. 25(2015), no. 4, 889--903.
• Gürses, N., Akbıyık, M. and Yüce, S., One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane C_{J}, Adv. Appl. Clifford Algebr. 26(2016), no. 1, 115--136.
• Harkin, A. A. and Harkin, J. B., Geometry of Generalized Complex Numbers, Math. Mag., 77(2004), no. 2, 118--129.
• Hunt, K. H., Kinematic Geometry of Mechanisms, Oxford University Press, New York, 1978.
• Klein, F., Über die sogenante nicht-Euklidische Geometrie, Gesammelte Mathematische Abhandlungen, (1921), 254--305.
• Klein, F., Vorlesungen über nicht-Euklidische Geometrie, Springer, Berlin, 1928.
• Koetsier, T., Euler and Kinematics, Leonhard Euler: Life Work and Legacy, Elseiver, (2007), 167--194.
• Masal, M., Tosun, M. and Pirdal, A. Z., Euler Savary Formula for the One Parameter Motions in the Complex Plane C, Int. J. Phys. Sci., 5(2010), no. 1, 6--10.
• Muminagić, A., Bobillierova formula, Osjećka Matematićka Śkola, 4(2004), 77--81.
• Müller, H. R., Kinematik, Sammlung Göschen, Walter de Gruyter, Berlin, 1963.
• Sandor, G. N., Erdman, A. G., Hunt, L., and Raghavacharyulu, E., New Complex-Number Forms of the Euler-Savary Equation in a Computer-Oriented Treatment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact, Asme J. Mech Des., 104(1982), 227--238.
• Sandor, G. N., Arthur, G. E. and Raghavacharyulu, E., Double Valued Solution of the Euler-Savary Equation and Its Counterpart in Bobillier's Construction, Mech. Mach. Theory, 20(1985), no. 2, 145--178.
• Yaglom, I. M., Complex Numbers in Geometry, Academic, Press, New York, 1968.
• Yaglom, I. M., A Simple non-Euclidean Geometry and its Physical Basis, Springer-Verlag, New-York, 1979.