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Year 2019, Volume: 68 Issue: 1, 111 - 124, 01.02.2019
https://doi.org/10.31801/cfsuasmas.443657

Abstract

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.

On the Construction of Generalized Bobillier Formula

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

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.
There are 28 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Tülay Erisir 0000-0001-6444-1460

Mehmet Ali Güngör 0000-0003-1863-3183

Soley Ersoy This is me 0000-0002-7183-7081

Publication Date February 1, 2019
Submission Date April 24, 2016
Acceptance Date October 13, 2017
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Erisir, T., Güngör, M. A., & Ersoy, S. (2019). On the Construction of Generalized Bobillier Formula. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 111-124. https://doi.org/10.31801/cfsuasmas.443657
AMA Erisir T, Güngör MA, Ersoy S. On the Construction of Generalized Bobillier Formula. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):111-124. doi:10.31801/cfsuasmas.443657
Chicago Erisir, Tülay, Mehmet Ali Güngör, and Soley Ersoy. “On the Construction of Generalized Bobillier Formula”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 111-24. https://doi.org/10.31801/cfsuasmas.443657.
EndNote Erisir T, Güngör MA, Ersoy S (February 1, 2019) On the Construction of Generalized Bobillier Formula. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 111–124.
IEEE T. Erisir, M. A. Güngör, and S. Ersoy, “On the Construction of Generalized Bobillier Formula”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 111–124, 2019, doi: 10.31801/cfsuasmas.443657.
ISNAD Erisir, Tülay et al. “On the Construction of Generalized Bobillier Formula”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 111-124. https://doi.org/10.31801/cfsuasmas.443657.
JAMA Erisir T, Güngör MA, Ersoy S. On the Construction of Generalized Bobillier Formula. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:111–124.
MLA Erisir, Tülay et al. “On the Construction of Generalized Bobillier Formula”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 111-24, doi:10.31801/cfsuasmas.443657.
Vancouver Erisir T, Güngör MA, Ersoy S. On the Construction of Generalized Bobillier Formula. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):111-24.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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