• Eren, Kemal (Department of Mathematics, Faculty of Arts and Sciences, Sakarya University) ;
  • Ersoy, Soley (Department of Mathematics, Faculty of Arts and Sciences, Sakarya University)
  • Received : 2017.12.18
  • Accepted : 2018.02.13
  • Published : 2018.03.25


In this paper, we study the instantaneous geometric properties of motion of rigid bodies in the Lorentzian plane. For this purpose we define Lorentzian form of Bottemas instantaneous invariants. In these regards, we obtain the necessary and sufficient condition of a Lorentzian plane to be at Cardan position with respect to these invariants.


Lorentzian plane motion;instantaneous invariants;Cardan positions


  1. H. Dorrie, 100 Great Problems of Elementary Mathematics: Their History and Solutions, New York: Dover, 1965.
  2. K. Rauh, H. Marks, M. Bundgens and K. Otto, Kardanbewegung und Koppelbewegung (Cardan Motion and Coupler Motion), Schriftenreihe Praktische Getriebetechnik, Heft 2, Berlin: VDI-Verlag, 1938.
  3. P. de. La Hire, Traite des Roulettes, Academie des Sciences, Memoires, (edition of 1707, 350-352.), (1706), 340-349.
  4. C. B. Boyer, Note on Epicycles & Ellipse from Copernicus to Lahire, Isis, 38(1/2) (1947), 54-56.
  5. N. N. Buchholz, Basic Course of Theoretical Mechanics, Vol. 1; Kinematics, Statics, Dynamics of a Material Point (in Russian). Science, Moscow, 1965.
  6. G. Glaeser, H. Stachel, Open geometry: OpenGL + Advanced geometry, Springer, New York, 1999.
  7. O. Bottema and B. Roth, Theoretical Kinematics, North-Holland, Amsterdam, 1979.
  8. O. Bottema, On Cardan Positions for the Plane Motion of a Rigid Body, Indag. Math., V. XI, Fasc. 3, 1949.
  9. F. Freudenstein, The Cardan Positions of a Plane, Trans. Sixth Conf. Mechanisms, Purdue University, West Lafayette, Ind., October 1960, 129-133.
  10. O. Bottema, Cardan Motion in Elliptic Geometry, Can. J. Math. XXVII (1) (1975), 37-43.
  11. C. C. Lee, J. M. Herve, On the Helical Cardan Motion and Related Paradoxical Chains, Mech. Mach. Theory, 52(2012) 94-105.
  12. M. Ergut, A. P. Aydin and N. Bildik, The Geometry of the Canonical Relative System and One-Parameter Motions in 2-Lorentzian Space, The J. of Firat Uni., 3(1)(1988), 113-122.
  13. A. A. Ergin, On the One-parameter Lorentzian Motion, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 40(1991), 59-66.
  14. A. Tutar, N. Kuruoglu, and M. Duldul, On the Moving Coordinate System and Pole Points on the Lorentzian Plane, Int. J. of Appl. Math., 7(4)(2001), 439-445.
  15. M. A. Gungor, A. Z. Pirdal and M. Tosun, Euler-Savary Formula for the Lorentzian Planar Homothetic Motions, Int. J. Math. Comb. 2(2010) 102-111.
  16. A. G. Horvath, Constructive Curves in Non-Euclidean Planes, Stud. Univ. Zilina, 28(2016), 13-42.
  17. V. Balestro, A. G. Horvath, H. Martini, Angle Measures, General Rotations, and Roulettes in Normed Planes, Anal. Math. Phys., 7(4) (2017), 549-575.
  18. B. O'Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983
  19. E. Nesovic, Hyperbolic Angle Function in the Lorentzian Plane, Kragujevac J. Math. 28 (2005), 139-144.