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Some Results on Point-Line Trajectories in Lorentz 3-space

Year 2016, Volume: 9 Issue: 2, 44 - 49, 30.10.2016
https://doi.org/10.36890/iejg.584581

Abstract

In this paper, we study curvature theory of point-line trajectories in Lorentz 3-space. We give
the characterization by indicatrix, directrix and their relationship in Lorentz 3-space. We use this
characterization and relationship to depict a point-line trajectory.

References

  • [1] Aydoğmus¸, Ö., Kula, L. and Yaylı, Y., On point-line displacement in Minkowski 3-space. Differential Geometry-Dynamical Systems, 10(2008), 32-43.
  • [2] Bottema, O., and Roth, B., Theoretical Kinematics, Dover Publications, New York, 1990.
  • [3] Hunt, K. H., Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, England, 465 p., 1978.
  • [4] Kirson, Y., Higher order curvature theory in space kinematics. Ph.D dissertation, University of California at Berkeley, 140 p., 1975.
  • [5] Kim, J. H., Ruyh, B. S., and Pennock, G. R., Development of a Trajectory Generation Method for a Five-Axis NC Machine. Mech. Mach.Theory, 36 (2001), 983-996.
  • [6] McCarthy, J. M., and Roth, B., The Curvature Theory of Line Trajectories in Spatial Kinematics. ASME J. Mech. Des., 103 (1981), 718-724.
  • [7] McCarthy, J. M., The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion. ASME J. Mech., Transm., Autom. Des., 109 (1987), 98-100.
  • [8] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
  • [9] Ryuh, B. S., Robot trajectory planning using the curvature theory of ruled surfaces. Doctoral dissertation, Purdue University, West Lafayette, Ind, USA, 1989.
  • [10] Ryuh, B. S., and Pennock, G. R., Accurate Motion of a Robot End-Effector Using the Curvature Theory of Ruled Surface. ASME J. Mech.Transm. Autom. Des., 110 (1988), 383-388.
  • [11] Roth, B., Finding Geometric Invariants From Time-Based Invariants for Spherical and Spatial Motions. ASME Journal of Mechanical Design, 127 (2005), 227-231.
  • [12] Stachel, H., Instantaneous Spatial Kinematics and the Invariants of the Axodes, Proceedings Ball 2000 Symposium, Cambridge University Press, London, 23 (2000).
  • [13] Ting, K. L., Zhang, Y., and Bunduwongse, R., Characterization and Coordination of Point-line Trajectories. ASME Journal of Mechanical Design, 127 (2005), 502-505.
  • [14] Yücesan, A., Özkan, G., Generalized relaxed elastic line on a non-degenerate surface. International Conference: Mathematical Science andApplications, 26-30 December 2012, Abu Dhabi, UAE.
Year 2016, Volume: 9 Issue: 2, 44 - 49, 30.10.2016
https://doi.org/10.36890/iejg.584581

Abstract

References

  • [1] Aydoğmus¸, Ö., Kula, L. and Yaylı, Y., On point-line displacement in Minkowski 3-space. Differential Geometry-Dynamical Systems, 10(2008), 32-43.
  • [2] Bottema, O., and Roth, B., Theoretical Kinematics, Dover Publications, New York, 1990.
  • [3] Hunt, K. H., Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, England, 465 p., 1978.
  • [4] Kirson, Y., Higher order curvature theory in space kinematics. Ph.D dissertation, University of California at Berkeley, 140 p., 1975.
  • [5] Kim, J. H., Ruyh, B. S., and Pennock, G. R., Development of a Trajectory Generation Method for a Five-Axis NC Machine. Mech. Mach.Theory, 36 (2001), 983-996.
  • [6] McCarthy, J. M., and Roth, B., The Curvature Theory of Line Trajectories in Spatial Kinematics. ASME J. Mech. Des., 103 (1981), 718-724.
  • [7] McCarthy, J. M., The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion. ASME J. Mech., Transm., Autom. Des., 109 (1987), 98-100.
  • [8] O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
  • [9] Ryuh, B. S., Robot trajectory planning using the curvature theory of ruled surfaces. Doctoral dissertation, Purdue University, West Lafayette, Ind, USA, 1989.
  • [10] Ryuh, B. S., and Pennock, G. R., Accurate Motion of a Robot End-Effector Using the Curvature Theory of Ruled Surface. ASME J. Mech.Transm. Autom. Des., 110 (1988), 383-388.
  • [11] Roth, B., Finding Geometric Invariants From Time-Based Invariants for Spherical and Spatial Motions. ASME Journal of Mechanical Design, 127 (2005), 227-231.
  • [12] Stachel, H., Instantaneous Spatial Kinematics and the Invariants of the Axodes, Proceedings Ball 2000 Symposium, Cambridge University Press, London, 23 (2000).
  • [13] Ting, K. L., Zhang, Y., and Bunduwongse, R., Characterization and Coordination of Point-line Trajectories. ASME Journal of Mechanical Design, 127 (2005), 502-505.
  • [14] Yücesan, A., Özkan, G., Generalized relaxed elastic line on a non-degenerate surface. International Conference: Mathematical Science andApplications, 26-30 December 2012, Abu Dhabi, UAE.
There are 14 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Tunahan Turhan This is me

Vildan Özdemir This is me

Nihat Ayyıldız

Publication Date October 30, 2016
Published in Issue Year 2016 Volume: 9 Issue: 2

Cite

APA Turhan, T., Özdemir, V., & Ayyıldız, N. (2016). Some Results on Point-Line Trajectories in Lorentz 3-space. International Electronic Journal of Geometry, 9(2), 44-49. https://doi.org/10.36890/iejg.584581
AMA Turhan T, Özdemir V, Ayyıldız N. Some Results on Point-Line Trajectories in Lorentz 3-space. Int. Electron. J. Geom. October 2016;9(2):44-49. doi:10.36890/iejg.584581
Chicago Turhan, Tunahan, Vildan Özdemir, and Nihat Ayyıldız. “Some Results on Point-Line Trajectories in Lorentz 3-Space”. International Electronic Journal of Geometry 9, no. 2 (October 2016): 44-49. https://doi.org/10.36890/iejg.584581.
EndNote Turhan T, Özdemir V, Ayyıldız N (October 1, 2016) Some Results on Point-Line Trajectories in Lorentz 3-space. International Electronic Journal of Geometry 9 2 44–49.
IEEE T. Turhan, V. Özdemir, and N. Ayyıldız, “Some Results on Point-Line Trajectories in Lorentz 3-space”, Int. Electron. J. Geom., vol. 9, no. 2, pp. 44–49, 2016, doi: 10.36890/iejg.584581.
ISNAD Turhan, Tunahan et al. “Some Results on Point-Line Trajectories in Lorentz 3-Space”. International Electronic Journal of Geometry 9/2 (October 2016), 44-49. https://doi.org/10.36890/iejg.584581.
JAMA Turhan T, Özdemir V, Ayyıldız N. Some Results on Point-Line Trajectories in Lorentz 3-space. Int. Electron. J. Geom. 2016;9:44–49.
MLA Turhan, Tunahan et al. “Some Results on Point-Line Trajectories in Lorentz 3-Space”. International Electronic Journal of Geometry, vol. 9, no. 2, 2016, pp. 44-49, doi:10.36890/iejg.584581.
Vancouver Turhan T, Özdemir V, Ayyıldız N. Some Results on Point-Line Trajectories in Lorentz 3-space. Int. Electron. J. Geom. 2016;9(2):44-9.