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Ruled surfaces corresponding to hyper-dual curves

Year 2022, Volume: 51 Issue: 1, 187 - 198, 14.02.2022

Selahattin Aslan Murat Bekar Yusuf Yaylı

https://doi.org/10.15672/hujms.988245
Cited By: 1

Abstract

In this paper, we give the definition of the concept of unit hyper-dual sphere. We take a subset of this sphere and show that each curve on this subset represents two ruled surfaces in three dimensional real vector space such that these ruled surfaces have a common base curve and their rulings are perpendicular. Finally, we give some examples to illustrate the applications of our main results. 

Keywords

Hyper-dual numbers Unit hyper-dual sphere Hyper-dual curves Ruled surfaces Principal curve

References

  • [1] O.P. Agrawal, Hamilton operators and dual-number-quaternions in spatial kinematics, Mech. Mach. Theory, 22 (6), 569-575, 1987.
  • [2] W.K. Clifford, Preliminary sketch of biquaternions, Proc. London Math. Soc. 4 (64), 381-395, 1873.
  • [3] A. Cohen and M. Shoham, Application of hyper-dual numbers to multi-body kinemat- ics, J. Mech. Robot., 8 (1), 011015, (4 pages), 2016.
  • [4] A. Cohen and M. Shoham, Application of hyper-dual numbers to rigid bodies equations of motion, Mech. Mach. Theory 111, 76-84, 2017.
  • [5] A. Cohen and M. Shoham, Principle of transference-An extension to hyper-dual num- bers, Mech. Mach. Theory 125, 101-110, 2018.
  • [6] J.A. Fike, Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, 2009.
  • [7] J.A. Fike and J.J. Alonso, The development of hyper-dual numbers for exact second- derivative calculations, 49th AIAA Aerodpace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 4-7, 2011.
  • [8] J.A. Fike and J.J. Alonso, Automatic differentiation through the use of hyper-dual numbers for second derivatives, in: Lecture Notes in Computational Science and Engineering, 87 (201), 163-173, 2011.
  • [9] J.A. Fike, S. Jongsma, J.J. Alonso, and E. van der Weida, Optimization with gradi- ent and hessian information calculated using hyper-dual numbers, 29 AIAA Applied Aerodynamics Conference, 2011.
  • [10] F. Hathout, M. Bekar, and Y. Yaylı, Ruled surfaces and tangent bundle of unit 2- sphere, Int. J. Geom. Methods Mod. Phys., 14 (10), 1750145, 2017.
  • [11] A.P. Kotelnikov, Screw calculus and some applications to geometry and mechanics, Annal. Imp. Univ. Kazan, Russia, 1895.
  • [12] J.M. McCarthy, The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion, ASME J. Mech., Transm., Autom. Des. 109 (1), 95-100, 1987.
  • [13] J.M. McCarthy, On the Scalar and Dual Formulations of the Curvature Theory of Line Trajectories, ASME J. Mech., Transm., Autom. Des. 109, 101-106, 1987.
  • [14] Y.S. Oh, P. Abhishesh, and B.S. Ryuh, Study on Robot Trajectory Planning by Robot EndEffector Using Dual Curvature Theory of the Ruled Surface, World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering 11 (3), 577-582, 2017.
  • [15] B. O’Neill, Elementary Differential Geometry, Revised 2nd edition, Academic Press, USA, 2006.
  • [16] M. Önder and H.H. Uğurlu, Normal and Spherical Curves in Dual Space $\mathbb{D}^{3}$, Mediterr. J. Math. 10, 1527-1537, 2013.
  • [17] B.S. Ryuh and G.R. Pennock, Accurate motion of a robot end-effector using the curva- ture theory of ruled surfaces, Journal of Mechanisms, Transmissions and Automation in Design 110 (4), 383-388, 1988.
  • [18] K. Sprott and B. Ravani, Kinematic generation of ruled surfaces, Adv. Comput. Math. 17, 115-133, 2002.
  • [19] E. Study, Geometry der Dynamen, Leipzig, 1901.
  • [20] G.R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mech. Mach. Theory 11 (2), 141-156, 1976.

Year 2022, Volume: 51 Issue: 1, 187 - 198, 14.02.2022

Selahattin Aslan Murat Bekar Yusuf Yaylı

https://doi.org/10.15672/hujms.988245
Cited By: 1

Abstract

References

  • [1] O.P. Agrawal, Hamilton operators and dual-number-quaternions in spatial kinematics, Mech. Mach. Theory, 22 (6), 569-575, 1987.
  • [2] W.K. Clifford, Preliminary sketch of biquaternions, Proc. London Math. Soc. 4 (64), 381-395, 1873.
  • [3] A. Cohen and M. Shoham, Application of hyper-dual numbers to multi-body kinemat- ics, J. Mech. Robot., 8 (1), 011015, (4 pages), 2016.
  • [4] A. Cohen and M. Shoham, Application of hyper-dual numbers to rigid bodies equations of motion, Mech. Mach. Theory 111, 76-84, 2017.
  • [5] A. Cohen and M. Shoham, Principle of transference-An extension to hyper-dual num- bers, Mech. Mach. Theory 125, 101-110, 2018.
  • [6] J.A. Fike, Numerically exact derivative calculations using hyper-dual numbers, 3rd Annual Student Joint Workshop in Simulation-Based Engineering and Design, 2009.
  • [7] J.A. Fike and J.J. Alonso, The development of hyper-dual numbers for exact second- derivative calculations, 49th AIAA Aerodpace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 4-7, 2011.
  • [8] J.A. Fike and J.J. Alonso, Automatic differentiation through the use of hyper-dual numbers for second derivatives, in: Lecture Notes in Computational Science and Engineering, 87 (201), 163-173, 2011.
  • [9] J.A. Fike, S. Jongsma, J.J. Alonso, and E. van der Weida, Optimization with gradi- ent and hessian information calculated using hyper-dual numbers, 29 AIAA Applied Aerodynamics Conference, 2011.
  • [10] F. Hathout, M. Bekar, and Y. Yaylı, Ruled surfaces and tangent bundle of unit 2- sphere, Int. J. Geom. Methods Mod. Phys., 14 (10), 1750145, 2017.
  • [11] A.P. Kotelnikov, Screw calculus and some applications to geometry and mechanics, Annal. Imp. Univ. Kazan, Russia, 1895.
  • [12] J.M. McCarthy, The Instantaneous Kinematics of Line Trajectories in Terms of a Kinematic Mapping of Spatial Rigid Motion, ASME J. Mech., Transm., Autom. Des. 109 (1), 95-100, 1987.
  • [13] J.M. McCarthy, On the Scalar and Dual Formulations of the Curvature Theory of Line Trajectories, ASME J. Mech., Transm., Autom. Des. 109, 101-106, 1987.
  • [14] Y.S. Oh, P. Abhishesh, and B.S. Ryuh, Study on Robot Trajectory Planning by Robot EndEffector Using Dual Curvature Theory of the Ruled Surface, World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering 11 (3), 577-582, 2017.
  • [15] B. O’Neill, Elementary Differential Geometry, Revised 2nd edition, Academic Press, USA, 2006.
  • [16] M. Önder and H.H. Uğurlu, Normal and Spherical Curves in Dual Space $\mathbb{D}^{3}$, Mediterr. J. Math. 10, 1527-1537, 2013.
  • [17] B.S. Ryuh and G.R. Pennock, Accurate motion of a robot end-effector using the curva- ture theory of ruled surfaces, Journal of Mechanisms, Transmissions and Automation in Design 110 (4), 383-388, 1988.
  • [18] K. Sprott and B. Ravani, Kinematic generation of ruled surfaces, Adv. Comput. Math. 17, 115-133, 2002.
  • [19] E. Study, Geometry der Dynamen, Leipzig, 1901.
  • [20] G.R. Veldkamp, On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mech. Mach. Theory 11 (2), 141-156, 1976.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Selahattin Aslan 0000-0001-5322-3265

Murat Bekar 0000-0002-0382-430X

Yusuf Yaylı 0000-0003-4398-3855

Publication Date February 14, 2022
Published in Issue Year 2022 Volume: 51 Issue: 1

Cite

APA Aslan, S., Bekar, M., & Yaylı, Y. (2022). Ruled surfaces corresponding to hyper-dual curves. Hacettepe Journal of Mathematics and Statistics, 51(1), 187-198. https://doi.org/10.15672/hujms.988245
AMA Aslan S, Bekar M, Yaylı Y. Ruled surfaces corresponding to hyper-dual curves. Hacettepe Journal of Mathematics and Statistics. February 2022;51(1):187-198. doi:10.15672/hujms.988245
Chicago Aslan, Selahattin, Murat Bekar, and Yusuf Yaylı. “Ruled Surfaces Corresponding to Hyper-Dual Curves”. Hacettepe Journal of Mathematics and Statistics 51, no. 1 (February 2022): 187-98. https://doi.org/10.15672/hujms.988245.
EndNote Aslan S, Bekar M, Yaylı Y (February 1, 2022) Ruled surfaces corresponding to hyper-dual curves. Hacettepe Journal of Mathematics and Statistics 51 1 187–198.
IEEE S. Aslan, M. Bekar, and Y. Yaylı, “Ruled surfaces corresponding to hyper-dual curves”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, pp. 187–198, 2022, doi: 10.15672/hujms.988245.
ISNAD Aslan, Selahattin et al. “Ruled Surfaces Corresponding to Hyper-Dual Curves”. Hacettepe Journal of Mathematics and Statistics 51/1 (February 2022), 187-198. https://doi.org/10.15672/hujms.988245.
JAMA Aslan S, Bekar M, Yaylı Y. Ruled surfaces corresponding to hyper-dual curves. Hacettepe Journal of Mathematics and Statistics. 2022;51:187–198.
MLA Aslan, Selahattin et al. “Ruled Surfaces Corresponding to Hyper-Dual Curves”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, 2022, pp. 187-98, doi:10.15672/hujms.988245.
Vancouver Aslan S, Bekar M, Yaylı Y. Ruled surfaces corresponding to hyper-dual curves. Hacettepe Journal of Mathematics and Statistics. 2022;51(1):187-98.

Cited By

Kinematic Differential Geometry of a Line Trajectory in Spatial Movement
Axioms
https://doi.org/10.3390/axioms12050472

https://dergipark.org.tr/en/pub/hujms