Abstract
In this study, we show that the quaternion product of quaternionic operator whose scalar part is a real parameter and vector part is a curve in \({\mathbb {R}}^{3}\) and a spherical striction curve represents a slant ruled surface in \({\mathbb {R}}^{3}\) if the vector part of the quaternionic operator is perpendicular to the position vector of the spherical striction curve. In \({\mathbb {R}}^{3}\), exploitting this operator, we define the slant ruled surface corresponding to the natural lift curve on the subset of the tangent bundle of unit 2-sphere, \(T{\bar{M}}.\) Then, we classify \(\vec {{\bar{q}}}-, \vec {{\bar{h}}}-\) and \(\vec {{\bar{a}}}-\) slant ruled surfaces. Furthermore, these surfaces can also be expressed with 2- parameter homothetic motions. Finally, we give the geometric interpretations of this operator with some examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
The author declares that the data supporting the findings of this study are available within the article and its supplementary information files.
References
Altın, M., Kazan, A., Karadağ, H.B.: Ruled surfaces constructed by planar curves in Euclidean 3-Space with density. Celal Bayar Univ. J. Sci. 16, 81–88 (2020)
Aslan, S., Bekar, M., Yaylı, Y.: Ruled surfaces constructed by quaternions. J. Geom. Phys. 161, 1–9 (2021)
Aslan, S., Yaylı, Y.: Canal surfaces with quaternions. Adv. Appl. Clifford Algebras 26, 31–38 (2016)
Babaarslan, M., Yaylı, Y.: A new approach to constant slope surfaces with quaternion. ISRN Geom. (2012). https://doi.org/10.5402/2012/126358
Bekar, M., Hathout, F., Yaylı, Y.: Tangent bundle of pseudo-sphere and ruled surfaces in Minkowski 3-space. Gen. Lett. Math. 5, 58–70 (2018)
Çalışkan, M., Karaca, E.: Dual spherical curves of natural lift curve and tangent bundle of unit 2-sphere. J. Sci. Arts. 3, 561–574 (2019)
Çanakçı, Z., Tuncer, O.O., Gök, İ, Yaylı, Y.: The construction of circular surfaces with quaternions. Asian-Eur. J. Math. 12, 1–14 (2019)
Do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall Inc, Englewood Cliffs (1976)
Düldül, M.: Two and three dimensional regions from homothetic motions. Appl. Math. ENotes. 10, 86–93 (2010)
Ergün, E., Bilici, M., Çalışkan, M.: The Frenet vector fields and the curvatures of the natural lift curve. Bull. Soc. Math. Serv. Standards 2, 38–43 (2012)
Ergün, E., Çalışkan, M.: On natural lift of a curve. Pure Math. Sci. 2, 81–85 (2012)
Fischer, I.S.: Dual-Number Methods in Kinematics. Statics and Dynamics, CRC Press, Boca Raton (1999)
Gök, İ: Quaternionic approach of canal surfaces constructed by some new ideas. Adv. Appl. Clifford Algebras 27, 1175–1190 (2017)
Hamilton, W.R.: On quaternions; or on a new system of imagniaries in algebra. Lond. Edinb. Dublin Philos. Magn. J. Sci. 25(3), 489–495 (1844)
Hathout, F., Bekar, M., Yaylı, Y.: Ruled surfaces and tangent bundle of unit 2-sphere. Int. J. Geom. Methods Mod. Phys. 2 (2017)
Izumiya, S., Takeuchi, N.: New special curves and developable surfaces. Turk. J. Math. 28, 531–537 (2004)
Karaca, E., Çalışkan, M.: Tangent bundle of unit 2-sphere and slant ruled surfaces. Filomat. (Submitted) (2020)
Karaca, E., Çalışkan, M.: Ruled surfaces and tangent bundle of pseudo-sphere of natural lift curves. J. Sci. Arts. 20, 583–586 (2020)
Karaca, E., Çalışkan, M.: Ruled surfaces and tangent bundle of unit 2-sphere of natural lift curves. Gazi Univ. J. Sci. 33, 751–759 (2020)
Karakaş, B., Gündoğan, H.: A relation among \(DS^{2}\), \(TS^{2}\) and non-cylindirical ruled surfaces. Math. Commun. 8, 9–14 (2003)
Kaya, O., Önder, M.: Characterizations of slant ruled surfaces in the Euclidean 3-space. Caspian J. Math. Sci. 6, 31–46 (2017)
Mert, T., Atçeken, M.: Special ruled surfaces in de-Sitter 3-Space. Fund. J. Math. Appl. 4, 195–209 (2021)
Önder, M.: Non-null slant ruled surfaces. arXiv:1604.03813 (2016)
Önder, M.: Slant ruled surfaces. arXiv:1311.0627v2 (4 November 2013)
Önder, M., Uğurlu, H.H.: Frenet frames and invariants of timelike ruled surfaces. Eng. Phys. Math. 4, 507–513 (2013)
Shoemake, K.: Animating rotation with quaternion curves. In: Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques (SIG-GRAPH’85). ACM New York, USA, 19, 245–254 (1985)
Thorpe, J.A.: Elementary Topics in Differential Geometry. Springer, New York (1979)
Ward, J.P.: Quaternions and Cayley Numbers. Kluwer Academic Publishers, Boston (1997)
Acknowledgements
The authors would like to thank referee(s) for their precious suggestions and comments that helped to improve the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author did not receive support from any organization for the submitted work.
Additional information
Communicated by Uwe Kaehler.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Karaca, E. Quaternionic Approach of Slant Ruled Surfaces. Adv. Appl. Clifford Algebras 32, 42 (2022). https://doi.org/10.1007/s00006-022-01225-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00006-022-01225-z