Abstract
In this paper, firstly the ruled surface drawn by the Darboux vector is expressed as a quaternion. Then, the spatial quaternionic definition of the striction curve is given and the integral invariants of the surface are calculated. Finally, the ruled surface which corresponds to a dual curve drawn by a dual Darboux vector is derived with the help of dual spatial quaternions and dual integral invariants of the ruled surface are obtained.
Keywords
Darboux vector distribution parameter dual spherical curve dual angle of pitch quaternion spatial quaternion ruled surface.
References
- [1] Aslan, S., Yaylı, Y., Quaternionic shape operator, Adv. Appl. Clifford Algebras, 27(2017), 2921–2931.
- [2] Babaarslan, M., Yaylı, Y., A new approach to constant slope surfaces with quaternions, ISRN Geom., 8(2012), article ID 126358.
- [3] Bharathi, K., Nagaraj, M., Quaternion valued function of a real variable Serret-Frenet formulae, Ind. J. P. Appl. Math., 18(1987), 507–511.
- [4] Çalışkan, A., S¸enyurt, S., The dual spatial quaternionic expression of ruled surfaces, Thermal Science, 23(1)(2019), 403–411.
- [5] Çalışkan, A., Spatial Quaternionic Curves and Ruled Surfaces, Ph.D. Thesis, Ordu University, Ordu, Turkey, 2020.
- [6] Do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice-Hall, USA, 1976.
- [7] Hamilton, W.R., Elements of Quaternions, New York, 1899.
- [8] Hacısalihoğlu, H.H., Motion Geometry and Quaternions Theory (in Turkish), University of Gazi Press, Turkey, 1983.
- [9] Hanson, A.J., Visualing Quaternions, Elsevier, USA, 2006.
- [10] Shoemake, K., Animating rotation with quaternion curves, Siggraph Computer Graphics, 19(1985), 245–254.
- [11] Sivridağ, A.˙I., Güneş, R., Keles¸, S., The Serret-Frenet formulae for dual quaternion-valued functions of a single real variable, Mechanism and Machine Theory, 29(5)(1994), 749-754.
- [12] Stoker, J.J., Differential Geometry, Wiley-Interscience, New York, 1969.
- [13] Şenyurt, S., Cevahir, C., Altun, Y., On spatial quaternionic involute curve a new view, Adv. Appl. Clifford Algebr., (2016), 1–10.
- [14] Şenyurt, S., Çalışkan, A., The quaternionic expression of ruled surfaces, Filomat, 32(16)(2018), 5753-5766.
Abstract
References
- [1] Aslan, S., Yaylı, Y., Quaternionic shape operator, Adv. Appl. Clifford Algebras, 27(2017), 2921–2931.
- [2] Babaarslan, M., Yaylı, Y., A new approach to constant slope surfaces with quaternions, ISRN Geom., 8(2012), article ID 126358.
- [3] Bharathi, K., Nagaraj, M., Quaternion valued function of a real variable Serret-Frenet formulae, Ind. J. P. Appl. Math., 18(1987), 507–511.
- [4] Çalışkan, A., S¸enyurt, S., The dual spatial quaternionic expression of ruled surfaces, Thermal Science, 23(1)(2019), 403–411.
- [5] Çalışkan, A., Spatial Quaternionic Curves and Ruled Surfaces, Ph.D. Thesis, Ordu University, Ordu, Turkey, 2020.
- [6] Do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice-Hall, USA, 1976.
- [7] Hamilton, W.R., Elements of Quaternions, New York, 1899.
- [8] Hacısalihoğlu, H.H., Motion Geometry and Quaternions Theory (in Turkish), University of Gazi Press, Turkey, 1983.
- [9] Hanson, A.J., Visualing Quaternions, Elsevier, USA, 2006.
- [10] Shoemake, K., Animating rotation with quaternion curves, Siggraph Computer Graphics, 19(1985), 245–254.
- [11] Sivridağ, A.˙I., Güneş, R., Keles¸, S., The Serret-Frenet formulae for dual quaternion-valued functions of a single real variable, Mechanism and Machine Theory, 29(5)(1994), 749-754.
- [12] Stoker, J.J., Differential Geometry, Wiley-Interscience, New York, 1969.
- [13] Şenyurt, S., Cevahir, C., Altun, Y., On spatial quaternionic involute curve a new view, Adv. Appl. Clifford Algebr., (2016), 1–10.
- [14] Şenyurt, S., Çalışkan, A., The quaternionic expression of ruled surfaces, Filomat, 32(16)(2018), 5753-5766.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Articles |
| Authors | |
| Publication Date | June 30, 2021 |
| Published in Issue | Year 2021 Volume: 13 Issue: 1 |