Canal Surfaces Through a Null Quaternionic Spine Curve

Yıl 2022, , 721 - 729, 31.12.2022

Gül Tuğ

https://doi.org/10.35193/bseufbd.1059044

Öz

In this study, we give the parameterizations of the canal surfaces trough a null quaternionic spine curve by using the pseudo-spheres in ℝ𝟏𝟒. We also calculate the Gauss and Mean curvatures and obtain some corollaries related to the Cartan curvatures of the null quaternionic curve.

Anahtar Kelimeler

Canal Surfaces, Cartan Frame, Null Quaternionic Curve, Semi Real Quaternions, Spine Curve, Kanal Yüzeyler, Cartan Çatısı, Null Kuaterniyonik Eğri, Yarı Reel Kuaterniyonlar, Omurga Eğrisi

Teşekkür

The author sincerely thanks to the editor and the referees for helpful suggestions.

Null Kuaterniyonik Omurga Eğrisi Boyunca Kanal Yüzeyler

Öz

In this study, we give the parameterizations of the canal surfaces through a null quaternionic spine curve by using the pseudo-spheres in R_1^4. Besides, we give formulas for the Gauss and Mean curvatures and some corollaries related to the Cartan curvatures of the null quaternionic curve.

Anahtar Kelimeler

Kanal Yüzeyler, Cartan Çatısı, Null Kuaterniyonik Eğri, Yarı Reel Kuaterniyonlar, Omurga Eğrisi

Kaynakça

• Hamilton, W. R. (1844). On quaternions; or on a new system of imagniaries in algebra. Lond. Edinb. Dublin. Philos. Mag. J. Sci., 25 (3), 489-495.
• Aslan, S & Yaylı, Y. (2016). Canal surfaces with quaternions. Adv. Appl. Cliffor.d Algebras, 26, 31–8.
• Gök, İ. (2017). Quaternionic approach of canal surfaces constructed by some new ideas. Adv. Appl. Clifford Algebras, 27, 1175-1190.
• Shoemake, K. (1985). Animating rotation with quaternion curves. Proceedings of the 12th annual conference on computer graphics and interactive techniques (SIG-GRAPH 85), 19, New York, NY, USA, 245-254.
• Haralick, R. M. (2017). Longitudinal & Scalar Waves: Biquaternion generalized maxwell equations. Computer Science, Graduate Center, 172.
• Rosenfeld, B. (1997). Geometry of Lie groups. Kluwer Academic Publishers, Netherlands, 397.
• Dyachkova, M. (2007). On Hopf bundle analogue for semiquaternion algebra. 10th International Conference DGA, Olomouc, Czech Republic, 45-47.
• Bharathi, K. & Nagaraj, M. (1987). Quaternion valued function of a real variable Serret-Frenet formula. Indian J. Pure Appl. Math., 18, 507-511.
• Çöken, A. C. & Tuna, A. (2004). On the quaternionic inclined curves in the semi-Euclidean space E42. Appl. Math. Comput., 155, 373–389.
• Gök, İ., Okuyucu, O. Z., Kahraman, F. & Hacisalihoglu, H. H. (2011). On the quaternionic B2-slant helix in the Euclidean space E4. Adv. Appl. Clifford Algebras, 21, 707–719.
• Körpınar, T. & Baş, S. (2016). Characterization of Quaternionic Curves by Inextensible Flows. Prespacetime Journal, 7, 1680-1684.
• Okuyucu, O. Z. (2013). Characterizations of the quaternionic Mannheim curves in Euclidean space E4. Mathematical Combinatorics, 2, 44-53.
• Kecilioglu, O. & Ilarslan, K. (2013). Quaternionic Bertrand Curves in Euclidean 4-Space. Bull. Math. Anal. Appl., 5 (3), 27-38.
• Önder, M. (2020). Quaternionic Salkowski Curves and Quaternionic Similar Curves. Proc. Natl. Acad. Sci. India, Sect. A Phys. Sci., 90 (3), 447-456.
• Shoemake, K . (1985). Animating rotation with quaternion curves. Proceedings of the 12th annual conference on computer graphics and interactive techniques (SIG-GRAPH 85), 19, New York, NY, USA, 245-254 .
• Duggal, K. L. & Bejancu, A. (1996), Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer Academic Publisher, Dordrecht / Boston / London, 303.
• Ferrandez A., Gimenez A. & Lucas P. (2001). Null Helices in Lorentzian Space Forms. Int. J. Mod. Phys. A, 16 (30), 4845-4863.
• Çoken, A. C. & Ciftci, U. (2005). On the Cartan curvatures of a null curve in Minkowski spacetime. Geom. Dedicata, 114, 71-78.
• Kahraman, T. (2019). Null Quaternionic Slant Helices in Minkowski Spaces. International J.Math. Combin., 1, 45-52.
• Aksoy, A. T. (2016). Pseudo-Spherical Null Quaternionic Curves in Minkowski Space R^4. Acta Physica Polonica A, 130, 259-261.
• Özel, Ş., Külahçı, M. A. & Bektaş, M. (2021). The Characterizations Of Null Quaternionic Curves In Minkowski 3-Space. Turkish Journal of Science and Technology, 16 (2), 261-267.
• Aksoy, A. T. & Çöken, A. C. (2015). Null Quaternionic Curves in Semi-Euclidean 3-Space of Index ν. Acta Physica Polonica A, 128, B-286-B-289.
• Maekawa, T., Patrikalakis, M. N., Sakkalis, T. & Yu, G. (1998). Analysis and applications of pipe surfaces. Comput. Aided Geom. Des., 15, 437–58 .
• Shani, U. & Ballard, D. H. (1984). Splines as embeddings for generalized cylinders. Computer Vision , Graphics and Image Processing, 27, 129-156.
• Wang, L., Ming, C. L. & Blackmore, D. (1995). Generating sweep solids for NC verification using the SEDE method. Proceedings of the Fourth Symposium on Solid Modeling and Applications, Atlanta, 364-375.
• Farouki, R. T. & Sverrissor, R. (1996). Approximation of rolling-ball blends for free-form parametric surfaces. Computer-Aided Design, 28, 871-878.
• Xu, Z., Feng, R. & Sun, JG. (2006). Analytic and algebraic properties of canal surfaces. Journal of Computational and Applied Mathematics, 195 (1-2), 220-228.
• Jinhua Qian, J., Tian, X., Fu, X. & Kim, Y. H. (2020). Classifications of Canal Surfaces with the Gauss Maps in Minkowski 3-Space. Mathematics, 8, 1453.
• Gray, A., Abbana, E. & Salamon, S. (2006). Modern Differential Geometry of Curves and Surfaces with Mathematica (3rd edition), Studies in Advanced Mathematics, Chapman and Hall/CRC, Boca Raton, FL, 1016.
• Ates, F. (2021). Tubular Surfaces Around a Null Curve and Its Spherical Images, FUJMA, 4 (3), 210-220.