Abstract
Bu çalışmada 3-boyutlu Öklid uzayında parametrik denklemi ile verilen yüzey üzerinde eğriliği sıfırdan farklı olan bir eğrinin Bertrand B-çiftinin bu yüzey üzerinde isogeodezik olması için gerekli ve yeterli şartlar elde edilerek, ortak Bertrand-B isogeodezik eğrili yüzey aileleri problemi ele alınmıştır.
Bu çalışmada 3-boyutlu Öklid uzayında parametrik denklemi ile verilen yüzey üzerinde eğriliği sıfırdan farklı olan bir eğrinin Bertrand B-çiftinin bu yüzey üzerinde isogeodezik olması için gerekli ve yeterli şartlar elde edilerek, ortak Bertrand-B isogeodezik eğrili yüzey aileleri problemi ele alınmıştır.
References
- Atalay GŞ, Kasap E, 2016. Surfaces family with common Smarandache geodesic curve according to Bishop frame in Euclidean space, Mathematical Science and Applications E-Notes., 4 , 1.
- Atalay GŞ, E. Kasap, 2017. Surfaces family with common Smarandache geodesic curve, Journal of Science and Arts (JOSA), 4, 41.
- Atalay GŞ, 2018. Surfaces family with a common Mannheim geodesic curve. Journal of Applied Mathematics and Computation, 2(4),155-165.
- Atalay GŞ, 2018. Surfaces family with a common Mannheim asymptotic curve. Journal of Applied Mathematics and Computation, 2(4),145-154.
- Ayvacı KH, 2019. Ortak Mannheim-B İsogeodezikli ve İsoasimptotikli Yüzey Ailesi, Ondokuz Mayıs Üniversitesi Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi (Basılmış).
- Bayram E, Kasap E, 2014. Hypersurface family with a common isogeodesic, Studies and Research Series Mathematics and Informatics, 24, 2, 12.
- Bertrand J, 1850. Mémoire sur la théorie des courbes à double courbure, Comptes Rendus 36, Journal de Mathématiques Pures et Appliquées 15, 332–350.
- Bishop RL, 1975. There is more than one way to Frame a curve. The American Mathematical Monthly, 82(3), 246.
- Ergün E, Bayram E, 2016. Surface family with a common natural geodesic lift, Int. J. Math. Combin., 1, 2.
- Kasap E, Akyildiz FT, Orbay K, 2008. A generalization of surfaces family with common spatial geodesic, Applied Mathematics and Computation, 201, 781-789.
- Millman RS, George DP, 1977. Elements of Differential Geometry, Prentice-Hall.
- O’Neill B, 1966. Elementary Differential Geometry, Academic Press Inc., New York.
- Wang GJ, Tang K, Tai CL, 2004. Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des. 36 (5), 447-459.
- Yerlikaya F, Karaahmetoğlu S, Aydemir İ, 2016. On the Bertrand B-Pair Curve in 3-Dimensional Euclidean Space, Journal of Science and Arts, 3(36), 215-224.
- Yılmaz S, Turgut M, 2010. A new version of Bishop frame and an application to spherical images, Journal of Mathematical Analysis and Applications, 371(2), 764-776.
Abstract
In this paper, we construct a surface family possessing a Bertrand B pair of a given curve as an geodesic curve. Using the Bishop-2 frame frame of the given Bertrand B curves, we present the surface as a linear combination of this frame and analyse the necessary and sufficient condition for a given curve such that its Bertrand B pairs is both parametric and geodesic on a parametric surface. Finally, we present some interesting examples to show the validity of this study.
In this paper, we construct a surface family possessing a Bertrand B pair of a given curve as an geodesic curve. Using the Bishop-2 frame frame of the given Bertrand B curves, we present the surface as a linear combination of this frame and analyse the necessary and sufficient condition for a given curve such that its Bertrand B pairs is both parametric and geodesic on a parametric surface. Finally, we present some interesting examples to show the validity of this study.
References
- Atalay GŞ, Kasap E, 2016. Surfaces family with common Smarandache geodesic curve according to Bishop frame in Euclidean space, Mathematical Science and Applications E-Notes., 4 , 1.
- Atalay GŞ, E. Kasap, 2017. Surfaces family with common Smarandache geodesic curve, Journal of Science and Arts (JOSA), 4, 41.
- Atalay GŞ, 2018. Surfaces family with a common Mannheim geodesic curve. Journal of Applied Mathematics and Computation, 2(4),155-165.
- Atalay GŞ, 2018. Surfaces family with a common Mannheim asymptotic curve. Journal of Applied Mathematics and Computation, 2(4),145-154.
- Ayvacı KH, 2019. Ortak Mannheim-B İsogeodezikli ve İsoasimptotikli Yüzey Ailesi, Ondokuz Mayıs Üniversitesi Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi (Basılmış).
- Bayram E, Kasap E, 2014. Hypersurface family with a common isogeodesic, Studies and Research Series Mathematics and Informatics, 24, 2, 12.
- Bertrand J, 1850. Mémoire sur la théorie des courbes à double courbure, Comptes Rendus 36, Journal de Mathématiques Pures et Appliquées 15, 332–350.
- Bishop RL, 1975. There is more than one way to Frame a curve. The American Mathematical Monthly, 82(3), 246.
- Ergün E, Bayram E, 2016. Surface family with a common natural geodesic lift, Int. J. Math. Combin., 1, 2.
- Kasap E, Akyildiz FT, Orbay K, 2008. A generalization of surfaces family with common spatial geodesic, Applied Mathematics and Computation, 201, 781-789.
- Millman RS, George DP, 1977. Elements of Differential Geometry, Prentice-Hall.
- O’Neill B, 1966. Elementary Differential Geometry, Academic Press Inc., New York.
- Wang GJ, Tang K, Tai CL, 2004. Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des. 36 (5), 447-459.
- Yerlikaya F, Karaahmetoğlu S, Aydemir İ, 2016. On the Bertrand B-Pair Curve in 3-Dimensional Euclidean Space, Journal of Science and Arts, 3(36), 215-224.
- Yılmaz S, Turgut M, 2010. A new version of Bishop frame and an application to spherical images, Journal of Mathematical Analysis and Applications, 371(2), 764-776.
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Matematik / Mathematics |
| Authors | |
| Publication Date | September 1, 2020 |
| Submission Date | March 1, 2020 |
| Acceptance Date | May 12, 2020 |
| Published in Issue | Year 2020 Volume: 10 Issue: 3 |