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4-Boyutlu Lorentz Minkowski Uzayında Null Olmayan Ortak İzoasimptotik Eğrili Hiperyüzey Aileleri

Year 2022, Issue: 38, 99 - 109, 31.08.2022
https://doi.org/10.31590/ejosat.1093177

Abstract

Bu çalışmada, E_^4 Lorentz Minkowski uzayında null olmayan Frenet vektörlere sahip spacelike ve timelike eğrilerinden geçen hiperyüzeylerin parametrik denklemi, bu eğrinin Frenet çatısı yardımıyla ifade edildi. Ayrıca E_^4 de spacelike ve timelike eğrilerinin hiperyüzey üzerinde ortak izoasimptotik olması için gerekli ve yeterli koşullar verilerek hiperyüzey aileleri oluşturuldu. Daha sonra sapma fonksiyonları yardımıyla elde edilen bu koşullar sadeleştirildi. Son olarak, örneklerle çalışma desteklendi ve belli izdüşüm metodları kullanılarak örneklerin grafikleri çizildi.

References

  • Abdel-Baky, R. A., (2016). A surface family with a common asymptotic curve in the Euclidean 3-space. Asian Journal of Mathematics an Applications, 201(6) 12.
  • Ali, A. T., (2010). Time-like Smarandache curves derived from a Space-like Helix. Journal of Dynamical Systems and Geometric Theories, 8(1), 93-100.
  • Altın, M., Kazan, A. & Karadağ, H.B., 2021. Hypersurface families with Smarandache curves in Galilean 4-space. Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 70(2), 744–761.
  • Altın, M. & Küçükarslan Yüzbaşı, Z., 2020. Surfaces using Smarandache asymptotic curves in Galilean Space. International Journal of Mathematical Combinatorics, (3), 1–15.
  • Atalay, G. Ş. & Kasap, E., (2016). Surfaces family with common Smarandache geodesic curve according to Bishop frame in Euclidean space. Mathematical Sciences and Applications E-Notes, 4(1), 164-174.
  • Ayvacı, K. H. & Atalay, G. Ş., (2020). Ortak Bertrand-B isogeodezik eğriye sahip yüzey aileleri. Journal of the Institute of Science and Technology, 10(3), 1975-1983.
  • Bayram, E., Güler, F. & Kasap, E., (2012). Parametric representation of a surface pencil with a common asymptotic curve. Computer-Aided Design, 44(7), 637-643.
  • Bayram, E. & Kasap, E., (2014a). Hypersurface family with a common isoasymptotic curve. Geometry, 2014, 1-6.
  • Bayram, E. & Kasap, E., (2014b). Hypersurface family with a common isogeodesic. Scientific Studies and Research, 24(2), 5-24.
  • Bejancu, A. & Duggal, K. L., (1995). Lightlike submanifolds of semi-Riemannian manifolds. Acta Applicandae Mathematica, 38(2), 197-215.
  • Contopoulos, G., (1990). Asymptotic curves and escapes in Hamiltonian systems. Astronomy and Astrophysics, 231, 41-55.
  • Turan, Ç., Altın, M. & Karadağ, H. B., (2022). Hypersurface families with common non-null geodesic in Minkowski 4-space. Advanced Studies: Euro-Tbilisi Mathematical Journal, 15(1), 167-180.
  • Ergün, E. & Bayram, E., (2019). 3-Boyutlu Minkowski Uzayında Timelike Binormalli Spacelike Eğrinin Tabii Liftini Asimptotik Kabul Eden Yüzey Ailesi. Journal of the Institute of Science and Technology, 9(2), 1082-1089.
  • Ergün, E., Bilici, M. & Çaliskan, M., (2015). The Natural Lift Curve of the Spherical Indicatrix of a Spacelike Curve in Minkowski 4-Space. Journal of Science and Arts, 15(1), 39.
  • Farin, G., (1988). Curves and surfaces for computer aided geometric design: A practical guide. San Diego, CA, Academic Press, Inc., 348.
  • Garcia, R., Gutierrez, C. & Sotomayor, J., (1999). Structural stability of asymptotic lines on surfaces immersed in R3. Bulletin des sciences mathematiques, 123(8), 599-622.
  • Garcia, R. A. & Tello, J. M. S., (1997). Structurall stability of parabolic points and periodic asymptotic lines. Brasil, 39, 84-102.
  • Hartman, P. & Wintner, A., (1951). On the asymptotic curves of a surface. American Journal of Mathematics, 73(1), 149-172.
  • Kasap, E., Akyildiz, F. T. & Orbay, K., (2008). A generalization of surfaces family with common spatial geodesic. Applied Mathematics and Computation, 201(1-2), 781-789.
  • Kasap, E. & Akyildiz, F. T., (2006). Surfaces with common geodesic in Minkowski 3-space. Applied mathematics and computation, 177(1), 260- 270.
  • Kitagawa, Y., (1988). Periodicity of the asymptotic curves on flat tori in S3. Journal of the Mathematical Society of Japan, 40(3), 457-476.
  • Kocayigit, H. & Çiçek, Z., (2015). Some characterizations of constant breadth spacelike curves in Minkowski 4-space E14. New Trends in Mathematical Sciences, 3(2), 1-12.
  • Latifi, S., (2015). Numerical solution of geodesic differential equations on a surface in R3. In 8th Seminar on Geometry and Topology, 20, 450.
  • O’Neill, B. (2014). The geometry of Kerr black holes. Courier Corporation, 400.
  • Şaffak, G. and Kasap, E., (2009). Family of surface with a common null geodesic. Fizik Bilimleri Dergisi, 4(8), 428-433.
  • Şaffak, G., Bayram, E. & Kasap, E., (2013). Surfaces with a common asymptotic curve in Minkowski 3-space. arXiv preprint arXiv:1305.0382.
  • Şenyurt, S., Ayvacı, K. H. & Canlı, D., (2020). Ortak Mannheim-D isogeodezik eğriye sahip yüzeyler. Ordu Üniversitesi Bilim ve Teknoloji Dergisi, 10(2), 105-116.
  • Thorpe, J. A., (1994). Elementary topics in differential geometry. Springer Science and Business Media, 256.
  • Tozak, H., (2010). Minkowski 4-uzayında eğriler ve hareketlerin geometrisi. Yükseklisans Tezi, Pamukkale Üniversitesi Fen Bilimleri Enstitüsü, Denizli, 113.
  • Walrave, J. (1995). Curves and surfaces in Minkowski space, Thesis (Ph.D.), Katholieke Universiteit Leuven (Belgium).
  • Turgut, M. & Yilmaz, S., (2008). On the Frenet frame and a characterization of space-like involute-evolute curve couple in Minkowski space-time. In Int. Math. Forum, 3(16), 793-801.
  • Turgut, M. & Yilmaz, S., (2009). Some characterizations of type-3 slant helices in Minkowski space-time. Involve, a Journal of Mathematics, 2(1), 115-120.
  • Yoon, D. W. & Yüzbasi, Z. K., (2018). An approach for Hypersurface family with common geodesic curve in the 4D Galilean space G 4. The Pure and Applied Mathematics, 25(4), 229-241.
  • Yüzbaşı, Z. K. & Bektaş, M., (2016). On the construction of a surface family with common geodesic in Galilean space G3. Open Physics, 14(1), 360-363.
  • Yüzbaşı, Z. K., (2016). On a family of surfaces with common asymptotic curve in the Galilean space G3. J. Nonlinear Sci, (9), 518-523.
  • Wang, G. J., Tang, K. & Tai, C. L., (2004). Parametric representation of a surface pencil with a common spatial geodesic. Computer-Aided Design, 36(5), 447-459.

Hypersurfaces Families with Common Non-Null Isoasymptotic Curve in Lorentz Minkowski 4-Space

Year 2022, Issue: 38, 99 - 109, 31.08.2022
https://doi.org/10.31590/ejosat.1093177

Abstract

In this study, the parametric equation of hypersurfaces passing through the spacelike and timelike curves that has non-null Frenet vectors in E_1^4 Lorentz Minkowski space was expressed with the help of this curve is Frenet frame. Furthermore, hypersurface families were created by giving necessary and sufficient conditions so that the spacelike and timelike curves on E_1^4 is common isoasymptotic on the hypersurface. Then, these conditions obtained with the help of deviation marching-scale functions were simplified. Finally, the study was supported with examples and the graphics were drawn using certain projection methods.

References

  • Abdel-Baky, R. A., (2016). A surface family with a common asymptotic curve in the Euclidean 3-space. Asian Journal of Mathematics an Applications, 201(6) 12.
  • Ali, A. T., (2010). Time-like Smarandache curves derived from a Space-like Helix. Journal of Dynamical Systems and Geometric Theories, 8(1), 93-100.
  • Altın, M., Kazan, A. & Karadağ, H.B., 2021. Hypersurface families with Smarandache curves in Galilean 4-space. Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 70(2), 744–761.
  • Altın, M. & Küçükarslan Yüzbaşı, Z., 2020. Surfaces using Smarandache asymptotic curves in Galilean Space. International Journal of Mathematical Combinatorics, (3), 1–15.
  • Atalay, G. Ş. & Kasap, E., (2016). Surfaces family with common Smarandache geodesic curve according to Bishop frame in Euclidean space. Mathematical Sciences and Applications E-Notes, 4(1), 164-174.
  • Ayvacı, K. H. & Atalay, G. Ş., (2020). Ortak Bertrand-B isogeodezik eğriye sahip yüzey aileleri. Journal of the Institute of Science and Technology, 10(3), 1975-1983.
  • Bayram, E., Güler, F. & Kasap, E., (2012). Parametric representation of a surface pencil with a common asymptotic curve. Computer-Aided Design, 44(7), 637-643.
  • Bayram, E. & Kasap, E., (2014a). Hypersurface family with a common isoasymptotic curve. Geometry, 2014, 1-6.
  • Bayram, E. & Kasap, E., (2014b). Hypersurface family with a common isogeodesic. Scientific Studies and Research, 24(2), 5-24.
  • Bejancu, A. & Duggal, K. L., (1995). Lightlike submanifolds of semi-Riemannian manifolds. Acta Applicandae Mathematica, 38(2), 197-215.
  • Contopoulos, G., (1990). Asymptotic curves and escapes in Hamiltonian systems. Astronomy and Astrophysics, 231, 41-55.
  • Turan, Ç., Altın, M. & Karadağ, H. B., (2022). Hypersurface families with common non-null geodesic in Minkowski 4-space. Advanced Studies: Euro-Tbilisi Mathematical Journal, 15(1), 167-180.
  • Ergün, E. & Bayram, E., (2019). 3-Boyutlu Minkowski Uzayında Timelike Binormalli Spacelike Eğrinin Tabii Liftini Asimptotik Kabul Eden Yüzey Ailesi. Journal of the Institute of Science and Technology, 9(2), 1082-1089.
  • Ergün, E., Bilici, M. & Çaliskan, M., (2015). The Natural Lift Curve of the Spherical Indicatrix of a Spacelike Curve in Minkowski 4-Space. Journal of Science and Arts, 15(1), 39.
  • Farin, G., (1988). Curves and surfaces for computer aided geometric design: A practical guide. San Diego, CA, Academic Press, Inc., 348.
  • Garcia, R., Gutierrez, C. & Sotomayor, J., (1999). Structural stability of asymptotic lines on surfaces immersed in R3. Bulletin des sciences mathematiques, 123(8), 599-622.
  • Garcia, R. A. & Tello, J. M. S., (1997). Structurall stability of parabolic points and periodic asymptotic lines. Brasil, 39, 84-102.
  • Hartman, P. & Wintner, A., (1951). On the asymptotic curves of a surface. American Journal of Mathematics, 73(1), 149-172.
  • Kasap, E., Akyildiz, F. T. & Orbay, K., (2008). A generalization of surfaces family with common spatial geodesic. Applied Mathematics and Computation, 201(1-2), 781-789.
  • Kasap, E. & Akyildiz, F. T., (2006). Surfaces with common geodesic in Minkowski 3-space. Applied mathematics and computation, 177(1), 260- 270.
  • Kitagawa, Y., (1988). Periodicity of the asymptotic curves on flat tori in S3. Journal of the Mathematical Society of Japan, 40(3), 457-476.
  • Kocayigit, H. & Çiçek, Z., (2015). Some characterizations of constant breadth spacelike curves in Minkowski 4-space E14. New Trends in Mathematical Sciences, 3(2), 1-12.
  • Latifi, S., (2015). Numerical solution of geodesic differential equations on a surface in R3. In 8th Seminar on Geometry and Topology, 20, 450.
  • O’Neill, B. (2014). The geometry of Kerr black holes. Courier Corporation, 400.
  • Şaffak, G. and Kasap, E., (2009). Family of surface with a common null geodesic. Fizik Bilimleri Dergisi, 4(8), 428-433.
  • Şaffak, G., Bayram, E. & Kasap, E., (2013). Surfaces with a common asymptotic curve in Minkowski 3-space. arXiv preprint arXiv:1305.0382.
  • Şenyurt, S., Ayvacı, K. H. & Canlı, D., (2020). Ortak Mannheim-D isogeodezik eğriye sahip yüzeyler. Ordu Üniversitesi Bilim ve Teknoloji Dergisi, 10(2), 105-116.
  • Thorpe, J. A., (1994). Elementary topics in differential geometry. Springer Science and Business Media, 256.
  • Tozak, H., (2010). Minkowski 4-uzayında eğriler ve hareketlerin geometrisi. Yükseklisans Tezi, Pamukkale Üniversitesi Fen Bilimleri Enstitüsü, Denizli, 113.
  • Walrave, J. (1995). Curves and surfaces in Minkowski space, Thesis (Ph.D.), Katholieke Universiteit Leuven (Belgium).
  • Turgut, M. & Yilmaz, S., (2008). On the Frenet frame and a characterization of space-like involute-evolute curve couple in Minkowski space-time. In Int. Math. Forum, 3(16), 793-801.
  • Turgut, M. & Yilmaz, S., (2009). Some characterizations of type-3 slant helices in Minkowski space-time. Involve, a Journal of Mathematics, 2(1), 115-120.
  • Yoon, D. W. & Yüzbasi, Z. K., (2018). An approach for Hypersurface family with common geodesic curve in the 4D Galilean space G 4. The Pure and Applied Mathematics, 25(4), 229-241.
  • Yüzbaşı, Z. K. & Bektaş, M., (2016). On the construction of a surface family with common geodesic in Galilean space G3. Open Physics, 14(1), 360-363.
  • Yüzbaşı, Z. K., (2016). On a family of surfaces with common asymptotic curve in the Galilean space G3. J. Nonlinear Sci, (9), 518-523.
  • Wang, G. J., Tang, K. & Tai, C. L., (2004). Parametric representation of a surface pencil with a common spatial geodesic. Computer-Aided Design, 36(5), 447-459.
There are 36 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section Articles
Authors

Çiğdem Turan 0000-0002-5224-730X

Mustafa Altın 0000-0001-5544-5910

Hacı Bayram Karadağ 0000-0001-6474-877X

Early Pub Date July 26, 2022
Publication Date August 31, 2022
Published in Issue Year 2022 Issue: 38

Cite

APA Turan, Ç., Altın, M., & Karadağ, H. B. (2022). 4-Boyutlu Lorentz Minkowski Uzayında Null Olmayan Ortak İzoasimptotik Eğrili Hiperyüzey Aileleri. Avrupa Bilim Ve Teknoloji Dergisi(38), 99-109. https://doi.org/10.31590/ejosat.1093177