Research Article
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On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group

Year 2022, Issue: 40, 82 - 89, 30.09.2022

Zuhal Kucukarslan Yuzbasi Sevinç Taze

https://doi.org/10.53570/jnt.1165809

Abstract

This paper finds sufficient conditions to determine a surface whose mean curvature along a given Smarandache curve is constant in a three-dimensional Lie group. This is accomplished by using the Frenet frames of the specified curve to express surfaces that span the TN, NB, and TB Smarandache curves parametrically. In terms of the curvatures of given Smarandache curves, marching scale functions, and their partial derivatives, the mean curvatures of these surfaces along the given TN, NB, and TB Smarandache curves are determined. Sufficient conditions are found to maintain the provided mean curvatures of the resulting surfaces at a constant value. Finally, some examples are provided.

Keywords

Smarandache Curve Mean curvature Lie group

References

  • A. T. Ali, Special Smarandache Curves in the Euclidean space, International Journal of Mathematical Combinatorics 2 (2010) 30–36.
  • M. Turgut, S. Yılmaz, Smarandache Curves in Minkowski Space-Time, International Journal of Mathematical Combinatorics 3 (2008) 51–55.
  • C. Değirmen, O. Z. Okuyucu, Ö. G. Yıldız, Smarandache Curves in Three-Dimensional Lie Groups, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 (2019) 1175–1185.
  • Ü. Çiftçi, A Generalization of Lancret’s Theorem, Journal of Geometry and Physics 59 (2009) 1597–1603.
  • O. Z. Okuyucu, I. Gök, Y. Yaylı, N. Ekmekci, Slant Helices in Three Dimensional Lie Groups, Applied Mathematics and Computation 221 (2013) 672–683.
  • D. W. Yoon, General Helices of AW(k)-Type in the Lie Group, Journal of Applied Mathematics 2012 (2012) Article ID 535123 pp. 10.
  • D. W. Yoon, Z. K. Yüzbaşı, M. Bektaş, An Approach for Surfaces Using an Asymptotic Curve in Lie Group, Journal of Advanced Physics 6 (4) (2017) 586–590.
  • D. W. Yoon, Z. K. Yüzbaşı, On Constructions of Surfaces Using A Geodesic in Lie Group, Journal of Geometry 110 (2) (2019) 1–10.
  • G. J. Wang, K. Tang, C. L. Tai, Parametric Representation of a Surface Pencil with a Common Spatial Geodesic, Computer-Aided Design 36 (2004) 447–459.
  • C. Y. Li, R. H. Wang, C. G. Zhu, Parametric Representation of a Surface Pencil with a Common Line of Curvature, Computer-Aided Design 43 (2011) 1110–1117.
  • E. Kasap, F. T. Akyıldız, Surfaces with a Common Geodesic in Minkowski 3-space, Applied Mathematics and Computation 177 (2006) 260–270.
  • M. Altın, A. Kazan, H. B. Karadağ, Hypersurface Families with Smarandache curves in Galilean 4-space, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 (2021) 744-761.
  • E. Bayram, Construction of Surfaces with Constant Mean Curvature Along a Timelike Curve, Journal of Polytechnic 1 (2022) pp. 7.
  • H. Coşanoğlu, E. Bayram, Construction of Surfaces with Constant Mean Curvature along a Curve in E^3, Journal of Natural and Applied Sciences 24 (3) (2020) 533–538.
  • E. Abbena, S. Salamon, A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition, 1998.

Year 2022, Issue: 40, 82 - 89, 30.09.2022

Zuhal Kucukarslan Yuzbasi Sevinç Taze

https://doi.org/10.53570/jnt.1165809

Abstract

References

  • A. T. Ali, Special Smarandache Curves in the Euclidean space, International Journal of Mathematical Combinatorics 2 (2010) 30–36.
  • M. Turgut, S. Yılmaz, Smarandache Curves in Minkowski Space-Time, International Journal of Mathematical Combinatorics 3 (2008) 51–55.
  • C. Değirmen, O. Z. Okuyucu, Ö. G. Yıldız, Smarandache Curves in Three-Dimensional Lie Groups, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 (2019) 1175–1185.
  • Ü. Çiftçi, A Generalization of Lancret’s Theorem, Journal of Geometry and Physics 59 (2009) 1597–1603.
  • O. Z. Okuyucu, I. Gök, Y. Yaylı, N. Ekmekci, Slant Helices in Three Dimensional Lie Groups, Applied Mathematics and Computation 221 (2013) 672–683.
  • D. W. Yoon, General Helices of AW(k)-Type in the Lie Group, Journal of Applied Mathematics 2012 (2012) Article ID 535123 pp. 10.
  • D. W. Yoon, Z. K. Yüzbaşı, M. Bektaş, An Approach for Surfaces Using an Asymptotic Curve in Lie Group, Journal of Advanced Physics 6 (4) (2017) 586–590.
  • D. W. Yoon, Z. K. Yüzbaşı, On Constructions of Surfaces Using A Geodesic in Lie Group, Journal of Geometry 110 (2) (2019) 1–10.
  • G. J. Wang, K. Tang, C. L. Tai, Parametric Representation of a Surface Pencil with a Common Spatial Geodesic, Computer-Aided Design 36 (2004) 447–459.
  • C. Y. Li, R. H. Wang, C. G. Zhu, Parametric Representation of a Surface Pencil with a Common Line of Curvature, Computer-Aided Design 43 (2011) 1110–1117.
  • E. Kasap, F. T. Akyıldız, Surfaces with a Common Geodesic in Minkowski 3-space, Applied Mathematics and Computation 177 (2006) 260–270.
  • M. Altın, A. Kazan, H. B. Karadağ, Hypersurface Families with Smarandache curves in Galilean 4-space, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 (2021) 744-761.
  • E. Bayram, Construction of Surfaces with Constant Mean Curvature Along a Timelike Curve, Journal of Polytechnic 1 (2022) pp. 7.
  • H. Coşanoğlu, E. Bayram, Construction of Surfaces with Constant Mean Curvature along a Curve in E^3, Journal of Natural and Applied Sciences 24 (3) (2020) 533–538.
  • E. Abbena, S. Salamon, A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition, 1998.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Zuhal Kucukarslan Yuzbasi 0000-0001-7630-5490

Sevinç Taze 0000-0001-6892-1760

Publication Date September 30, 2022
Submission Date August 23, 2022
Published in Issue Year 2022 Issue: 40

Cite

APA Kucukarslan Yuzbasi, Z., & Taze, S. (2022). On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group. Journal of New Theory(40), 82-89. https://doi.org/10.53570/jnt.1165809
AMA Kucukarslan Yuzbasi Z, Taze S. On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group. JNT. September 2022;(40):82-89. doi:10.53570/jnt.1165809
Chicago Kucukarslan Yuzbasi, Zuhal, and Sevinç Taze. “On Parametric Surfaces With Constant Mean Curvature Along Given Smarandache Curves in Lie Group”. Journal of New Theory, no. 40 (September 2022): 82-89. https://doi.org/10.53570/jnt.1165809.
EndNote Kucukarslan Yuzbasi Z, Taze S (September 1, 2022) On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group. Journal of New Theory 40 82–89.
IEEE Z. Kucukarslan Yuzbasi and S. Taze, “On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group”, JNT, no. 40, pp. 82–89, September 2022, doi: 10.53570/jnt.1165809.
ISNAD Kucukarslan Yuzbasi, Zuhal - Taze, Sevinç. “On Parametric Surfaces With Constant Mean Curvature Along Given Smarandache Curves in Lie Group”. Journal of New Theory 40 (September 2022), 82-89. https://doi.org/10.53570/jnt.1165809.
JAMA Kucukarslan Yuzbasi Z, Taze S. On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group. JNT. 2022;:82–89.
MLA Kucukarslan Yuzbasi, Zuhal and Sevinç Taze. “On Parametric Surfaces With Constant Mean Curvature Along Given Smarandache Curves in Lie Group”. Journal of New Theory, no. 40, 2022, pp. 82-89, doi:10.53570/jnt.1165809.
Vancouver Kucukarslan Yuzbasi Z, Taze S. On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group. JNT. 2022(40):82-9.
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