Research Article
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Year 2024, , 79 - 91, 24.03.2024
https://doi.org/10.17798/bitlisfen.1345438

Abstract

References

  • [1] H. H. Hacısalihoğlu, Diferansiyel Geometri. İnönü Üniversitesi Yayınları, Malatya, 1983.
  • [2] P. D. Scofield, "Curves of constant precessions," The American mathematical monthly, vol. 102, pp. 531–537, 1995.
  • [3] R.L. Bishop, "There is more than one way to frame a curve," The American Mathematical Monthly, vol. 82, pp. 246–251, 1975.
  • [4] N. Clauvelin, W. K. Olson and I. Tobias, "Characterizations of the geometry and topology of DNA pictured as a discrete collection of atoms," Journal of Chemical Theory and Computation, vol. 8, pp. 1092–1107, 2012.
  • [5] C. Y. Han, "Nonexistence of rational rotation-minimizing frames on cubic curves," Computer Aided Geometric Design, vol. 25, pp. 298–304, 2008.
  • [6] S. Yılmaz and M. Turgut, "A new version of Bishop frame and an application to spherical images," Journal of Mathematical Analysis and Applications, vol. 371, pp. 764–776, 2010.
  • [7] O. Keskin and Y. Yaylı, "An application of N-Bishop frame to spherical images for direction curves," International Journal of Geometric Methods in Modern Physics, vol. 14, pp. 1750162, 2017.
  • [8] B. Bükcü and M. K. Karacan, "Special Bishop motion and Bishop Darboux rotation axis of the space curve," Journal of Dynamical Systems and Geometric Theories, vol. 6, pp. 27–34, 2008.
  • [9] B. Bükcü and M. K. Karacan, "The slant helices according to Bishop frame," International Journal of Computational and Mathematical Sciences, vol. 3, pp. 67–70, 2009.
  • [10] E. Damar, N. Yüksel and A. T. Vanlı, "The ruled surfaces according to the type-2 Bishop frame in " International Mathematical Forum, vol. 12, pp. 133–143, 2017.
  • [11] A. Kelleci, M. Bektaş and M. Ergüt, "The Hasimoto surface according to bishop frame," Adıyaman Üniversitesi Fen Bilimleri Dergisi, vol. 9, pp. 13–22, 2019.
  • [12] S. Kızıltuğ, S. Kaya and O. Tarakcı, "The slant helices according to the type-2 Bishop frame in Euclidean 3-space," International Journal of Pure and Applied Mathematics, vol. 2, pp. 211–222, 2013.
  • [13] A. Çakmak and V. Şahin, "Characterizations of Adjoint Curves According to the alternativeMoving Frame", Fundamental Journal of Mathematics and Applications, vol. 5, pp. 42-50, 2022.
  • [14] M. Masal and A. Azak, "The Ruled Surfaces According to the type-2 Bishop Frame in the Euclidean 3-Space ," Mathematical Sciences and Applications E-Notes, vol. 3, pp. 74–83, 2015.
  • [15] M. Masal and A. Azak, "Ruled surfaces according to Bishop frame in the Euclidean 3 –spaces," Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, vol. 89, pp. 415–424, 2019.
  • [16] S. Ouarab, A. Ouazzani and M. Izıd, "Ruled surfaces with alternative moving frame in Euclidean 3- space," International Journal of Mathematical Sciences and Engineering Applications, vol. 12, pp. 43–58, 2018.
  • [17] H. K. Samancı and M. İncesu, "Investigating a quadratic Bezier curve due to NCW and N-Bishop frames," Turkish Journal of Mathematics and Computer Science, vol. 12, pp. 120–127, 2020.
  • [18] H. K. Samancı and M. Sevinç, "N-Bishop Çatısına Göre Regle Yüzeylerin Bazı Karakterizasyonları," Karadeniz Fen Bilimleri Dergisi, vol. 12, pp. 113–134, 2022.
  • [19] B. Uzunoğlu, İ. Gök, and Y. Yaylı, "A New approach on curves of constant precession," Applied Mathematics and Computation, vol. 275, pp. 317–323, 2016.
  • [20] S. Yılmaz and Ü. Z. Savcı, "A New Version Darboux Vector and Characterization Some Special Curves According to the type-2 Bishop Frame in ," Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, vol. 87, pp. 355–362, 2017.
  • [21] S. Şenyurt, S. Gür Mazlum, D. Canlı and E. Çan, "Some special Smarandache ruled surfaces according to the alternativeframe in ," Maejo International Journal of Science and Technology, vol. 17, pp. 138–153, 2023.
  • [22] E. Salkowski, "Zur transformation von raumkurven," Mathematische Annalen, vol. 66, pp. 517–557, 1909.
  • [23] J. Monterde, "Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, "Computer Aided Geometric Design, vol. 26, pp. 271–278, 2009.
  • [24] S. Gür Mazlum, S. Şenyurt and M. Bektaş, "Salkowski Curves and Their Modified Orthogonal Frames in ," Journal of New Theory, vol. 40, pp. 12–26, 2022.
  • [25] S. Şenyurt and B. Öztürk, "Smarandache curves according to the Sabban frame belonging to spherical indicatrix curve of the Salkowski curve," Tbilisi Mathematical Journal, vol. 13, pp. 111–131, 2020.
  • [26] S. Gür and S. Şenyurt, "Frenet vectors and geodesic curvatures of spheric indicatrix curves of Salkowski curve in ," Hadronic Journal, vol. 33, pp. 485–512, 2010.
  • [27] J. Monterde, "The Bertrand curve associated to a Salkowski curve," Journal of Geometry, vol. 111, pp. 21, 2020.
  • [28] S. Deshmukh, A. Alghanemi and R. T. Farouki, "Space curves defined by curvature–torsion relations and associated helices," Filomat, vol. 33, pp. 4951–4966, 2019.

Bishop Frames of Salkowski Curves in E3

Year 2024, , 79 - 91, 24.03.2024
https://doi.org/10.17798/bitlisfen.1345438

Abstract

In this study, alternative, type-1 Bishop, type-2 Bishop and N-Bishop frames of Salkowski curves in E3 are calculated. Moreover, curvatures, Darboux and pol vectors of these frames are found. Also, relationships between the Bishop frames, Darboux vectors and pole vectors are given.

References

  • [1] H. H. Hacısalihoğlu, Diferansiyel Geometri. İnönü Üniversitesi Yayınları, Malatya, 1983.
  • [2] P. D. Scofield, "Curves of constant precessions," The American mathematical monthly, vol. 102, pp. 531–537, 1995.
  • [3] R.L. Bishop, "There is more than one way to frame a curve," The American Mathematical Monthly, vol. 82, pp. 246–251, 1975.
  • [4] N. Clauvelin, W. K. Olson and I. Tobias, "Characterizations of the geometry and topology of DNA pictured as a discrete collection of atoms," Journal of Chemical Theory and Computation, vol. 8, pp. 1092–1107, 2012.
  • [5] C. Y. Han, "Nonexistence of rational rotation-minimizing frames on cubic curves," Computer Aided Geometric Design, vol. 25, pp. 298–304, 2008.
  • [6] S. Yılmaz and M. Turgut, "A new version of Bishop frame and an application to spherical images," Journal of Mathematical Analysis and Applications, vol. 371, pp. 764–776, 2010.
  • [7] O. Keskin and Y. Yaylı, "An application of N-Bishop frame to spherical images for direction curves," International Journal of Geometric Methods in Modern Physics, vol. 14, pp. 1750162, 2017.
  • [8] B. Bükcü and M. K. Karacan, "Special Bishop motion and Bishop Darboux rotation axis of the space curve," Journal of Dynamical Systems and Geometric Theories, vol. 6, pp. 27–34, 2008.
  • [9] B. Bükcü and M. K. Karacan, "The slant helices according to Bishop frame," International Journal of Computational and Mathematical Sciences, vol. 3, pp. 67–70, 2009.
  • [10] E. Damar, N. Yüksel and A. T. Vanlı, "The ruled surfaces according to the type-2 Bishop frame in " International Mathematical Forum, vol. 12, pp. 133–143, 2017.
  • [11] A. Kelleci, M. Bektaş and M. Ergüt, "The Hasimoto surface according to bishop frame," Adıyaman Üniversitesi Fen Bilimleri Dergisi, vol. 9, pp. 13–22, 2019.
  • [12] S. Kızıltuğ, S. Kaya and O. Tarakcı, "The slant helices according to the type-2 Bishop frame in Euclidean 3-space," International Journal of Pure and Applied Mathematics, vol. 2, pp. 211–222, 2013.
  • [13] A. Çakmak and V. Şahin, "Characterizations of Adjoint Curves According to the alternativeMoving Frame", Fundamental Journal of Mathematics and Applications, vol. 5, pp. 42-50, 2022.
  • [14] M. Masal and A. Azak, "The Ruled Surfaces According to the type-2 Bishop Frame in the Euclidean 3-Space ," Mathematical Sciences and Applications E-Notes, vol. 3, pp. 74–83, 2015.
  • [15] M. Masal and A. Azak, "Ruled surfaces according to Bishop frame in the Euclidean 3 –spaces," Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, vol. 89, pp. 415–424, 2019.
  • [16] S. Ouarab, A. Ouazzani and M. Izıd, "Ruled surfaces with alternative moving frame in Euclidean 3- space," International Journal of Mathematical Sciences and Engineering Applications, vol. 12, pp. 43–58, 2018.
  • [17] H. K. Samancı and M. İncesu, "Investigating a quadratic Bezier curve due to NCW and N-Bishop frames," Turkish Journal of Mathematics and Computer Science, vol. 12, pp. 120–127, 2020.
  • [18] H. K. Samancı and M. Sevinç, "N-Bishop Çatısına Göre Regle Yüzeylerin Bazı Karakterizasyonları," Karadeniz Fen Bilimleri Dergisi, vol. 12, pp. 113–134, 2022.
  • [19] B. Uzunoğlu, İ. Gök, and Y. Yaylı, "A New approach on curves of constant precession," Applied Mathematics and Computation, vol. 275, pp. 317–323, 2016.
  • [20] S. Yılmaz and Ü. Z. Savcı, "A New Version Darboux Vector and Characterization Some Special Curves According to the type-2 Bishop Frame in ," Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, vol. 87, pp. 355–362, 2017.
  • [21] S. Şenyurt, S. Gür Mazlum, D. Canlı and E. Çan, "Some special Smarandache ruled surfaces according to the alternativeframe in ," Maejo International Journal of Science and Technology, vol. 17, pp. 138–153, 2023.
  • [22] E. Salkowski, "Zur transformation von raumkurven," Mathematische Annalen, vol. 66, pp. 517–557, 1909.
  • [23] J. Monterde, "Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, "Computer Aided Geometric Design, vol. 26, pp. 271–278, 2009.
  • [24] S. Gür Mazlum, S. Şenyurt and M. Bektaş, "Salkowski Curves and Their Modified Orthogonal Frames in ," Journal of New Theory, vol. 40, pp. 12–26, 2022.
  • [25] S. Şenyurt and B. Öztürk, "Smarandache curves according to the Sabban frame belonging to spherical indicatrix curve of the Salkowski curve," Tbilisi Mathematical Journal, vol. 13, pp. 111–131, 2020.
  • [26] S. Gür and S. Şenyurt, "Frenet vectors and geodesic curvatures of spheric indicatrix curves of Salkowski curve in ," Hadronic Journal, vol. 33, pp. 485–512, 2010.
  • [27] J. Monterde, "The Bertrand curve associated to a Salkowski curve," Journal of Geometry, vol. 111, pp. 21, 2020.
  • [28] S. Deshmukh, A. Alghanemi and R. T. Farouki, "Space curves defined by curvature–torsion relations and associated helices," Filomat, vol. 33, pp. 4951–4966, 2019.
There are 28 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Araştırma Makalesi
Authors

Sümeyye Gür Mazlum 0000-0003-2471-1627

Early Pub Date March 21, 2024
Publication Date March 24, 2024
Submission Date August 18, 2023
Acceptance Date October 11, 2023
Published in Issue Year 2024

Cite

IEEE S. Gür Mazlum, “Bishop Frames of Salkowski Curves in E3”, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol. 13, no. 1, pp. 79–91, 2024, doi: 10.17798/bitlisfen.1345438.



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Fen Bilimleri Dergisi Editörlüğü

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E-posta: fbe@beu.edu.tr