Research Article
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Year 2023, Volume: 72 Issue: 2, 307 - 330, 23.06.2023

Emre Öztürk

https://doi.org/10.31801/cfsuasmas.1083750

Abstract

References

  • Ali, A.T, Position vectors of slant helices in Euclidean 3-space, Journal of the Egyptian Mathematical Society, 20 (2012), 1-6. doi: 10.1016/j.joems.2011.12.005
  • Bhat, V. S., Haribaskar, R., A pair of kinematically related space curves, International Journal of Geometric Methods in Modern Physics, 15(1850180) (2018), 17 pp. doi:10.1142/S0219887818501803
  • Blum, R., A remarkable class of Mannheim curves, Canad. Math. Bull., 9 (1966), 223–228. https://doi.org/10.4153/CMB-1966-030-9
  • Bottema, O., Roth, B., Theoretical Kinematics, New York, Dover Publications, 1990.
  • Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Translated from the Portuguese, Englewood Cliffs, NJ, USA, Prentice-Hall, Inc., 1976.
  • Deshmukh, S., Alghanemi, A., Farouki, R. T., Space curves defined by curvaturetorsion relations and associated helices, Filomat Journal, 33 (2019), 4951–4966. doi:10.2298/FIL1915951D
  • Eisenhart, L. P., An Introduction to Differential Geometry with Use of the Tensor Calculus, Princeton, Princeton University Press, 1947.
  • Honda, S., Takahashi, M., Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turkish Journal of Mathematics, 44 (2020), 883–899. doi:10.3906/mat-1905-63
  • Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28 (2004), 153–163.
  • Kahveci, D., Yaylı, Y., Persistent rigid-body motions on slant helices, International Journal of Geometric Methods in Modern Physics, 16(1950193) (2019), 15 pp. doi:10.1142/S0219887819501937
  • Kim, D. S., Chung, H. S., Cho, K. H., Space curves satisfying $\tau/\kappa=as+b$, Honam Math. J., 1 (1993), 5–9.
  • Liu, H., Wang, F., Mannheim partner curves in 3-space, Journal of Geometry, 88 (2008), 120–126. doi: 10.1007/s00022-007-1949-0
  • Menninger, A., Characterization of the slant helix as successor curve of the general helix, International Electronic Journal of Geometry, 7 (2014), 84–91. doi: 10.36890/iejg.593986
  • Monterde, J., Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Computer Aided Geometric Design, 26 (2009), 271–278. doi:10.1016/j.cagd.2008.10.002
  • Orbay, K., Kasap, E., On Mannheim partner curves in E3, International Journal of Physical Sciences, 4 (2009), 261–264.
  • Öztürk, E., Mannheim curves in 3−dimensional Euclidean space, International Scientific and Vocational Journal, 4 (2020), 86–89. doi: 10.47897/bilmes.818723
  • Öztürk, E., Geometric elements of constant precession curve, Hagia Sophia Journal of Geometry, 2 (2020), 48-55.
  • Öztürk, E., Yaylı, Y., W−curves in Lorentz-Minkowski space, Mathematical Sciences and Applications e-Notes, 5 (2017), 76-88. doi: 10.36753/mathenot.421740
  • Salkowski, E., Zur Transformation von Raumkurven, Mathematische Annalen, 66 (1909), 517–557.
  • Selig, J. M., Carricato, M., Persistent rigid-body motions and Study’s “Ribaucor” problem, Journal of Geometry, 108 (2017), 149–169. doi: 10.1007/s00022-016-0331-5
  • Struik, D. J., Lectures on Classical Differential Geometry, Reprint of the Second Edition, New York, NY, USA: Dover Publications, Inc., 1988.
  • Uzunoğlu, B., Gök, İ., Yaylı, Y., A new approach on curves of constant precession, Applied Mathematics and Computation, 275 (2016), 317–323. doi: 10.1016/j.amc.2015.11.083
  • Wang, Y., Chang, Y., Mannheim curves and spherical curves, International Journal of Geometric Methods in Modern Physics, 17(2050101) (2020), 15 pp. doi:10.1142/S0219887820501017

A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space

Year 2023, Volume: 72 Issue: 2, 307 - 330, 23.06.2023

Emre Öztürk

https://doi.org/10.31801/cfsuasmas.1083750

Abstract

In this paper, we define a nonlinear transformation between space curves which preserves the ratio of $\tau/\kappa$ of the given curve in 3−dimensional Euclidean space $E^3$. We investigate invariant and associated curves of this transformation by the help of curvature and torsion functions of the base curve. Moreover, we define a new curve (family) so-called quasi-slant helix, and we obtain some characterizations in terms of the curvatures of this curve. Finally, we examine some curves in the kinematics, and give the pictures of some special curves and their images with respect to the transformation.

Keywords

Mannheim curve constant pitch curve nonlinear transformation slant helix

References

  • Ali, A.T, Position vectors of slant helices in Euclidean 3-space, Journal of the Egyptian Mathematical Society, 20 (2012), 1-6. doi: 10.1016/j.joems.2011.12.005
  • Bhat, V. S., Haribaskar, R., A pair of kinematically related space curves, International Journal of Geometric Methods in Modern Physics, 15(1850180) (2018), 17 pp. doi:10.1142/S0219887818501803
  • Blum, R., A remarkable class of Mannheim curves, Canad. Math. Bull., 9 (1966), 223–228. https://doi.org/10.4153/CMB-1966-030-9
  • Bottema, O., Roth, B., Theoretical Kinematics, New York, Dover Publications, 1990.
  • Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Translated from the Portuguese, Englewood Cliffs, NJ, USA, Prentice-Hall, Inc., 1976.
  • Deshmukh, S., Alghanemi, A., Farouki, R. T., Space curves defined by curvaturetorsion relations and associated helices, Filomat Journal, 33 (2019), 4951–4966. doi:10.2298/FIL1915951D
  • Eisenhart, L. P., An Introduction to Differential Geometry with Use of the Tensor Calculus, Princeton, Princeton University Press, 1947.
  • Honda, S., Takahashi, M., Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turkish Journal of Mathematics, 44 (2020), 883–899. doi:10.3906/mat-1905-63
  • Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28 (2004), 153–163.
  • Kahveci, D., Yaylı, Y., Persistent rigid-body motions on slant helices, International Journal of Geometric Methods in Modern Physics, 16(1950193) (2019), 15 pp. doi:10.1142/S0219887819501937
  • Kim, D. S., Chung, H. S., Cho, K. H., Space curves satisfying $\tau/\kappa=as+b$, Honam Math. J., 1 (1993), 5–9.
  • Liu, H., Wang, F., Mannheim partner curves in 3-space, Journal of Geometry, 88 (2008), 120–126. doi: 10.1007/s00022-007-1949-0
  • Menninger, A., Characterization of the slant helix as successor curve of the general helix, International Electronic Journal of Geometry, 7 (2014), 84–91. doi: 10.36890/iejg.593986
  • Monterde, J., Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion, Computer Aided Geometric Design, 26 (2009), 271–278. doi:10.1016/j.cagd.2008.10.002
  • Orbay, K., Kasap, E., On Mannheim partner curves in E3, International Journal of Physical Sciences, 4 (2009), 261–264.
  • Öztürk, E., Mannheim curves in 3−dimensional Euclidean space, International Scientific and Vocational Journal, 4 (2020), 86–89. doi: 10.47897/bilmes.818723
  • Öztürk, E., Geometric elements of constant precession curve, Hagia Sophia Journal of Geometry, 2 (2020), 48-55.
  • Öztürk, E., Yaylı, Y., W−curves in Lorentz-Minkowski space, Mathematical Sciences and Applications e-Notes, 5 (2017), 76-88. doi: 10.36753/mathenot.421740
  • Salkowski, E., Zur Transformation von Raumkurven, Mathematische Annalen, 66 (1909), 517–557.
  • Selig, J. M., Carricato, M., Persistent rigid-body motions and Study’s “Ribaucor” problem, Journal of Geometry, 108 (2017), 149–169. doi: 10.1007/s00022-016-0331-5
  • Struik, D. J., Lectures on Classical Differential Geometry, Reprint of the Second Edition, New York, NY, USA: Dover Publications, Inc., 1988.
  • Uzunoğlu, B., Gök, İ., Yaylı, Y., A new approach on curves of constant precession, Applied Mathematics and Computation, 275 (2016), 317–323. doi: 10.1016/j.amc.2015.11.083
  • Wang, Y., Chang, Y., Mannheim curves and spherical curves, International Journal of Geometric Methods in Modern Physics, 17(2050101) (2020), 15 pp. doi:10.1142/S0219887820501017
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Emre Öztürk 0000-0001-6638-3233

Publication Date June 23, 2023
Submission Date March 6, 2022
Acceptance Date November 25, 2022
Published in Issue Year 2023 Volume: 72 Issue: 2

Cite

APA Öztürk, E. (2023). A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(2), 307-330. https://doi.org/10.31801/cfsuasmas.1083750
AMA Öztürk E. A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2023;72(2):307-330. doi:10.31801/cfsuasmas.1083750
Chicago Öztürk, Emre. “A Nonlinear Transformation Between Space Curves Defined by Curvature-Torsion Relations in 3-Dimensional Euclidean Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, no. 2 (June 2023): 307-30. https://doi.org/10.31801/cfsuasmas.1083750.
EndNote Öztürk E (June 1, 2023) A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 2 307–330.
IEEE E. Öztürk, “A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 72, no. 2, pp. 307–330, 2023, doi: 10.31801/cfsuasmas.1083750.
ISNAD Öztürk, Emre. “A Nonlinear Transformation Between Space Curves Defined by Curvature-Torsion Relations in 3-Dimensional Euclidean Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/2 (June 2023), 307-330. https://doi.org/10.31801/cfsuasmas.1083750.
JAMA Öztürk E. A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:307–330.
MLA Öztürk, Emre. “A Nonlinear Transformation Between Space Curves Defined by Curvature-Torsion Relations in 3-Dimensional Euclidean Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 72, no. 2, 2023, pp. 307-30, doi:10.31801/cfsuasmas.1083750.
Vancouver Öztürk E. A nonlinear transformation between space curves defined by curvature-torsion relations in 3-dimensional Euclidean space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(2):307-30.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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