Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space

Abstract

In this study, pedal curves defined by the Type-1, Type-2, and Type-3 Bishop frame vectors of a regular space curve in Euclidean 3-space are investigated. In addition, the pedal curves obtained from the unit Darboux vectors of the Bishop frames are also considered. The Frenet and Bishop frame vectors of these curves are determined, and their curvature and torsion functions are expressed in terms of the geometric invariants of the base curve. Furthermore, several examples are given, and the corresponding pedal curves are illustrated to present their geometric properties. The resulting curves form a new family of curves defined by the given frame structures in Euclidean 3-space.

Keywords

• Pedal curves
• Bishop frame
• Darboux vector
• Frenet frame
• Curvature
• Torsion

Ethical Statement

Ethics committee approval was not required for this study because of there was no study on animals or humans.

Thanks

This study is based on the master’s thesis of the second author.

References

1. Bayar, F., Tul, S., & Sarıoğlugil, A. (2019). Relations between the frames of a curve and its pedal in . Journal of Science and Arts, 46(1), 23–40.
2. Bishop, R. L. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246–251. https://doi.org/10.2307/2319846
3. Canlı, D., Şenyurt, S., Ertem Kaya, F., & Grilli, L. (2024). The pedal curves generated by alternative frame vectors and their Smarandache curves. Symmetry, 16(8), Article 1012. https://doi.org/10.3390/sym16081012
4. Ceylan, A. Y., & Kara, M. (2021). On pedal and contrapedal curves of Bézier curves. Konuralp Journal of Mathematics, 9(2), 217–221.
5. Damar, E. (2024). Adjoint curves of special Smarandache curves with respect to the Bishop frame. AIMS Mathematics, 9(12), 35355–35376. https://doi.org/10.3934/math.20241680
6. Damar, E. (2025). A novel approach to ruled surfaces using adjoint curve. Symmetry, 17(7), Article 1018. https://doi.org/10.3390/sym17071018
7. do Carmo, M. P. (1976). Differential geometry of curves and surfaces. Prentice-Hall.
8. Ertem Kaya, F. (2024). Differential geometric aspects of pedal curves on surfaces. Journal of Mathematics, 2024, Article 6237091. https://doi.org/10.1155/2024/6237091

9. Gür Mazlum, S. (2024). Öklid 3-uzayında herhangi bir regüler eğrinin Bishop çatıları üzerine. Afyon Kocatepe Üniversitesi Fen ve Mühendislik Bilimleri Dergisi, 24(1), 23–33. https://doi.org/10.35414/akufemubid.1343172
10. Lockwood, E. H. (1961). A book of curves. Cambridge University Press.
11. O’Neill, B. (1996). Elementary differential geometry (2nd ed.). Academic Press.
12. Özdemir, M. (2020). Diferansiyel geometri. Altın Nokta Yayınevi.
13. Şenyurt, S., Ertem Kaya, F., & Canlı, D. (2024). Pedal curves obtained from Frenet vector of a space curve and Smarandache curves belonging to these curves. AIMS Mathematics, 9(8), 20136–20162. https://doi.org/10.3934/math.2024981
14. Şenyurt, S., & Damar, E. (2026). Construction of new curves on $n$th degree polynomial-type curves using the Flc frame. Demonstratio Mathematica, 59(1), Article 20250264. https://doi.org/10.1515/dema-2025-0264
15. Soliman, M. A., Abdel-All, N. H., Hussien, R. A., & Youssef, T. (2019). Evolution of space curves using Type-3 Bishop frame. Caspian Journal of Mathematical Sciences, 8(1), 58–73. https://doi.org/10.22080/CJMS.2018.14455.1344
16. 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. https://doi.org/10.1016/j.jmaa.2010.06.002
17. Yılmaz, S., & Savcı, Ü. (2017). A new version Darboux vector and characterization of some special curves according to Type-2 Bishop frame in . Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 87(3), 355–362. https://doi.org/10.1007/s40010-017-0373-6