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

Özgür Boyacioğlu Kalkan; Şevval Çınar

doi:10.34248/bsengineering.2001123

EN TR

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

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.

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

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

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.
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
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
Ceylan, A. Y., & Kara, M. (2021). On pedal and contrapedal curves of Bézier curves. Konuralp Journal of Mathematics, 9(2), 217–221.
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
Damar, E. (2025). A novel approach to ruled surfaces using adjoint curve. Symmetry, 17(7), Article 1018. https://doi.org/10.3390/sym17071018
do Carmo, M. P. (1976). Differential geometry of curves and surfaces. Prentice-Hall.
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

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
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Özdemir, M. (2020). Diferansiyel geometri. Altın Nokta Yayınevi.
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
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
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
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
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

Details

Primary Language

English

Subjects

Algebraic and Differential Geometry

Journal Section

Research Article

Authors

Özgür Boyacioğlu Kalkan ^*
0000-0003-1665-233X
Türkiye

Şevval Çınar
0009-0001-0809-8183
Türkiye

Publication Date

November 15, 2026

Submission Date

July 23, 2026

Acceptance Date

October 8, 2026

Published in Issue

Year 2026 Volume: 9 Number: 6

DOI

https://doi.org/10.34248/bsengineering.2001123

IZ

https://izlik.org/JA74SH62TL

Cite

RIS / Bibtex

APA

Boyacioğlu Kalkan, Ö., & Çınar, Ş. (2026). Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space. Black Sea Journal of Engineering and Science, 9(6), 3372-3389. https://doi.org/10.34248/bsengineering.2001123

AMA

1.Boyacioğlu Kalkan Ö, Çınar Ş. Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space. BSJ Eng. Sci. 2026;9(6):3372-3389. doi:10.34248/bsengineering.2001123

Chicago

Boyacioğlu Kalkan, Özgür, and Şevval Çınar. 2026. “Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space”. Black Sea Journal of Engineering and Science 9 (6): 3372-89. https://doi.org/10.34248/bsengineering.2001123.

EndNote

Boyacioğlu Kalkan Ö, Çınar Ş (November 15, 2026) Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space. Black Sea Journal of Engineering and Science 9 6 3372–3389.

IEEE

[1]Ö. Boyacioğlu Kalkan and Ş. Çınar, “Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space”, BSJ Eng. Sci., vol. 9, no. 6, pp. 3372–3389, Nov. 2026, doi: 10.34248/bsengineering.2001123.

ISNAD

Boyacioğlu Kalkan, Özgür - Çınar, Şevval. “Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space”. Black Sea Journal of Engineering and Science 9/6 (November 15, 2026): 3372-3389. https://doi.org/10.34248/bsengineering.2001123.

JAMA

1.Boyacioğlu Kalkan Ö, Çınar Ş. Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space. BSJ Eng. Sci. 2026;9:3372–3389.

MLA

Boyacioğlu Kalkan, Özgür, and Şevval Çınar. “Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space”. Black Sea Journal of Engineering and Science, vol. 9, no. 6, Nov. 2026, pp. 3372-89, doi:10.34248/bsengineering.2001123.

Vancouver

1.Özgür Boyacioğlu Kalkan, Şevval Çınar. Construction of Pedal Curves Using Different Types of Bishop Frames in Euclidean 3-Space. BSJ Eng. Sci. 2026 Nov. 15;9(6):3372-89. doi:10.34248/bsengineering.2001123