On the Differential Geometry of Tq - Pedal, Nq - Pedal and Bq - Pedal Curves Derived Via Quasi-Frames in Euclidean 3-Space E^3

Abstract

This paper investigates the differential geometric properties of generalized pedal curves in Euclidean 3-space , constructed using the Quasi-frame apparatus along a space curve α(s). By projecting the position vector onto the quasi-frame field , we define the pedal (tangent-quasi pedal), pedal (normal-quasi pedal), and pedal (binormal-quasi pedal) curves with respect to a given origin O. We establish the explicit derivative formulas, tangent vectors, arclength parameters, principal curvatures, and torsions for each associated pedal curve. Furthermore, we analyze the geometric conditions under which these pedal curves become degenerate, planar, or straight line and explore the relationships between their quasi-curvature functions and the original curve's geometry.

Keywords

• Quasi-frame (q-frame)
• Tq-pedal curves
• nq-pedal curves
• Pedal curves
• Quasi-pedal curvature
• Quasi-pedal torsion

Project Number

No project number

Ethical Statement

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

References

1. As, E., & Sarıoğlugil, A. (2015). On the pedal surfaces of 2-d surfaces with the constant support function in E4. Pure Mathematical Sciences, 4, 105–120. https://doi.org/10.12988/pms.2015.545
2. 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
3. Ceylan, A. Y., & Kara, M. (2021). On pedal and contrapedal curves of Bézier curves. Konuralp Journal of Mathematics, 9(2), 217–221.
4. Coquillart, S. (1987). Computing offsets of B-spline curves. Computer-Aided Design, 19(6), 305–309. https://doi.org/10.1016/0010-4485(87)90264-5
5. Damar, E. (2024). Adjoint curves of special Smarandache curves with respect to Bishop frame. AIMS Mathematics, 9(12), 35355–35376. https://doi.org/10.3934/math.20241712
6. Damar, E. (2025). Tzitzeica curves according to modified orthogonal frames. Cumhuriyet Science Journal, 46(3), 595–600.
7. Dede, M., Ekici, C., & Tozak, H. (2015). Directional tubular surfaces. International Journal of Algebra, 9(12), 527–535. https://doi.org/10.12988/ija.2015.51060
8. do Carmo, M. P. (1976). Differential geometry of curves and surfaces. Prentice-Hall.

9. Elshenhab, A. M., Moaaz, O., Dassios, I., & Elsharkawy, A. (2022). Motion along a space curve with a quasi-frame in Euclidean 3-space: Acceleration and jerk. Symmetry, 14(8), Article 1610. https://doi.org/10.3390/sym14081610
10. Elsharkawy, A., Tashkandy, Y., Emam, W., Cesarano, C., & Elsharkawy, N. (2023). On some quasi-curves in Galilean three-space. Axioms, 12(9), Article 823. https://doi.org/10.3390/axioms12090823
11. Elsharkawy, A., Elsayied, H. K., Desouky, M. E., & Cesarano, C. (2025). T-pedal ruled surface with the Frenet frame of the original curve in E3. AIMS Mathematics, 10(11), 25606–25623. https://doi.org/10.3934/math.20251134
12. 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
13. Ertem Kaya, F. (2025). Gaussian and mean curvatures of the surface obtained along T-pedal curve of a given curve. Turkish Journal of Science, 10(2), 105–120.
14. Ertem Kaya, F. (2026). Differential geometry of parallel curves derived from pedal curves of a given curve in Euclidean 3-space. AIMS Mathematics, 11(8), 24397–24418. https://doi.org/10.3934/math.2026985
15. Gray, A., Abbena, E., & Salamon, S. (2006). Modern differential geometry of curves and surfaces with Mathematica (3rd ed.). Chapman & Hall/CRC.
16. Kaymanlı, G. U., & Ekici, C. (2022). Evolutions of the ruled surfaces along a spacelike space curve. Punjab University Journal of Mathematics, 54(4), 221–232. https://doi.org/10.52280/pujm.2022.540401
17. Kuruoğlu, N., & Sarıoğlugil, A. (1993). On the characteristic properties of the a pedal surfaces in the Euclidean space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 42(1–2), 19–25. https://doi.org/10.1501/Commua1_0000000179
18. Li, Y., & Pei, D. (2016). Pedal curves of fronts in the sphere. Journal of Nonlinear Sciences and Applications, 9(3), 836–844. https://doi.org/10.22436/jnsa.009.03.12
19. Li, Y., & Pei, D. (2018). Pedal curves of frontals in the Euclidean plane. Mathematical Methods in the Applied Sciences, 41(5), 1988–1997. https://doi.org/10.1002/mma.4724
20. Li, Y., & Tuncer, O. O. (2023). On (contra)pedals and (anti)orthotomics of frontals in de Sitter 2-space. Mathematical Methods in the Applied Sciences, 46(10), 11157–11171. https://doi.org/10.1002/mma.9173
21. Li, M., Yao, K., Li, P., & Pei, D. (2022). Pedal curves of non-lightlike curves in Minkowski 3-space. Symmetry, 14(1), Article 59. https://doi.org/10.3390/sym14010059
22. Nishimura, T. (2008). Normal forms for singularities of pedal curves produced by non-singular dual curve germs in Sn. Geometriae Dedicata, 133(1), 59–66. https://doi.org/10.1007/s10711-008-9233-5
23. O'Neill, B. (2006). Elementary differential geometry (Revised 2nd ed.). Elsevier Academic Press.
24. Özdemir, M. (2020). Differential geometry. Altın Nokta Publishing House.
25. 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.2024980
26. Şenyurt, S., & Damar, E. (2026). Construction of new curves on nth degree polynomial-type curves using the FLC frame. Demonstratio Mathematica, 59(1), Article 20250264. https://doi.org/10.1515/dema-2025-0264
27. Tuncer, O. O., Ceyhan, H., Gök, İ., & Ekmekci, F. N. (2018). Notes on pedal and contrapedal curves of fronts in the Euclidean plane. Mathematical Methods in the Applied Sciences, 41(13), 5096–5111. https://doi.org/10.1002/mma.5056