On non-Newtonian Helices in Multiplicative Euclidean Space

Abstract

In this article, spherical indicatrices of a curve and helices are re-examined using both the algebraic structure and the geometric structure of non-Newtonian (multiplicative) Euclidean space. Indicatrices of a multiplicative curve on the multiplicative sphere in multiplicative space are obtained. In addition, multiplicative general helix, multiplicative slant helix and multiplicative clad and multiplicative g-clad helix characterizations are provided. Finally, examples and drawings are given.

Keywords

• Non-Newtonian calculus
• spherical indicatrices
• helices
• multiplicative differential geometry.

Thanks

We would like to thank the esteemed editors and authors for their contributions to the article. I would also like to thank TÜBITAK, a scholarship holder, for supporting me in every field.

References

1. Aniszewska D., Multiplicative Runge-Kutta methods, Nonlinear Dynamics, 50, 262-272, 2007.
2. Aydın M.E., Has A., Yılmaz B., A non-Newtonian approach in differential geometry of curves: Multiplicative rectifying curves, Bulletin of the Korean Mathematical Society, 61(3), 849-866, 2024.
3. Aydın M.E., Has A., Yılmaz B., Multiplicative rectifying submanifolds of multiplicative Euclidean space, Mathematical Methods in the Applied Sciences, 48(1), 329-339, 2025.
4. Bashirov A.E., Kurpınar E.M., Özyapıcı A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337, 36-48, 2008.
5. Bashirov A.E., Mısırlı E., Tandoğdu Y., Özyapıcı A., On modeling with multiplicative differential equations, Applied Mathematics-A Journal of Chinese Universities, 26(4), 425-438, 2011.
6. Bashirov A.E., Rıza M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1), 75-85, 2011.
7. Boruah K., Hazarika B., Some basic properties of bigeometric calculus and its applications in numerical analysis, Afrika Matematica, 32, 211-227, 2021.
8. Ceyhan H., Özdemir Z., Nurkan S.K., Gürgil I., A non-Newtonian approach to geometric phase through optic fiber via multiplicative quaternions, Revista Mexicana de Física, 70(6), 061301, 2024.

9. Foley J.D., Dam A.V., Feiner S.K., Hughes J.F., Computer Graphics: Principles and Practice, Addison-Wesley Professional Publishing, 2013.
10. Georgiev S.G., Multiplicative Differential Geometry, Chapman and Hall/CRC, 2022.
11. Georgiev S.G., Zennir K., Multiplicative Differential Calculus, Chapman and Hall/CRC., 2022.
12. Georgiev S.G., Zennir K., Boukarou A., Multiplicative Analytic Geometry, Chapman and Hall/CRC, 2022.
13. Göktaş S., Kemaloglu H., Yılmaz E., Multiplicative conformable fractional Dirac system, Turkish Journal of Mathematics, 46, 973–990, 2022.
14. Grossman M., Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foundation, 1983.
15. Grossman M., Katz R., Non-Newtonian Calculus, Lee Press, 1972.
16. Gülsen T., Yılmaz E., Göktaş S., Multiplicative Dirac system, Kuwait Journal of Science, 49(3), 1-11, 2022.
17. Has A., Yılmaz B., A non-Newtonian conics in multiplicative analytic geometry, Turkish Journal of Mathematics, 48(5), 976-994, 2024.
18. Has A., Yılmaz B., A non-Newtonian magnetic curves in multiplicative Riemann manifolds, Physica Scripta, 99(4), 045239, 2024.
19. Has A., Yılmaz B., Yıldırım H., A non-Newtonian perspective on multiplicative Lorentz–Minkowski space L^3, Mathematical Methods in the Applied Sciences, 47(18), 13875-13888, 2024.
20. Izumuya S., Takeuchi N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28, 153-163, 2024.
21. Kaya S., Ateş O., Gök I., Yaylı Y., Timelike clad helices and developable surfaces in Minkowski 3-space, Rendiconti del Circolo Matematico di Palermo Series 2, 68, 259–273, 2019.
22. Mak M., Framed clad helices in Euclidean 3-space, Filomat, 37(28), 9627-9640, 2023.
23. Nurkan S.K., Gürgil I., Karacan M.K., Vector properties of geometric calculus, Mathematical Methods in the Applied Sciences, 46(17), 17672-17691, 2023.
24. Rybaczuk M., Stoppel P., The fractal growth of fatigue defects in materials, International Journal of Fracture, 103, 71-94, 2000.
25. Rybaczuk M., Zielinski W., The concept of physical and fractal dimension I. The projective dimensions, Chaos, Solitons and Fractals, 12(13), 2517-2535, 2001.
26. Sadraey M.H., Aircraft Design: A Systems Engineering Approach, Wiley, 2013.
27. Stanley D., A multiplicative calculus, Primus, 9(4), 310-326, 1999.
28. Takahashi T., Takeuchi N., Clad helices and developable surface, Tokyo Gakugei University Bulletin; Natural Science, 66, 1-9, 2014.
29. Uzer A., Multiplicative type complex calculus as an alternative to the classical calculus, Computers and Mathematics with Applications, 60, 2725-2737, 2010.
30. Volterra V., Hostinsky B., Operations Infinitesimales Lineares, Herman, 1938.
31. Waseem M., Noor N.A., Shah F.A., Noor K.I., An efficient technique to solve nonlinear equations using multiplicative calculus, Turkish Journal of Mathematics, 42, 679-691, 2018.
32. Watson J.D., Baker T., Stephen P.B., Gann A., Michael L., et al., Molecular Biology of the Gene, Pearson Publishing, 2013.
33. Yalçın N., Çelik E., Solution of multiplicative homogeneous linear differential equations with constant exponentials, New Trends in Mathematical Sciences, 6(2), 58-67, 2018.
34. Yazici M., Selvitopi H., Numerical methods for the multiplicative partial differential equations, Open Mathematics, 15, 1344-1350, 2017.
35. Yılmaz B., Has A., New approach to slant helix, International Electronic Journal of Geometry, 12, 111-115, 2019.