On non-Newtonian Helices in Multiplicative Euclidean Space

Year 2025, Volume: 6 Issue: 2, 196 - 217, 30.07.2025

Aykut Has , Beyhan Yılmaz

https://doi.org/10.54974/fcmathsci.1644427

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

• Aniszewska D., Multiplicative Runge-Kutta methods, Nonlinear Dynamics, 50, 262-272, 2007.
• 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.
• 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.
• Bashirov A.E., Kurpınar E.M., Özyapıcı A., Multiplicative calculus and its applications, Journal of Mathematical Analysis and Applications, 337, 36-48, 2008.
• 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.
• Bashirov A.E., Rıza M., On complex multiplicative differentiation, TWMS Journal of Applied and Engineering Mathematics, 1(1), 75-85, 2011.
• Boruah K., Hazarika B., Some basic properties of bigeometric calculus and its applications in numerical analysis, Afrika Matematica, 32, 211-227, 2021.
• 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.
• Foley J.D., Dam A.V., Feiner S.K., Hughes J.F., Computer Graphics: Principles and Practice, Addison-Wesley Professional Publishing, 2013.
• Georgiev S.G., Multiplicative Differential Geometry, Chapman and Hall/CRC, 2022.
• Georgiev S.G., Zennir K., Multiplicative Differential Calculus, Chapman and Hall/CRC., 2022.
• Georgiev S.G., Zennir K., Boukarou A., Multiplicative Analytic Geometry, Chapman and Hall/CRC, 2022.
• Göktaş S., Kemaloglu H., Yılmaz E., Multiplicative conformable fractional Dirac system, Turkish Journal of Mathematics, 46, 973–990, 2022.
• Grossman M., Bigeometric Calculus: A System with a Scale-Free Derivative, Archimedes Foundation, 1983.
• Grossman M., Katz R., Non-Newtonian Calculus, Lee Press, 1972.
• Gülsen T., Yılmaz E., Göktaş S., Multiplicative Dirac system, Kuwait Journal of Science, 49(3), 1-11, 2022.
• Has A., Yılmaz B., A non-Newtonian conics in multiplicative analytic geometry, Turkish Journal of Mathematics, 48(5), 976-994, 2024.
• Has A., Yılmaz B., A non-Newtonian magnetic curves in multiplicative Riemann manifolds, Physica Scripta, 99(4), 045239, 2024.
• 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.
• Izumuya S., Takeuchi N., New special curves and developable surfaces, Turkish Journal of Mathematics, 28, 153-163, 2024.
• 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.
• Mak M., Framed clad helices in Euclidean 3-space, Filomat, 37(28), 9627-9640, 2023.
• Nurkan S.K., Gürgil I., Karacan M.K., Vector properties of geometric calculus, Mathematical Methods in the Applied Sciences, 46(17), 17672-17691, 2023.
• Rybaczuk M., Stoppel P., The fractal growth of fatigue defects in materials, International Journal of Fracture, 103, 71-94, 2000.
• Rybaczuk M., Zielinski W., The concept of physical and fractal dimension I. The projective dimensions, Chaos, Solitons and Fractals, 12(13), 2517-2535, 2001.
• Sadraey M.H., Aircraft Design: A Systems Engineering Approach, Wiley, 2013.
• Stanley D., A multiplicative calculus, Primus, 9(4), 310-326, 1999.
• Takahashi T., Takeuchi N., Clad helices and developable surface, Tokyo Gakugei University Bulletin; Natural Science, 66, 1-9, 2014.
• Uzer A., Multiplicative type complex calculus as an alternative to the classical calculus, Computers and Mathematics with Applications, 60, 2725-2737, 2010.
• Volterra V., Hostinsky B., Operations Infinitesimales Lineares, Herman, 1938.
• 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.
• Watson J.D., Baker T., Stephen P.B., Gann A., Michael L., et al., Molecular Biology of the Gene, Pearson Publishing, 2013.
• Yalçın N., Çelik E., Solution of multiplicative homogeneous linear differential equations with constant exponentials, New Trends in Mathematical Sciences, 6(2), 58-67, 2018.
• Yazici M., Selvitopi H., Numerical methods for the multiplicative partial differential equations, Open Mathematics, 15, 1344-1350, 2017.
• Yılmaz B., Has A., New approach to slant helix, International Electronic Journal of Geometry, 12, 111-115, 2019.