On Hybrid Curves

Abstract

In this paper, we first define the vector product in a special analog Minkowski Geometry (R^3,<>) which is identified with the space of spatial hybrids. Next, we derive the Frenet-Serret frame formulae for a three dimensional non-parabolic curve by using the spatial hybrids and the vector product. However, we present the Frenet-Serret frame formulae of a non-lightlike hybrid curve in R^4 and an illustrative example for all theorems of the paper with MATLAB 2016a codes.

Keywords

• Hybrid numbers
• Curvatures
• Frenet-Serret frame formulae

References

1. [1] Bharathi, K., Nagaraj, M., "Quaternion valued function of a real variable Serret–Frenet formulae", Indian J. Pure Appl. Math. 16 (1985) : 741–756.
2. [2] Coken, A.C., Tuna, A., "On the quaternionic inclined curves in the semi-Euclidean space ", Appl. Math. Comput. 155 (2004) : 373-389.
3. [3] Dağdeviren, A., Yüce, S., "Dual quaternions and dual quaternionic curves", Filomat 33(4) (2019) : 1037–1046.
4. [4] Ohashi, M., "G 2-Congruence theorem for curves in purely imaginary octonions and its application", Geom Dedicata 163 (2013) : 1–17.
5. [5] Özdemir, M., "Introduction to Hybrid Numbers", Adv. Appl. Clifford Algebras 28(11) (2018).
6. [6] Özdemir, M., "Finding Roots of a 2×2 real matrix using De Moivre’s formula", Advances in Applied Clifford Algebras 29(2) (2019).
7. [7] Öztürk, İ., Özdemir, M., "Similarity of hybrid numbers", Mathematical Methods in Applied Sciences 43(15) (2020): 8867-8881.
8. [8] Akbıyık, M., S. Yamaç Akbıyık, E. Karaca, F. Yılmaz, "De Moivre’s and Euler Formulas for matrices of hybrid numbers", Axioms 2021, 10, 213.

9. [9] Szynal-Liana, A., "The Horadam Hybrid Numbers", Discussiones Mathematicae General Algebra and Applications 38 (2018) : 91–98.
10. [10] Kızılateş, C., "A new generalization of Fibonacci hybrid and Lucas hybrid numbers", Chaos, Solitons & Fractals 130 (2020) : 109449.