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A new quaternion valued frame of curves with an application

Cansu Gizem (Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey), gcansu@ankara.edu.tr
Yaylı Yusuf (Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey), yayli@science.ankara.edu.tr
Gök İsmail (Department of Mathematics, Faculty of Science, Ankara University, Ankara, Turkey), igok@science.ankara.edu.tr

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

Keywords: Quaternionic curve, Quaternionic helix, Frenet formulas