Frenet-Serret Formulas and Frenet Elements of the n-Dimensional Quaternionic and Spatial Quaternionic Spaces

Abstract

Four-dimensional number systems are of great importance in physics, robotics, and computer-aided engineering. The quaternions give a chance to make examinations on 4-dimensional structures and also 3-dimensional calculations as spatial quaternions. The major goal of this research is to extend the Frenet elements for the Frenet-Serret n-vectors for spatial quaternionic and quaternionic curves constructed in H^n using real variable quaternion-valued vector functions. For this, the mathematical formulations of the Frenet elements in H_P^n are evaluated, and the results in H^n are presented by using them. Finally, exemplifications of spatial quaternion and quaternion curves with n-vectors are provided to support the conclusions.