Spatial Generalized Octonionic Curves

This study investigates curves in a 7-dimensional space, represented by spatial generalized octonion-valued functions of a single variable, where the general octonions include real, split, semi, split semi, quasi, split quasi, and para octonions. We begin by constructing a new frame, referred to as the $G_2$-frame, for spatial generalized octonionic curves, and subsequently derive the corresponding derivative formulas. We also present the connection between the $G_2$-frame and the standard orthonormal basis of spatial generalized octonions. Moreover, we verify that Frenet–Serret formulas hold for spatial generalized octonionic curves. We establish the $G_2$-congruence of two spatial generalized octonionic curves and present the correspondence between the Frenet–Serret frame and the $G_2$-frame. A key advantage of the $G_2$-frame is that the associated frame equations involve lower-order derivatives. This method is both time-efficient and computationally efficient. To demonstrate the theory, we present an example of a unit-speed spatial generalized octonionic curve and compute its $G_2$-frame and invariants using MATLAB.