On the quaternionic osculating direction curves

Quaternions are widely used in physics. Quaternions, an extension of complex numbers, are closely related to many fundamental concepts (e.g. Pauli matrices) in physics. The aim of this study is the geometric structure underlying the quaternions used in physics. In this paper, we have investigated a new structure of unit speed associated curves such as spatial quaternionic and quaternionic osculating direction curves. For this, we have assumed that the vector fields chi(rho) = nu(1)(rho)t(rho) + nu(2)(rho)n(rho), where nu(2)(1)(rho) + nu(2)(2)(rho) = 1 for the spatial quaternionic curve psi and chi(rho) = lambda(1)(rho) inverted perpendicular (rho) + lambda(2)(rho)eta(rho) + lambda(3)beta(2)(rho), where lambda(2)(1)(rho) + lambda(2)(2)(rho) + lambda(2)(3)(rho) = 1 for the quaternionic curve phi. Then, we have given the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we have examined in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. So, considering that helices also take place in electron physics, it is thought that this study will create a bridge between physics and geometry. Finally, we have given the examples and draw the figures of curves in the examples.