A New Construction of Rectifying Direction Curves for Quaternionic Space Q

Our article focuses on the study of quaternions topic introduced by Hamilton. Quaternions are a generalization of complex numbers and have multiple applications in mathematical physics. Another application of quaternions is robotics because what generalizes the imaginary axis is the family i, j, k modeling Euler angles and rotations in space. The first part of the article we recall the different definitions of how the algebra of quaternions is well constructed. The main results are given in the third part and concern: spatial quaternionics rectifying-direction (sqRD) curves and and spatial quaternionic rectifying-donor (sqRDnr) curves. We study a new tip of unit speed associated curves in E3, which is also used in robotic systems and kinematics, like a spatial quaternionic rectifying-direction curve and spatial quaternionic rectifying-donor curve. Then, we achieve qualification for the curves. Moreover, we present applications of spatial quaternionic rectifying-direction to some specific curves like helix, slant helix, Salkowski and anti-Salkowski curves or rectifying curves. In addition, we establish different theorems which generalize the results obtained on the quaternionic curves in Q. Then, we give some examples are finally discussed. Consequently, Our paper is centered around theoretical analysis in geometry rather than experimental investigations. © 2025 World Scientific and Engineering Academy and Society. All rights reserved.