On The Darboux Frame Motion of a Developable Surface

Developable ruled surfaces and the rotation-minimizing frames (RMF) have a crucial importance in computer-aided geometric design (CAGD) because of being unfoldable (developable) into a plane without distortion. Also, the RMF of curves are generally used in computer graphics. In this paper, we discuss the relationship between spherical motion and a developable ruled surface that has a line of curvature as a base curve. On the other hand, we use spherical linear interpolation (SLERP) to analyze the relationship between the trajectory generated on the unit sphere due to this spherical motion and the inflection curve. Moreover, we examine some invariant values of this developable ruled surface. We also present the S.Frenet frame of this trajectory based on general quaternion frame equations