DIFFERENT PERSPECTIVE OF DARBOUX SLANT RULED SURFACES AND THEIR CONSTRUCTIONS VIA QUATERNIONS

In this paper, based on the definition of Darboux slant ruled surface in R^3, a novel method for the Darboux slant ruled surface obtained from the striction curve of the natural lift curve is described in R^3. Then, an isomorphism between the unit dual sphere, 〖DS〗^2, and the subset of the unit 2-sphere, TM ̅ is used to define this surface. Hence, the striction curve of the natural lift curve is corresponded to Darboux slant ruled surface in R^3by considering the isomorphism. Additionally, some fundamental theorems are proved to characterize this surface. Using the quaternion product of the quaternionic operator and a pure unit quaternion, the quaternionic expression for the Darboux slant ruled surface is defined. In the last, 2-parameter homothetic motion is expressed for this surface.