$\text{NC}$-Smarandache Ruled Surface and $\text{NW}$-Smarandache Ruled Surface according to Alternative Moving Frame in $E^{}$

Abstract

In this study, original definitions of NC − Smarandache ruled surface and NW − Smarandache ruled surface according to the alternative moving frame are introduced in E 3 . The main results of the study are presented in theorems that give necessary and sufficient conditions for those special surfaces to be developable and minimal. Finally, an example with illustrations is presented.

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

“…In addition, studies with Smarandache curves are available in [1, 4, 6, 24, 26-31, 34, 35]. Recently, in [18][19][20], generating the way of new ruled surfaces have been given by exploiting the idea of Smarandache geometry and using the Frenet, the Darboux or the alternative frame. In the light of all these informations in this study, we obtain the ruled surfaces from the direction vector obtained from the Frenet and Darboux vectors of any curve and from the Smarandache base curve obtained in the same way.…”