Smarandache Ruled Surfaces according to Darboux Frame in $E^3$

Abstract

This paper presents a new approach of constructing special ruled surfaces and aims to study their developability and minimalist conditions. Our concept opens opportunities for application in engineering, surface modeling, and architectural design. The principle of our study is to introduce the original definitions of Smarandache ruled surfaces according to Darboux frame of a curve lying on an arbitrary regular surface in E 3 . It concerns T g -Smarandache ruled surface, T n -smarandache ruled surface, and g n -Smarandache ruled surface. New theorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. 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.…”