Smarandache-Based Ruled Surfaces with the Darboux Vector According to Frenet Frame in E3

Abstract

The paper introduces a new kind of special ruled surface. The base of each ruled surface is taken to be one of the Smarandache curves of a given curve according to Frenet frame, and the generator (ruling) is chosen to be the correspond- ing unit Darboux vector. The characteristics of these newly defined ruled surfaces are investigated by means of first and second fundamental forms and their corre- sponding curvatures. An example is provided by considering both the helix curve and the Viviani’s curve.