Smarandache-Based Ruled Surfaces with the Darboux Vector According to Frenet Frame in $E^3$

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 corresponding unit Darboux vector. The characteristics of these newly defined ruled surfaces are investigated by means of first and second fundamental forms and their corresponding curvatures. An example is provided by considering both the helix curve and the Viviani’s curve.