Relatively normal-slant helices lying on a surface and their characterizations

Abstract

In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame {T,V,U} along the curve, where T is the unit tangent vector field of the curve, U is the surface normal restricted to the curve and V = U x T. We define a new curve on a surface by using the Darboux frame. This new curve whose vector field V makes a constant angle with a fixed direction is called as relatively normal-slant helix. We give some characterizations for such curves and obtain their axis. Besides we give some relations between some special curves (general helices, integral curves, etc.) and relatively normal-slant helices. Moreover, when a regular surface is given by its implicit or parametric equation, we introduce the method for generating the relatively normal-slant helix with the chosen direction and constant angle on the given surface.