In this paper, we consider the notion of the Smarandache curves by considering the asymptotic orthonormal frames of curves lying fully on lightlike cone in Minkowski 3-space R. We give the relationships between Smarandache curves and curves lying on lightlike cone in R.

In this paper, by considering dual geodesic trihedron (dual Darboux frame) we define dual Smarandache curves lying fully on dual unit sphere and corresponding to ruled surfaces. We obtain the relationships between the elements of curvature of dual spherical curve (ruled surface) and its dual Smarandache curve (Smarandache ruled surface) and we give an example for dual Smarandache curves of a dual spherical curve.