In this talk, we consider the curves on the surface in de Sitter 3-space. For a regular curve on a surface in de Sitter 3-space, we have a moving frame along the curve which is called a de Sitter Darboux frame. We introduce a special vector field along the curve with respect to the de Sitter Darboux frame and investigate the singularities of the normal extrinsic flat pseudo great circular surface associated to the vector field.