Curves and surfaces of spacelike curves according to Bishop frame and their singularities

Legendrian dualities between pseudo-spherical images of spacelike curves in Minkowski 3-space are investigated by using the theory of Legendrian duality. Moreover, the singularities of parallel lightcone developables, dual surface, Bishop pseudospherical Darboux images and Bishop pseudo-spherical images, which are generated by spacelike curves, are classified from the viewpoints of wave fronts and caustics, and we also give some more detail descriptions on the conditions of those singularities. Finally, some properties of parallel slant helix are given.