Legendrian dual surfaces of a spacelike curve in the 3-dimensional lightcone

In this paper, we study Legendrian dual surfaces, lying in the 3-dimensional pseudo-(hyper)spheres of various radii in Lorentz-Minkowski 4-space, of a spacelike curve in the 3-dimensional lightcone. We point out that these surfaces are constructed by the extended Legendrian dualities between the 3-dimensional lightcone and these 3-dimensional pseudo-spheres. Moreover, we investigate the singularities of these surfaces and show the dualities of the singularities of a certain class of such a surface in the 3-dimensional lightcone. Furthermore, we deal with the singularities of all of these surfaces when they are non-flat.