Pseudo-spherical Darboux Images And Lightcone Images Of Principal-directional Curves In Minkowski 3-space

Choosing an alternative frame which is the Frenet frame of the principal-directional curve along a non-lightlike Prenet curve γ,we define de Sitter Darboux images,hyperbolic Darboux images and lightcone images generated by the principal directional curves of non-lightlike Frenet curves and investigate geometric proper-ties of these associated curves under considerations of singularity theory,contact and Legendrian duality.It is shown that pseudo-spherical Darboux images and lightcone images can occur singularities(ordinary cusp)characterized by some im-portant invariants.More interestingly,the cusp is closely related to the contact between non-lightlike Frenet curve γ and a slant helix,the principal-directional curve Ψ of γ and a helix or the principal-directional curve Ψ and a slant helix.In addition,some relations of Legendrian dualities between C-curves and pseudo-spherical Daxboux images or lightcone images are shown.Some concrete examples axe provided to illustrate our results.