Pseudo‐spherical Darboux images and lightcone images of principal‐directional curves of nonlightlike curves in Minkowski 3‐space

Choosing an alternative frame, which is the Frenet frame of the principal‐directional curve along a nonlightlike Frenet curve γ , we define de Sitter Darboux images, hyperbolic Darboux images, and lightcone images generated by the principal directional curves of nonlightlike Frenet curves and investigate geometric properties 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 important invariants. More interestingly, the cusp is closely related to the contact between nonlightlike 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 Darboux images or lightcone images are shown. Some concrete examples are provided to illustrate our results.