Lorentz-Darboux rectifying surfaces and osculating surfaces along frontal curves on timelike surfaces

In this paper, we consider two kinds of developable surfaces along a timelike frontal curve lying in a timelike surface in Minkowski 3-space, the Lorentz-Darboux rectifying surfaces and the Lorentz-Darboux osculating surfaces. Meanwhile, we also consider two curves generated by such a timelike frontal curve. We give two new invariants of the frontal curve which characterize singularities of the Lorentz-Darboux developable surfaces and Lorentz-Darboux rectifying and osculating curves. Unlike the regular curves, the frontal curves may have singular points. Using the methods of the unfolding theory in singularity theory, we complete the classifications of the singular points of these two surfaces and two curves. The main results indicate that compared with developable surfaces along a regular curve, there exists a more complicated construction for the singularities of the developable surfaces along a timelike frontal curve, there will appear an extra locus, arising by the singular point of the timelike frontal curve, for the singularities of the Lorentz-Darboux rectifying surface, whereas the Lorentz-Darboux osculating surface did not. In addition, we investigate the geometric properties of the timelike frontal curve, it is shown that the timelike frontal curve can be regarded as the envelope of a family of timelike frontal rectifying curves. Finally, we provide several examples to illustrate the theoretical results.