Singularities of Osculating Developable Surfaces of Timelike Surfaces along Curves

In this paper, we focus on a developable surface tangent to timelike along curve in Minkowski 3-space, which is called the osculating of curve. The ruling parallel Darboux vector field. main goal paper classify singularities surface. To end, two new invariants curves are defined characterize these singularities. Meanwhile, also research singular properties surfaces near their lightlike points. Moreover, give relation between fields and normal from viewpoint Legendrian dualities. Finally, some examples with symmetrical structures presented illustrate results.