In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and swallowtails. In order to characterize these singularities, two geometric invariants are discovered. At last, an example is given to demonstrate the main results.

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

In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and focal surfaces of timelike Sabban curves in de Sitter space. We find that de Sitter focal surfaces can be regarded as caustics and de Sitter evolutes corresponding to the locus of the polar vectors of osculating de Sitter subspaces. By using singularity theory, we classify the singularities of the de Sitter focal surfaces and de Sitter evolutes and show that there is a close relationship between a new geometric invariant and the types of singularities. Moreover, the Legendrian dual relationships between the hyperbolic tangent indicatrix of timelike Sabban curves and the focal surfaces are given. Finally, we provide an example to illustrate our main results.

In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.

In this paper, we address the topic of envelopes created by pseudo-circle families in the Minkowski plane, which exhibit some different properties when compared with the Euclidean case. We provide solutions to all four basic problems associated with these envelopes, namely the existence problem, representation problem, problem on the number of envelopes, and problem on the relationships of definitions.