Generalized focal surfaces of spacelike curves lying in lightlike surfaces

Abstract

The main work of this paper is to investigate two kinds of generalized focal surfaces and two kinds of evolutes generated by spacelike curve γ lying in lightlike surfaces in Minkowski three‐space. Applying the method of unfolding theory in singularity theory to our study, it is shown that there exist the cuspidal edge and the swallowtail types of singularities in each of two classes of generalized focal surfaces under certain conditions; the only cusps will appear in each of evolutes. Two new geometric invariants are presented to classify the singularities of generalized focal surfaces and evolutes. Much more importantly, we reveal the correspondence among the geometric invariants, the types of singularities on generalized focal surfaces and evolutes, the singularities of two kinds of evolutes, and the contact of γ with the osculating spheres. Finally, several examples are presented to demonstrate the correctness of the theoretical results.

On Generalized Darboux Frame of a Pseudo Null Curve Lying on a Lightlike Surface in Minkowski 3-space

“…Pseudo null curves lying on non-degenerate or degenerate surfaces in Minkowski 3space, defined in terms of the Darboux frame's vector fields, are introduced as k-type pseudo null slant helices and isophote curves in [10,8]. Darboux frame's vector fields are also used in characterizations of different types of curves, such as pseudo-spherical Darboux images and lightcone images of principal-directional curves ( [16]) and in investigations of generalized focal surfaces ( [6]) and lightlike surfaces along pseudo-spherical normal Darboux images of spacelike curves ( [14]).…”

On geometry of focal surfaces due to Flc frame in Euclidean 3-space

“…In addition, these curves can be used to create a skeleton of a surface, which Gordon 3 and Thirion and Gourdon 9 have used to compare surfaces. There are many studies on focal surfaces to date, 2,4,16,6,17 . Tube surfaces 8 .…”

Evolutes and focal surfaces of (1, k)‐type curves with respect to Bishop frame in Euclidean 3‐space

“…2 To solve the singularity problems, we can use the unfolding theory, Legendrian singularity theory or duality theory and so on, [3][4][5][6][7][8][9][10][11][12][13] for instance, Liu and Wang studied generalized focal surfaces and evolutes generated by a spacelike curve which are located in lightlike surfaces in Minkowski 3-space. 14 Then, they showed the types of their singularities under certain conditions according to the unfolding theory. The singularity problems of lightlike curves and lightlike surfaces can also be solved from the viewpoint of Legendrian dualities.…”