Singularities of Dual Hypersurfaces and Hyperbolic Focal Surfaces Along Spacelike Curves in Light Cone in Minkowski 5-Space

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

“…Many geometric objects including evolutes and focal surfaces may have singularities. 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.…”

Slant helix of order n and sequence of Darboux developables of principal‐directional curves

“…Some new results concerning the singularities of submanifolds were established by the second author and his collaborators. [12][13][14][15][16][17][18][19][20][21][22][23][24] As an application of singularity theory, in this paper, we study the singularities of the Darboux developable of nth principal-directional curve of a curve. It is demonstrated that the ratio of torsion to curvature of a curve play a key role in characterizing the singularities of the Darboux developables of the nth principal-directional curve of a curve .…”

“…Better understanding of the local topological structure of singularities of a manifold is very important. Some new results concerning the singularities of submanifolds were established by the second author and his collaborators 12‐24 . As an application of singularity theory, in this paper, we study the singularities of the Darboux developable of n th principal‐directional curve of a curve.…”

Generalized focal surfaces of spacelike curves lying in lightlike surfaces

“…Geometric objects in Minkowski space, regarding singularity, have been studied extensively by, among others, the second author and by previous researchers. [14][15][16][17][18][19][20][21][22][23][24][25][26] However, to the best of the authors' knowledge, the singularities of surfaces and curves as they relate to a curve lying in a general lightlike surface have not been considered in the literature, aside from the case of lightcone. Thus, the current study hopes to serve such a need.…”