Evolutes of fronts in the Minkowski plane

Abstract

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.

Direction curves of generalized Bertrand curves and involute-evolute curves in $E^{4}$

“…From ( 4), it follows T ⊥sp {N, B 2 }. Then, multiplying (9) with N and B 2 , we have l(s) = 0, n(s) = 0, respectively, and (9) becomes T (s) = p(s)T (s) + r(s)B 1 (s), (10) and from (10), it follows p 2 (s) + r 2 (s) = 1, since T is unit. Differentiating (10) with respect to s and using Frenet formulas (1), we get…”

“…Özyılmaz and Yılmaz studied involute-evolute of W -curves in Euclidean 4-space E 4 [16]. Li and Sun studied evolutes of fronts in the Minkowski Plane [9].…”

Anti-pedals and Primitives of Curves in Minkowski Plane

“…This theory is also used to link physics and mathematics. Many other sub-disciplines of mathematics, including differential geometry and algebra, utilize from it (Li and Sun 2019). The idea of combining differential geometry with singularity theory was proposed by Arnold (1990) and Thom (1956).…”

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.…”