Evolutes of dual spherical curves for ruled surfaces

Abstract

E. Study found that there is a one‐to‐one correspondence between the oriented lines in Euclidean three space and the dual points of the dual unit sphere in dual three space, and it has wide applications in Engineering. In this paper, we investigate a ruled surface as a curve on the dual unit sphere by using E. Study's theory. Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves. Finally, we give an example to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.

Evolutes of fronts in the Minkowski plane

“…Along with the moving frame, they defined a pair of smooth functions like as the curvature of a regular curve and called the pair the curvature of the Legendrian curve. By using the moving frame, they can give a new definition of evolute of the front . In this paper, we proceed with this way to investigate the evolutes of smooth curves in the Minkowski plane.…”

“…By using the moving frame, they can give a new definition of evolute of the front. 4,7 In this paper, we proceed with this way to investigate the evolutes of smooth curves in the Minkowski plane. To the best of the authors knowledge, no literature exists regarding the evolute of fronts in the Minkowski plane.…”

Tubular surfaces of center curves on spacelike surfaces in Lorentz‐Minkowski 3‐space

“…It is well known that there exist three kinds of submanifolds, that is, spacelike submanifolds, timelike submanifolds, and lightlike submanifolds in Lorentz‐Minkowski space. Most researchers focus on the submanifolds immersed in Lorentzian space forms with constant curvature from the viewpoint of singularity, the geometric properties of these submanifolds have been explored extensively by the other scholars and the second author . However, the causalities (ie, the structure as a Lorentzian manifold) of submanifolds with nonconstant curvature are quite different from those of the Lorentzian space forms with constant curvature, therefore there will be some different and interesting results when we consider a submanifold immersed in a higher dimensional submanifold from the viewpoint of singularity, for instance, Ito and Izumiya investigate regular curves on a spacelike surfaces in Lorentz‐Minkowski 3‐space, they introduce five special curves that are in Lorentzian subspace forms with constant curvature along the curve associated to the Lorentzian Darboux frame and obtain some good results .…”

“…Most researchers focus on the submanifolds immersed in Lorentzian space forms with constant curvature from the viewpoint of singularity, the geometric properties of these submanifolds have been explored extensively by the other scholars and the second author. [8][9][10][11][12][13][14][15][16][17] However, the causalities (ie, the structure as a Lorentzian manifold) of submanifolds with nonconstant curvature are quite different from those of the Lorentzian space forms with constant curvature, therefore there will be some different and interesting results when we consider a submanifold immersed in a higher dimensional submanifold from the viewpoint of singularity, for instance, Ito and Izumiya investigate regular curves on a spacelike surfaces in Lorentz-Minkowski 3-space, they introduce five special curves that are in Lorentzian subspace forms with constant curvature along the curve associated to the Lorentzian Darboux frame and obtain some good results. 6 However, to the best of the authors' knowledge, we can not find any literature on the study for regarding curves lying in spacelike surfaces as the original objects and considering the singularities of surfaces generated by these curves.…”

Null surfaces of null Cartan curves in Anti-de Sitter 3-space

“…Bonnor [1] gave a method for the general study of the geometry of null curves in Lorentz manifolds and more generally, in semi-Riemannian manifolds. Ferrandez et al [6][7][8] have generalized the Cartan frame to semi-Riemannian space forms. They proved the fundamental existence and uniqueness theorems and obtained values of the Cartan curvatures in higher dimensions.…”