Slant helices: a new approximation

Abstract

In this paper, we study a weaker version of classic slant helices in Euclidean space R 3 or Minkowski space R 3 1 , which will be called general slant helices. We show that any classic slant helix is a general slant helix but the converse is not true. We also obtain equations involving the curvature and torsion that characterize this family of curves.

On Null Cartan Rectifying Isophotic and Rectifying Silhouette Curves Lying on a Timelike Surface in Minkowski Space $\mathbb{E}^3_1$

“…The vector fields T , ζ and η satisfying relation (2.10) are called constant vector fields with respect to the Darboux frame ( [18]). Throughout the next sections, let R 0 denote R\ {0}.…”

“…The notion of the helix in E 3 is generalized in [18] in terms of the vector field that is constant with respect to the Frenet frame of the curve and forms a constant angle with some fixed direction. If the mentioned vector field lies in a normal, rectifying, or osculating plane, the curve is called a normal, rectifying and osculating helix, respectively.…”

Constant pseudo-angle lightlike surfaces

“…or a Cartan slant helix (for further information on these helices see [15], [19], [2], [17], [16], [5]). We classify such ruled surfaces in three cases depending on type of the corresponding helices.…”

Slant helices in Minkowski 3-space E31 with Sasai’s modified frame fields

“…A regular curve ϕ(s) in E 3 1 is said to be a general slant helix if there is a Killing vector field U along with constant length such that it make constant angle with principal normal vector N [14]. U is called an axis of the general slant helix.…”

“…where v is the velocity of the curve ϕ. From [14], it is easy to see that U(s) is a Killing vector field along a spacelike curve ϕ with spacelike principal normal if and only if it satisfies the following conditions:…”