Spacelike developable surfaces through a common line of curvature in Minkowski 3-space

Abstract

This paper studies the problem to construct a spacelike developable surface possessing a given non-null parametric curve as the line of curvature in Minkowski 3-space. By using Frenet frame to express the surface, we derive the necessary and sufficient conditions when the resulting spacelike developable surface is cylinder, cone or tangent surface. Finally, we give some representative examples.

A generalization for surfaces using a line of curvature in Lie group

“…In [7], the authors proposed a new method to construct a developable surface with a given curve as the line of curvature of it. After, in [5,9] authors gave the necessary and sufficient conditions for the given non-null curve to be the line of curvature on a spacelike or timelike surface in Minkowski space and investigated the problem of constructing a surface from a given non-null line of curvature.…”

K-type slant helices on spacelike and timelike surfaces

“…The Darboux frame vector field V K cannot be constant because if it is constant, then the relation V k , U = C is always satisfied, and there is nothing left to calculate. κ n e κnτg κg ds ds = 0, (9) where κ n = 0, κ g = 0, and τ g = 0.…”

“…Substituting equation ( 12) into the first part of the equation ( 11), we get (9), where κ n = 0, κ g = 0 and τ g = 0.…”

“…Conversely, assume that γ is a spacelike curve with a spacelike normal lying on a spacelike surface M . Let there exists a vector U defined by κ n e κnτg κg ds ds Z (13) and satisfying condition (9), where κ n = 0, κ g = 0, τ g = 0, and C ∈ R. Now, differentiating the vector U and using the given condition, we obtain U = 0. Hence U is a fixed direction vector such that, T, U = C. Thus γ is a 1-type spacelike slant helix with an axis U , not orthogonal to T .…”