A Surface Family with a Common Asymptotic Null Curve in Minkowski 3-Space ${\mathbb{E}}_1^3$

Abstract

This approach is on constructing a surface family with a common asymptotic null curve. It has provided the necessary and sufficient condition for the curve to be an asymptotic null curve and extended the study to ruled and developable surfaces. Subsequently, the study has examined the Bertrand offsets of a surface family with a common asymptotic null curve. Lastly, we support the results of this approach by some examples.

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

“…It must be pointed out that, in Equations ( 25) and ( 26), there exist two geodesic T L curves crossing through every point on the curves ω(v)(ω(v)), where one is ω itself and the other is a geodesic T L line in the orientation e(v), as given in Equation (23). Every constituent of the isoparametric ruled T L surface bundle with the joint T L geodesic ω is defined by two set functions, γ(v) and β(v) ̸ = 0.…”

“…Similar studies into the same kind of the space were performed but in the case of a given spacelike or timelike line of curvature [22]. Recently, a common asymptotic null curve is used to build a surface family, as well as providing the necessary and sufficient conditions for them to be ruled and developable surfaces [23]. In addition, using the Cartan frame, a surfaces family was constructed with an asymptotic curve where the surface was presented as a linear combination of this frame [24,25].…”