Surfaces with a common asymptotic curve in Minkowski 3-space

TL;DR

This paper characterizes surfaces in Minkowski 3-space that share a common asymptotic curve, providing parametric expressions and conditions based on the Frenet frame, with illustrative examples.

Contribution

It introduces a method to construct surfaces with a specified asymptotic curve in Minkowski space using Frenet frames, including necessary and sufficient conditions.

Findings

Derived parametric expressions for such surfaces

Established conditions for the Frenet frame coefficients

Provided examples validating the theoretical results

Abstract

In this paper, we express surfaces parametrically through a given spacelike (timelike) asymptotic curve using the Frenet frame of the curve in Minkowski 3-space. Necessary and sufficient conditions for the coefficients of the Frenet frame to satisfy both parametric and asymptotic requirements are derived. We also present some interesting examples to show the validity of this study.