Surface pencil with a common line of curvature in Minkowski 3-space

TL;DR

This paper develops a method to construct surface pencils in Minkowski 3-space from a given spacelike or timelike line of curvature, using the Frenet frame to derive conditions for the surface to have the specified line as a curvature line.

Contribution

It introduces a new approach to generate surface pencils with prescribed lines of curvature in Minkowski 3-space using Frenet frames.

Findings

Derived necessary and sufficient conditions for the coefficients of the surface pencil.

Provided examples illustrating the construction of surface pencils with lines of curvature.

Established a framework for analyzing curvature lines in Minkowski 3-space.

Abstract

In this paper, we analyze the problem of constructing a surface pencil from a given spacelike (timelike) line of curvature. By using the Frenet frame of the given curve in Minkowski 3-space, we express the surface pencil as a linear combination of this frame and derive the necessary and sufficient conditions for the coefficients to satisfy the line of curvature requirement. To illustrate the method some examples showing members of the surface pencil with their line of curvature are given.