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

In this study, we investigate how to construct surfaces using a line of curvature in a 3-dimensional Lie group. Then, by utilizing the Frenet frame, we give the conditions that a curve becomes a line of curvature on a surface when the marching-scale functions are more general expressions. After then, we provide some crucial examples of how efficient our method is on these surfaces.