Ruled surfaces according to alternative moving frame

TL;DR

This paper introduces a new class of ruled surfaces based on the alternative frame of a base curve and explores their geometric properties, including curvature and torsion, with illustrative examples.

Contribution

It defines a novel type of ruled surface using the alternative frame and analyzes its differential geometric characteristics.

Findings

Derived formulas for striction line and distribution parameter.

Calculated Gaussian and mean curvatures of the new ruled surface.

Provided examples illustrating the geometric properties.

Abstract

In this paper, we define a new type of ruled surface called ruled surface by using the alternative frame of a base curve. Then, we study its differential geometric properties such as striction line, distribution parameter, fundamental forms, Gaussian and mean curvatures. Moreover, we find geodesic curvatures, normal curvatures, and geodesic torsions of the base curve and the striction line of the new ruled surface. Finally, we give two related examples.