Evolution of space curves and the special ruled surfaces with modified orthogonal frame

Abstract

In this study, we investigate the evolution of space curves and the special ruled surfaces with modified orthogonal frame. Firstly, we recall the relations between the Serret-Frenet frame and the modified orthogonal frame. Then we investigate the evolution of space curves relative to these relations. Moreover, we determine the first and second fundamental forms, the Gaussian and mean curvatures of the tangent, normal, and binormal ruled surfaces according to the modified frame. In these regards, we obtain the characterizations of the minimal and developable ruled surfaces based on modified orthogonal frame.

Keywords

• modified orthogonal frame
• evolution of curve
• smoke ring equations
• ruled surfaces
• gaussian curvature
• mean curvature