New Representation of Hasimoto Surfaces with the Modified Orthogonal Frame

Abstract

In this study, we investigate Hasimoto surfaces considering the modified orthogonal frame. Firstly, we recall the relations between the Frenet frame and the modified orthogonal frame, and then we give the evolution equations of the modified orthogonal frame. After that, the first and second fundamental forms, mean curvatures, and Gaussian curvatures of the Hasimoto surfaces are determined with respect to the modified orthogonal frame. We give the definitions and some new theorems about Hasimoto surfaces. Finally, we express the properties of parameter curves of Hasimoto surfaces with a modified orthogonal frame in Euclidean 3-space.

Keywords

• Binormal motion
• evolution of curves
• Hasimoto surfaces
• modified orthogonal frame
• smoke ring equations

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