A Study of the Hasimoto Surfaces Constructed by the Tangent Spherical Indicatrices in Euclidean 3-Space

In this study, we investigate the vortex filament equation formed by the tangent spherical indicatrix of a unit-speed moving curve in Euclidean 3-space. For this purpose, the relations between the Frenet frames of the space curve and its tangent spherical indicatrix are considered. Then we introduce how the Hasimoto surfaces are constructed by the tangent spherical indicatrices and put forth some characterizations via related new findings. In the meantime, we present the first and second fundamental forms, Gaussian, and mean curvatures of this type of Hasimoto surface. Finally, we express some properties of the parameter curves of these Hasimoto surfaces with respect to the characters of both the tangent indicatrix and the original curve.