Solutions of localized induction equation associated with involute-evolute curve pair

Abstract

Involute-evolute curve pairs enable us to forecast their behaviors in various situations because of their well-defined mathematical characteristics. A mathematical surface connected to the localized induction equation (LIE) is called the Hasimoto surface. The research gap in this context is the lack of a comprehensive understanding of the surfaces' differential geometric properties and behaviors in relation to solutions of the LIE for the involute-evolute curve pair. In this regard, we provide information on these surfaces' behavior by deriving them associated with involute-evolute curve pairs and calculating their certain curvatures. Notably, we establish a link between the curvatures of these curve pairs and the Gaussian and mean curvatures of Hasimoto surfaces. Our research further identifies the precise conditions under which the parameter curves of these surfaces assume the roles of geodesics, asymptotics, or lines of curvature on the surface. Finally, we provide some examples of LIEs' hierarchy.