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

Abstract

In this paper, we derive the surfaces, which are called Hasimoto surfaces, corresponding to solutions of the localized induction equation for involute evolute curves. We investigate the differential geometric properties of these surfaces associated with the involute evolute curves. Moreover, we calculate the Gaussian and mean curvatures of Hasimoto surfaces in terms of curvatures of the curve pairs. Then, we determine the conditions for these surfaces' parameter curves to be the geodesics, asymptotics, and/or lines of curvature of the surface. We have demonstrated the evolutions of involute evolute curves in examples.