Family of Surfaces with a Common Special Involute and Evolute Curves

Abstract

In this paper we define the necessary and sufficient conditions for both the involute and evolute of a given curve to be geodesic, asymptotic and curvature line on a parametric surface. Then, the first and second fundamental forms of these surfaces are calculated. By using the Gaussian and mean curvatures, the developability and minimality assumptions are drawn, as well. Moreover we extended the idea to the ruled surfaces. Finally, we provide a set of examples to illustrate the corresponding surfaces.