Family of Surfaces with a Common Bertrand D- Curve as Isogeodesic, Isoasymptotic and Line of Curvature

Abstract

In this paper, we establish the necessary and sufficient conditions to parameterize a surface family on which the Bertrand D-partner of any given curve lies as isogeodesic, isoasymptotic or curvature line in E-3. Then, we calculate the fundamental forms of these surfaces and determine the developability and minimality conditions with the Gaussian and mean curvatures. We also extend this idea on ruled surfaces and provide the required conditions for those to be developable. Finally, we present some examples and graph the corresponding surfaces.