Surface family with a common Mannheim B-pair asymptotic curve

Abstract

In this paper, we construct a surface family possessing a Mannheim B-pair of a given curve as an asymptotic curve. Using the Bishop frame of the given Mannheim B curves, we present the surface as a linear combination of this frame and analyze the necessary and sufficient condition for a given curve such that its Mannheim B-pairs are both isoparametric and asymptotic on a parametric surface. The extension to ruled surfaces is also outlined. In addition, necessary and sufficient conditions have been given for the development of this ruled surface family. Finally, we present some interesting examples to show the validity of this study.