Mannheim curves and spherical curves

Abstract

In this paper, we study cylindrical helices, Mannheim curves and Darboux developable surfaces associated to Mannheim curves. We give a method for constructing Mannheim curves from spherical curves and demonstrate that all Mannheim curves can be constructed by such a way. We also introduce the spherical Darboux images of Mannheim curves, and we give a classification of singularities of Darboux developable surfaces and spherical Darboux images. Moreover, we study the geometric properties of singularities as applications of singularity theory for spherical curves.