Tangent developables and Darboux developables of framed curves

Abstract

This paper generalizes the tangent developables of regular curves with linear independent condition to the tangent developables of framed curves and investigates the singularities of tangent developables in Euclidean 3-space. In contrast to tangent developables of regular curves with linear independent condition, the tangent developables of framed curves will appear cuspidal beaks, Scherbak surface and swallowtail. Employing the associated frame of the framed curves, an effective tool for analyzing the curves with singularities or with linearly dependent condition, we establish the relationships between types of singularities of tangent developables of framed curves and curvature functions via the classification approaches of the finite type so that we can characterize the singularities of tangent developables of framed curves via these functions. Moreover, some types of singularities of the Darboux developable are classified by means of the results of the tangent developables.