Directional developable surfaces and their singularities in Euclidean 3-Space

Abstract

The developable surface is a surface that can be unfolded on a plane without tearing or stretching, which is widely used in many fields of engineering and manufacturing. This work presents a new version of developable ruled surfaces in Euclidean 3-space. First, we establish an adapted frame along a spatial curve, denoted by the quasi-frame. We then introduce a parametric representation of a developable ruled surface and call it a directional developable ruled surface. At the core of this paper, we investigate the existence and uniqueness of such developable surfaces, then study their classification by singularity theory and unfolding method. Some examples are given in the final.