Developable normal surface pencil

Abstract

In this paper, we introduce a new class of surfaces, called as normal surface pencil. We parameterize a normal surface pencil by using the principal normal vector $\mathbf{n}$ and the binormal vector $\mathbf{b}$ of the Frenet frame of a space curve $\alpha(s)$ as follows $\varphi(s,t)=\alpha(s)+y(s,t)\mathbf{n}+z(s,t)\mathbf{b}.$ A well known example of normal surface pencil is a canal surface. Finally, we propose the sufficient conditions of a normal surface pencil being a developable surface. Then several new examples of developable normal surface pencil are constructed from these conditions.