A new method for designing a developable surface utilizing the surface pencil through a given curve

Abstract

This paper proposes a new method for designing a developable surface by constructing a surface pencil passing through a given curve, which is quite in accord with the practice in industry design and manufacture. By utilizing the Frenet trihedron frame, we derive the necessary and sufficient conditions to construct a developable surface through a given curve. Considering the requirements in shoemaking and garment-manufacture industries, we also study the special case of specifying the given curve as a geodesic. The given geodesic can be classified into three types corresponding to each type of developable surface. We also present the polynomial representation of the developable surface. The algorithm is convenient and efficient for applications in engineering.