Developable surface pencil pairs with special pairs as common asymptotes

Developable surface is widely used in the design and machining of industrial products. In this paper, developable surface pencil pairs with Natural pair and Conjugate pair (hereinafter referred to as special pairs) as common asymptotes are proposed and further applied to surface modeling. By establishing the Frenet frame of special pairs in three-dimensional Euclidean space, we express surface pencil pairs as a linear combination of Frenet frame, and derive the necessary and sufficient conditions of surface pencil pairs with special pairs as common asymptotes. Furthermore, the surface pencil pairs are extended to developable surface pencil pairs, as the selection of marching-scale functions are different, developable surface pencil pairs have good adjustability. Finally, when spatial curve is a cylindrical helix or a slant helix, examples of developable surface pencil pairs with special pairs as common asymptotes are given, respectively.