Geometric design and continuity conditions of developable λ-Bézier surfaces

In this paper, two explicit methods are presented for the computer-aided design of developable λ-Bézier surfaces associated with shape parameter. Based on the duality between points and planes in 3D projective space, a developable λ-Bézier surface associated with a shape parameter is designed by using a set of control planes with λ-Bézier basis functions. The shape of developable λ-Bézier surface can be easily adjusted by modifying the value of the shape parameter. When the shape parameter takes on different values, a family of developable λ-Bézier surfaces can be constructed, which keeps most of beneficial properties of traditional Bézier surfaces. In order to tackle the problem that an engineering complex developable surface is usually hard to be constructed by using a single developable surface, we also derive the necessary and sufficient conditions for G1 continuity, Farin-Boehm G2 continuity and G2 Beta continuity between two adjacent developable λ-Bézier surfaces. Finally, the properties and applications of developable λ-Bézier surfaces are discussed. The modeling examples show that the proposed method is effective and easy to implement, which greatly improve the problem-solving abilities in engineering appearance design by adjusting the position and shape of developable surfaces.