Developable Bézier-like surfaces with multiple shape parameters and its continuity conditions

Abstract

To solve the problems of shape adjustment and shape control of developable surfaces, we propose two direct explicit methods for the computer-aided design of developable Bézier-like surfaces with multiple shape parameters. Firstly, with the aim of constructing Bézier-like curves with multiple shape parameters, we present a class of novel Bernstein-like basis functions, which is an extension of classical Bernstein basis functions. Then, according to the important idea of duality between points and planes in 3D projective space, we design the developable Bézier-like surfaces with multiple shape parameters by using control planes with Bernstein-like basis functions. The shape of the developable Bézier-like surfaces can be adjusted by changing their three shape parameters. When the shape parameters take different values, a family of developable Bézier-like surfaces can be constructed and they retain the characteristics of Bézier surfaces. Finally, in order to tackle the problem that most complex developable surfaces in engineering often cannot be constructed by using a single developable surface, we derive the necessary and sufficient conditions for G1 continuity, Farin−Boehm G2 continuity and G2 Beta continuity between two adjacent developable Bézier-like surfaces. In addition, some properties and applications of the developable Bézier-like surfaces are discussed. The modeling examples show that the proposed methods are 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.