CE-Bézier可展曲面的设计与形状调整

Geometric Design and Adjustment of Shape for Developable CE-Bézier Surfaces

为了解决工程中可展曲面位置与形状难以调整和控制的问题,基于3D射影空间中点和平面间的对偶性这一重要思想,提出了2种直接、简单有效的可展曲面设计新方法.首先,构造了一组含有2个形状参数α、γ的三次多项式调配函数,并定义了一种带2个形状控制参数的CE-Bézier曲线族,然后利用这种带参数的CE-Bézier调配函数生成了具有CE-Bézier基的控制平面,并由该控制平面来进行可展曲面的设计,同时给出了在CE-Bézier基函数下可展曲面的参数表示形式.由新方法生成的可展曲面不仅具有灵活的局部形状可调性和更强的描述能力,而且保留了Bézier曲面的特性,特别是当α、γ都取值为1时,所生成的可展曲面即为Bézier可展曲面.2种可展曲面设计方法的应用实例表明,该设计方法不仅简单、有效,而且易于控制曲面形状,从而为可展曲面的设计提供了一种新途径.

To solve the problems in adjusting and controlling shapes of developable surfaces,following the important idea of duality between points and planes in 3D projective space,two direct explicit and efficient methods of computer-aided design for developable surfaces are proposed.Constructing a class of cubic extension Bézier（CE-Bézier） basis functions with two shape parameters α,γ,the corresponding developable CE-Bézier surfaces with multiple shape control parameters are represented.The developable CE-Bézier surfaces inherit the outstanding properties of the Bézier surfaces,with a good performance on adjusting their local shapes by changing the two shape control parameters.In the particular case where α and γ are equal to 1,the developable CE-Bézier surface is a developable Bézier surface.Finally,some applications in developable surfaces design are discussed.The modeling examples illustrate that the developable CE-Bézier surfaces provide two valuable ways for the design of developable surfaces.