可展曲面的几何设计与形状调节

Geometric Design of Adjustably Developable Surfaces

为了克服传统方法在可展曲面设计上的缺陷,方便地解决工程中经常遇到的可展曲面位置和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,提出了两种直接、简单的可展曲面设计方法。该方法将可展曲面用具有基函数的控制平面来表示,通过引入形状控制参数λ(0≤λ≤1),使生成的可展曲面在较大的范围内可进行调节和控制,增加了造型的自由度,通过调节参数λ,可以得到一族可展曲面,这族可展曲面保留了许多B样条曲面的特性,在λ取0.5的特殊情况下,所生成的可展曲面即为均匀B样条曲面。研究结果表明,该设计方法具有现有曲面设计方法的特征,算法简单、有效。

Based on the important idea of duality between plane and point geometries, two direct explicit methods of computer-aided design for developable surfaces were proposed, the developable surfaces were represented using control planes with appropriate basis functions. The shapes of devel opable surfaces can be adjusted using a control parameter λ （0 ≤ λ ≤1）. When adjusting parameter λ , a family of developable surfaces can be constructed, they can keep the characteristics of B-spline surfaces, and in the particular case of λ = 0.5, the developable surface is a uniform B-spline surface. We show that the techniques for the geometric design of developable surfaces herein have all the characteristics of existing approaches for curve design. The algorithms were explained in detail, and the examples demonstrate freedom and efficiency of the methods.