带多参数B样条上可展曲面的几何设计与形状调节

针对基于点和平面间对偶性的可展曲面设计方法中的不足，提出两种带多局部形状参数的可展曲面设计新方法. 首先构造一组含有两个形状参数的三次调配函数，并由此调配函数定义一种带多局部形状参数的B样条曲线族. 基于3D 射影空间中点和平面间的对偶性原理，利用这种带多形状参数的B样条调配函数生成具有带多参数B样条基的控制平面，并由该控制平面进行包络和脊线可展曲面的设计，同时给出在带多参数B样条基下两种可展曲面的参数表示形式. 该方法生成的可展曲面不仅具有灵活的局部形状可调性，而且保留了B样条曲面的许多特性. 特别是当形状参数都取1 时，所生成的可展曲面即为可展B样条曲面. 分析所设计的两种可展曲面的形状与性质，给出具体的应用实例. 实例表明所提方法不仅简单有效，而且易于控制可展曲面的形状，是可展曲面设计的有效新途径.

Geometric Design and Shape Adjustment for Developable B-Spline Surfaces with Multiple Shape Parameters

To solve the problems in adjusting and controlling shapes of developable surfaces,two explicit and efficient methods of computer-aided design for developable surfaces with multiple local shape parameters are proposed.A class of novel quasi-B-spline basis functions with two shape parameters is presented to construct Bspline curves with multiple shape parameters,which is an extension of the classical cubic uniform B-splinc basis functions.Following the idea of duality between points and planes in 3D projective space,the corresponding developable quasi-B-spline surfaces with multiple shape parameters are represented using control planes with quasi-B-spline basis functions.The developable quasi-B-spline surfaces inherit the outstanding properties of the B-spline surfaces,with good performance in adjusting the local shapes by changing the two shape parameters. In the particular case where shape parameters are both equal to 1,the developable quasi-B-spline surface is a developable B-spline surface.In addition,some properties of the developable quasi-B-spline surfaces and applications in developable surfaces design are discussed.Modeling examples illustrate that the developable quasi-B-spline surfaces provide two valuable ways for the design of developable surfaces.