沿整体C-Bézier曲线的运动  

Movement along an integral C-Bézier curve

作　　者：沈莞蔷[1] 李玲玉 汪国昭[2,3] Shen Wanqiang1, Li Lingyu1 , Wang Guozhao2,3(1. School of Science, Jiangnan University, Wuxi 214122, China; 2. Department of Mathematics, Zhejiang University, Hangzhou 310027, China; 3. State Key Laboratory of CAD＆CG, Zhejiang University, Hangzhou 310058, Chin)

机构地区：[1]江南大学理学院,无锡214122 [2]浙江大学数学系,杭州310027 [3]浙江大学CAD&CG国家重点实验室,杭州310058

出　　处：《中国图象图形学报》2018年第4期595-604,共10页Journal of Image and Graphics

基　　金：国家自然科学基金项目(61402201,61772013,61272300,11371174)

摘　　要：目的整体曲线包括传统有限闭区间(比如[0,α])上的内部段和该区间外的延拓段。在计算机辅助设计(CAD)中,构造整体曲线常用分段表示,存在冗余数据——为了减少冗余,需知道各分段间的关系,并判断它们是否在同一整体曲线上。由此,本文研究当整体C-Bézier曲线原参数域[0,α]在(-∞,+∞)上缩放变动时,曲线的控制顶点的变化情况。方法通过基函数的递推比较,寻找运动前后控制顶点之间的关系。首先考虑特殊细分情下线性插值。因插值后生成的NUAT-B样条基分段且具有支撑区间,它无法适应整体情况。因此用其与t轴间的区域面积取代它;接着进一步讨论了一般情况下沿整体C-Bézier运动的线性插值。由于C-Bézier参数区间长度要小于π,特殊细分情况下线性插值不能直接推广。不过虽然参数区间在变化,整体曲线上每点位置却不变。针对这点,使用两次递归,寻求得到以线性插值形式沿整体C-Bézier曲线运动的结果。结果只要保持参数区间的长度在(0,π)上,运动的曲线都可以写成传统的C-Bézier内部段的形式,且控制顶点可以表示为原始控制顶点直接的线性组合,或者逐步地线性插值(包括内插和外插)的形式。结论考虑整体曲线及沿整体曲线的运动,可以改变C-Bézier曲线的造型区间,减少造型过程中的冗余数据。不过,C-Bézier基由递归积分定义,其运动过程较慢。所以今后可以考虑加速运动的方法,也可以考虑其他类型的拟-Bézier曲线。Objective The parameter of a conventional C-Bezier curve is often limited in a closed interval. In this study, we focus on an integral one made up of the traditional inner segment in a finite closed interval （ such as [ 0, α ] ） and a part out of the interval. However, in computer-aided design, the modeling of an integral curve is often expressed as different stages and results in redundant data. In fact, when modeling an entire curve, the control points of different segments may have relations with one another. Therefore, if the control points of one segment and some shape parameters are stored, then the entire curve may be obtained, and the curve data may be decreased. We need to determine the relations among different segments and judge whether they are on the same integral curve to decrease the final redundancy. We raise two questions：1 ） Given an inner curve, can any segment of its integral curve be presented as an inner form？ and 2） Are two neighboring inner C-Brzier curves on the same integral curve？ We select a C-Bezier curve for our research. The focus of this study is to consider the changes in control vertices for the C-B6zier curve when the original parametric region [ 0, α] is scaled on （ - ∞ , ＋ ∞） . Method Any C-Brzier curve is divided into two arcs from geometric point of view： a center Brzier curve and a trigonometric part. On the basis of their movements, any segment of an integral C-Bezier curve can be represented as an inner form. We can analyze relations of control vertices from algebra perspective and give three forms （ direct, subdivision, and linear interpolation forms） between newly produced control points in the movement and old ones. First, we repre- sent certain segments of the integral C-B6zier curve as an inner form, consider basis functions recursively, and compare them to obtain the direct form of original control points. Second, one endpoint of the moving segment is considered, which relates to a subdivision scheme. The scheme subdivides the inner c

关 键 词：几何造型 C—Bezier曲线 控制顶点 参数区间 线性插值 

分 类 号：TP391.41[自动化与计算机技术—计算机应用技术]