基于全正基的非均匀三次加权B样条  

Non uniform weighted cubic B-spline based on all positive basis

作　　者：姬佩佩 张贵仓 汪凯 孟建军[2] Ji Peipei;Zhang Guicang;Wang Kai;Meng Jianjun(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China;Mechanical T&R Institute,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]西北师范大学数学与统计学院,兰州730070 [2]兰州交通大学机电技术研究所,兰州730070

出　　处：《中国图象图形学报》2021年第4期939-951,共13页Journal of Image and Graphics

基　　金：国家自然科学基金项目(61861040);甘肃省教育厅科技成果转化项目(2017D-09);甘肃省科技资助项目(17YF1FA119);兰州市科技计划项目(2018-4-35)。

摘　　要：目的对B样条的改进方法大多从增加局部参数和在三角函数空间定义基两个角度出发,但仍存在缺陷,原因是通过模型的控制顶点对曲线进行编辑和处理,存在控制顶点给定时曲线较为固定的不足。为此,本文构造了一类基于全正基的非均匀三次加权λαβ-B样条基。方法结合加权思想,首先证明三次有理基在相应空间上的全正性;其次对三次三角基和三次有理基同时进行扩展,得到新的λαβ-B样条基,新扩展基具有和经典B样条基相似的性质;最后对新扩展基进行线性组合,用得到的多项式构造非均匀三次加权λαβ-B样条基,并研究了曲线的定义及性质。结果实验结果表明,新曲线保留传统B样条曲线基本性质的同时,还具有局部调整性,可以改善只通过调整控制顶点改变曲线形状的不足。结论构造的新λαβ-B样条曲线可以有效克服传统方法在改进时的不足,适合曲线设计。Objective The construction of B-spline basis functions has always been the focus of computer-aided design. The purpose of its research is mainly to solve the problem that the curve generated by the traditional method is fixed relative to the control vertices. The transformation form mainly incorporates the shape into the constructed basis function parameters to increase the flexibility of the curve, that is, to introduce free parameters to the expression of the classic Bernstein basis function or the extendedBernstein basis function, and adjusts the value of the parameter to adjust the shape of the curve. In recent years, researchers have proposed a large number of B-spline improvements, and they are mainly focused on two function spaces, namely, polynomial function space and trigonometric function space. The spline basis functions constructed in these two function spaces have their own advantages in addition to local adjustments to the corresponding curves. The spline curve constructed in the polynomial function space can be degenerated into a classic B-spline curve and has the advantages of simple calculation. Conversely, the basis function constructed in the trigonometric function space has the advantage of the derivation and cyclability of the trigonometric function. Both have high-order continuity, enabling the accurate representation of circle, ellipse, parabola, sine, cosine, cylindrical spiral, etc. The main purpose of this study is to combine the advantages of constructing spline basis functions in these two function spaces and use the weighting method to integrate the basis functions constructed in the two function spaces. The newly introduced weighting factors can be used as global parameters to make new extensions, and the flexibility of the curve is further enhanced. However, from the above two perspectives, some defects still exist. The reason is that the curve is edited and processed through the control vertices of the model,similar to the traditional method. When the control vertices are given, th

关 键 词：λαβ-Bernstein基 加权λαβ-B样条 非均匀 全正性 局部调整性质 

分 类 号：TP391.[自动化与计算机技术—计算机应用技术]