基于全正基的三次均匀B样条曲线的扩展
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摘要
为了构造具有保形性的三次均匀B样条扩展曲线,首先运用拟扩展切比雪夫空间的理论框架证明现有文献中的三次Bézier曲线的扩展基,简称λ-Bézier基,恰为相应空间的规范B基。然后用λ-Bézier基的线性组合来表示三次均匀B样条曲线的扩展基,根据预设的曲线性质反推扩展基的性质,进而求出线性组合的系数。扩展基可表示成λ-Bézier基与一个转换矩阵的乘积,证明了转换矩阵的全正性及扩展基的全正性。由扩展基定义了基于3点分段的曲线,分析了曲线的性质,扩展基的全正性决定了曲线可以较好的模拟控制多边形的形态。简要介绍了由扩展基定义的基于16点分片的曲面。
This paper aims to construct a shape-preserving extended cubic uniform B-spline curve. Firstly, within the theoretical framework of quasi extended Chebyshev space, we prove that the existing extended basis of the cubic Bézier curve, λ-Bézier basis for short, is the normalized B-basis of the corresponding space. Then we use the linear combination of the λ-Bézier basis to express the extended basis of the cubic uniform B-spline curve. According to the preset properties of the curve, we deduce the properties of the extended basis, and then determine the coefficients of the linear combination. The extended basis can be represented as the product of the λ-Bézier basis and a conversion matrix. We prove the totally positive property of the matrix and the extended basis. By using this basis, we define a curve based on 3-point piecewise scheme and analyze its properties. The totally positive property makes the curve can simulate the shape of the control polygon. The surface based on 16-point piecewise scheme is briefly introduced.
This paper aims to construct a shape-preserving extended cubic uniform B-spline curve. Firstly, within the theoretical framework of quasi extended Chebyshev space, we prove that the existing extended basis of the cubic Bézier curve, λ-Bézier basis for short, is the normalized B-basis of the corresponding space. Then we use the linear combination of the λ-Bézier basis to express the extended basis of the cubic uniform B-spline curve. According to the preset properties of the curve, we deduce the properties of the extended basis, and then determine the coefficients of the linear combination. The extended basis can be represented as the product of the λ-Bézier basis and a conversion matrix. We prove the totally positive property of the matrix and the extended basis. By using this basis, we define a curve based on 3-point piecewise scheme and analyze its properties. The totally positive property makes the curve can simulate the shape of the control polygon. The surface based on 16-point piecewise scheme is briefly introduced.
引文
[1]吴晓勤,韩旭里.三次Bézier曲线的扩展[J].工程图学学报,2005,26(6):98-102.
[2]仇茹,杭后俊,潘俊超.带三参数的类四次Bézier曲线及其应用研究[J].计算机工程与应用,2014,50(20):158-162.
[3]Wang W T,Wang G Z.Bézier curves with shape parameter[J].Journal of Zhejiang University SCIENCE A,2005,6A(6):497-501.
[4]Chen J,Wang G J.A new type of the generalized Bezier curves[J].Applied Mathematics-A Journal of Chinese Universities,2011,26(1):47-56.
[5]Qin X Q,Hu G,Zhang N J,et al.A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters[J].Applied Mathematics and Computation,2013,223(10):1-16.
[6]韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报,2003,15(5):576-578.
[7]王树勋,叶正麟,陈作平.带最多独立形状参数的三阶三次均匀B样条曲线[J].计算机工程与应用,2010,46(15):142-145.
[8]Xu G,Wang G Z.Extended cubic uniform B-spline andα-B-spline[J].Acta Automatica Sinica,2008,34(8):980-984.
[9]耿紫星,张贵仓.Bézier曲线的三角扩展[J].计算机工程与应用,2008,44(26):59-62.
[10]Chen Q Y,Wang G Z.A class of Bézier-like curves[J].Computer Aided Geometric Design,2003,20(1):29-39.
[11]Han X L,Zhu Y P.Curve construction based on five trigonometric blending functions[J].BIT Numerical Mathematics,2012,52(4):953-979.
[12]Su B Y,Tan J Q.A family of quasi-cubic blended splines and applications[J].Journal of Zhejiang University SCIENCE A,2006,7(9):1550-1560.
[13]尹池江,檀结庆.带多形状参数的三角多项式均匀B样条曲线曲面[J].计算机辅助设计与图形学学报,2011,23(7):1131-1138.
[14]张锦秀,檀结庆.代数双曲Bézier曲线的扩展[J].工程图学学报,2011,32(1):31-38.
[15]邬弘毅,左华.多形状参数的二次非均匀双曲B样条曲线[J].计算机辅助设计与图形学学报,2007,19(7):876-883.
[16]Han X L,Zhu Y P.Total positivity of the cubic trigonometric Bézier basis[J].Journal of Applied Mathematics,2014,(3):1-5.
[17]Marie-Laurence.Which space for design[J].Numerische Mathematik,2008,110(3):357-392.
[2]仇茹,杭后俊,潘俊超.带三参数的类四次Bézier曲线及其应用研究[J].计算机工程与应用,2014,50(20):158-162.
[3]Wang W T,Wang G Z.Bézier curves with shape parameter[J].Journal of Zhejiang University SCIENCE A,2005,6A(6):497-501.
[4]Chen J,Wang G J.A new type of the generalized Bezier curves[J].Applied Mathematics-A Journal of Chinese Universities,2011,26(1):47-56.
[5]Qin X Q,Hu G,Zhang N J,et al.A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters[J].Applied Mathematics and Computation,2013,223(10):1-16.
[6]韩旭里,刘圣军.三次均匀B样条曲线的扩展[J].计算机辅助设计与图形学学报,2003,15(5):576-578.
[7]王树勋,叶正麟,陈作平.带最多独立形状参数的三阶三次均匀B样条曲线[J].计算机工程与应用,2010,46(15):142-145.
[8]Xu G,Wang G Z.Extended cubic uniform B-spline andα-B-spline[J].Acta Automatica Sinica,2008,34(8):980-984.
[9]耿紫星,张贵仓.Bézier曲线的三角扩展[J].计算机工程与应用,2008,44(26):59-62.
[10]Chen Q Y,Wang G Z.A class of Bézier-like curves[J].Computer Aided Geometric Design,2003,20(1):29-39.
[11]Han X L,Zhu Y P.Curve construction based on five trigonometric blending functions[J].BIT Numerical Mathematics,2012,52(4):953-979.
[12]Su B Y,Tan J Q.A family of quasi-cubic blended splines and applications[J].Journal of Zhejiang University SCIENCE A,2006,7(9):1550-1560.
[13]尹池江,檀结庆.带多形状参数的三角多项式均匀B样条曲线曲面[J].计算机辅助设计与图形学学报,2011,23(7):1131-1138.
[14]张锦秀,檀结庆.代数双曲Bézier曲线的扩展[J].工程图学学报,2011,32(1):31-38.
[15]邬弘毅,左华.多形状参数的二次非均匀双曲B样条曲线[J].计算机辅助设计与图形学学报,2007,19(7):876-883.
[16]Han X L,Zhu Y P.Total positivity of the cubic trigonometric Bézier basis[J].Journal of Applied Mathematics,2014,(3):1-5.
[17]Marie-Laurence.Which space for design[J].Numerische Mathematik,2008,110(3):357-392.