基于全正基的三次均匀B样条曲线的扩展

The Extended Cubic Uniform B-Spline Curve Based on Totally Positive Basis

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 B＆233;zier curve,λ-B＆233;zier basis for short, is the normalized B-basis of the corresponding space. Then we use the linear combination of theλ-B＆233;zier 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 theλ-B＆233;zier 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.