Totally positive bases for shape preserving curve design and optimality of B-splines

Normalized totally positive (NTP) bases present good shape preserving properties when they are used in Computer Aided Geometric Design. Here we characterize all the NTP bases of a space and obtain a test to know if they exist. Furthermore, we construct the NTP basis with optimal shape preserving properties in the sense of (Goodman and Said, 1991), that is, the shape of the control polygon of a curve with respect to the optimal basis resembles with the highest fidelity the shape of the curve among all the control polygons of the same curve corresponding to NTP bases. In particular, this is the case of the B-spline basis in the space of polynomial splines. Further examples are given.  

Joining rational curves smoothly

A theory is developed for projective-invariant construction from a sequenceof control points of a piecewise rational curve of arbitrary degree joining with continuity of certain geometric properties of the curve. In particular a recursive means of evaluation is derived which generalises the Cox-de Boor algorithm for B-splines.  

Representing circles with five control points

We show that five is the minimal dimension of a space required to draw a complete circle with a unique control polygon. We identify all five-dimensional spaces invariant under translations and reflections where we can find shape preserving representations of a circle parameterized by its arc length.  

Corner cutting algorithms associated with optimal shape preserving representations

Given a space of functions which admits shape preserving representations using control polygons, we construct a corner cutting algorithm which will be called B-algorithm. It is an evaluation algorithm satisfying important properties such as subdivision property and convergence to the curve of the resulting control polygons. Many examples are given.  

Harmonic rational Bézier curves, p-Bézier curves and trigonometric polynomials

In a recent article, Ge et al. (1997) identify a special class of rational curves (Harmonic Rational Bézier (HRB) curves) that can be reparameterized in sinusoidal form. Here we show how this family of curves strongly relates to the class of p-Bézier curves, curves easily expressible as single-valued in polar coordinates. Although both subsets do not coincide, the reparameterization needed in both cases is exactly the same, and the weights of a HRB curve are those corresponding to the representation of a circular arc as a p-Bézier curve. We also prove that a HRB curve can be written as a combination of its control points and certain Bernstein-like trigonometric basis functions. These functions form a normalized totally positive B-basis (that is, the basis with optimal shape preserving properties) of the space of trigonometric polynomials {1, sint, cost, …. sinmt, cosmt} defined on an interval of length < π.  

两类推广的渐近迭代逼近

在计算机辅助设计领域里,曲线或曲面的渐近迭代逼近(Progressive iterative approximation,PIA)性质在插值与拟合问题中有着广泛的应用,以前的文献对这一性质的讨论主要局限在标准全正基的情形.对于一般的非标准全正基,本文指出,其在适当的参数下也有可能同样具有这一优良的性质,并给出了相应的实例,从而拓宽了渐近迭代逼近的适用范围.与此同时,还讨论了权因子各不相同时,带权渐近迭代逼近的收敛性,使得迭代逼近曲线对不同的控制顶点,具有不同的加速收敛速度.  

3次均匀B样条曲线的保形扩展

为了在不增加计算复杂度的前提下，构造既具有凸包性，又具有保形性的类3次均匀B样条曲线。首先采用逆向思维法，通过预设的曲线性质来反推调配函数的性质，进而计算出调配函数的表达式。然后采用定性分析法，分别讨论当曲线具备凸包性、保单调性、保凸性、变差缩减性时，曲线中参数的取值范围，文中图例显示了分析结果的正确性。不同情况下所得参数取值范围的交集，即为最终确定的曲线中形状参数的可行域，在可行域内改变形状参数，可以在不破坏曲线保形性的前提下调整曲线对控制多边形的逼近程度。简要讨论了与曲线对应的张量积曲面，并给出了图例。  

带局部形状参数的λ-B曲线设计

目的 目前有很多研究B样条曲线的含参数扩展,给出的曲线都具备B样条曲线的局部形状控制性以及独立于控制顶点的形状可调性,但有些文献给出的参数是全局的,导致曲线不具备局部形状调整性,有些文献给出的调配函数不具有全正性,导致曲线不具备变差缩减性、保凸性。本文的出发点是构造同时具备保凸性、局部形状调整性、局部形状控制性的曲线。方法 首先运用拟扩展函数空间的理论框架证明了已有的3次Bézier曲线的扩展基,简称λμ-Bernstein基,恰好为所在空间中的规范B基。然后运用λμ-Bernstein基的线性组合来构造3次均匀B样条曲线的扩展基,根据预设的曲线性质反推出扩展基的性质,进而求出线性组合的系数,得出扩展基的表达式。扩展基可以表示成λμ-Bernstein基与一个转换矩阵的乘积,证明了转换矩阵的全正性,由扩展基定义了一种结构与3次B样条曲线相同的含一个局部形状参数的分段曲线。结果 转换矩阵的全正性决定了扩展基的全正性,扩展基的全正性决定了扩展曲线的变差缩减性、保凸性,形状参数的局部性决定了曲线的局部形状调整性,曲线的分段结构决定了曲线的局部形状控制性。结论 本文给出的构造具有全正性的B样条扩展基的方法具有一般性,与现有众多扩展曲线相比,本文方法构造的曲线因为具有变差缩减性和保凸性,从而为保形设计提供了一种有效方法。