带有能量约束的类B\'ezier 曲线的构造

本文提出了一类创新的类Bernstein基函数,这是经典Bernstein基函数的推广.我们分析了这类基函数的性质并定义了带有两个形状参数的类B\''ezier曲线.这类基函数和类B\''ezier曲线分别具有类似三次Bernstein基函数和三次B\''ezier曲线的性质.进一步地,我们构造了带有能量约束的类B\''ezier曲线并考虑了带有极小能量约束的$C^1$和$G^1$ Hermite插值问题.最后,给出了一些具有代表性的例子来说明我们方法的应用性和有效性.     

Curves and Surfaces Construction Based on New Basis with Exponential Functions

By incorporating two exponential functions into the cubic Bernstein basis functions, a new class of λμ-Bernstein basis functions is constructed. Based on these λμ-Bernstein basis functions, a kind of λμ-Bézier-like curve with two shape parameters, which include the cubic Bernstein-Bézier curve, is proposed. The C ¹ and C ² continuous conditions for joining two λμ-Bézier-like curves are given. By using tensor product method, a class of rectangular Bézier-like patches with four shape parameters is shown. The G ¹ and G ² continuous conditions for joining two rectangular Bézier-like patches are derived. By incorporating three exponential functions into the cubic Bernstein basis functions over triangular domain, a new class of λμη-Bernstein basis functions over triangular domain is also constructed. Based on the λμη-Bernstein basis functions, a kind of triangular λμη-Bézier-like patch with three shape parameters, which include the triangular Bernstein-Bézier cubic patch, is presented. The conditions for G ¹ continuous smooth joining two triangular λμη-Bézier-like patches are discussed. The shape parameters serve as tension parameters and have a predictable adjusting role on the curves and patches.     

Developable Bézier-like surfaces with multiple shape parameters and its continuity conditions

To solve the problems of shape adjustment and shape control of developable surfaces, we propose two direct explicit methods for the computer-aided design of developable Bézier-like surfaces with multiple shape parameters. Firstly, with the aim of constructing Bézier-like curves with multiple shape parameters, we present a class of novel Bernstein-like basis functions, which is an extension of classical Bernstein basis functions. Then, according to the important idea of duality between points and planes in 3D projective space, we design the developable Bézier-like surfaces with multiple shape parameters by using control planes with Bernstein-like basis functions. The shape of the developable Bézier-like surfaces can be adjusted by changing their three shape parameters. When the shape parameters take different values, a family of developable Bézier-like surfaces can be constructed and they retain the characteristics of Bézier surfaces. Finally, in order to tackle the problem that most complex developable surfaces in engineering often cannot be constructed by using a single developable surface, we derive the necessary and sufficient conditions for G¹ continuity, Farin−Boehm G² continuity and G² Beta continuity between two adjacent developable Bézier-like surfaces. In addition, some properties and applications of the developable Bézier-like surfaces are discussed. The modeling examples show that the proposed methods are effective and easy to implement, which greatly improve the problem-solving abilities in engineering appearance design by adjusting the position and shape of developable surfaces.     

An extension of the Bézier model

In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein-Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.     

可调形三次三角Cardinal插值样条曲线

在三次Cardinal插值样条曲线的基础上,引入了三角函数多项式,得到一组带调形参数的三次三角Cardinal样条基函数,以此构造一种可调形的三次三角Cardinal插值样条曲线.该插值样条可以精确表示直线、圆弧、椭圆以及自由曲线,改变调形参数可以调控插值曲线的形状.该插值样条避免了使用有理形式,其表达式较为简洁,计算量也相对较少,从而为多种线段的构造与处理提供了一种通用与简便的方法.     

带两个形状参数的五次Bézier曲线的扩展

给出了一组含有两个形状参数α,β的六次多项式基函数,是五次Bernstein基函数的扩展,分析了这组基的性质;基于这组基定义了带两个形状参数的多项式曲线,所定义的曲线具有五次Bézier曲线的性质,改变参数α,β的取值,曲线具有更灵活的形状可调性,而且能向上或从两侧逼近控制多边形.另外,经典的五次Bézier曲线和有关文献中带一个形状参数的曲线均是该文所定义曲线的特例.实例表明,定义的曲线为曲线/曲面的设计提供了一种有效的方法.     

Triangular domain extension of algebraic trigonometric Bézier-like basis

In computer aided geometric design (CAGD), Bézier-like bases receive more and more considerations as new modeling tools in recent years. But those existing Bézier-like bases are all defined over the rectangular domain. In this paper, we extend the algebraic trigonometric Bézier-like basis of order 4 to the triangular domain. The new basis functions defined over the triangular domain are proved to fulfill non-negativity, partition of unity, symmetry, boundary representation, linear independence and so on. We also prove some properties of the corresponding Bézier-like surfaces. Finally, some applications of the proposed basis are shown.     

The cubic trigonometric Bézier curve with two shape parameters

A cubic trigonometric Bézier curve analogous to the cubic Bézier curve, with two shape parameters, is presented in this work. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. With the shape parameters, the cubic trigonometric Bézier curves can be made close to the cubic Bézier curves or closer to the given control polygon than the cubic Bézier curves. The ellipses can be represented exactly using cubic trigonometric Bézier curves.     

On the Structure of Bézier Nets

The distance between two neighbouring multivariate Bézier nets is proved to be $O(m^{-2})$ in this paper. As a consequence, the sequence of Bézier nets is uniformly convergent with the optimal approximation order $O(m^{-1})$. Furthermore, the structures of Bézier nets are explored by investigating how the piecewise linear surface tends to the Bézier surface of $C^{\infty}$.     

带形状参数的T-Bézier曲线

给出了n阶带形状参数的三角多项式T-Bézier基函数.由带形状参数的三角多项式T-Bézier基组成的带形状参数的T-Bézier曲线,可通过改变形状参数的取值而调整曲线形状,随着形状参数的增加,带形状参数的T-Bézier曲线将接近于控制多边形,并且可以精确表示圆、螺旋线等曲线.阶数的升高,形状参数的取值范围将扩大.     

A Practical Method for Generating Trigonometric Polynomial Surfaces over Triangular Domains

Mediterranean Journal of Mathematics - A class of trigonometric polynomial basis functions over triangular domain with three shape parameters is constructed in this paper. Based on these new basis...     

A novel generalization of Bézier curve and surface

A new formulation for the representation and designing of curves and surfaces is presented. It is a novel generalization of Bézier curves and surfaces. Firstly, a class of polynomial basis functions with n$n$ adjustable shape parameters is present. It is a natural extension to classical Bernstein basis functions. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n$n$ shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon. It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. Some examples are given by figures.     

带两个形状参数的同次Bézier曲线

构造了一种带两个形状参数的Bézier型曲线,并研究了该曲线的性质、形状参数对曲线的影响及曲线的拼接.所提出的曲线是多项式Bezier曲线的一种同次新扩展,不仅具有传统Bézier曲线的诸多性质,而且可通过修改两个形状参数的取值对其形状进行调节.由于所提出的曲线是一种带有形状参数且与传统Bézier曲线具有相似性质的同次多项式模型,因此比现有的一些带形状参数的Bézier型曲线更有优势.     

三角域上带两个形状参数的Bézier曲面的扩展

给出了三角域上带双参数λ1,λ2的类三次Bernstein基函数,它是三角域上三次Bernstein基函数的扩展.分析了该组基的性质并定义了三角域上带有两个形状参数λ1,λ2的类三次Bernstein-Bézier(B-B)参数曲面.该基函数及参数曲面分别具有与三次Bernstein基函数及三次B-B参数曲面类似的性质.当λ1,λ2取特殊的值时,可分别得到三次Bernstein基函数及三次B-B参数曲面以及参考文献中所定义的类三次Bernstein基函数及类三次B-B参数曲面.由实例可知,通过改变形状参数的取值,可以调整曲面的形状.     

平面三次Pythagorean-Hodograph Hyperbolic曲线

本文基于Pythagorean-hodograph (PH)曲线和代数双曲线的良好几何特性,构造了Pythagorean-Hodograph Hyperbolic (PH-H)曲线,并给出了PH-H曲线的定义以及相应性质.同时,分别利用Hyperbolic基函数和Algebraic Hyperbolic (AH) B\''ezier基函数,得到了平面三次AH B\''ezier曲线为PH曲线的两个不同的充要条件.此外,三次PH-H曲线也被用于求解具有确定解的$G^1$ Hermite插值问题.文中给出了具体实例来说明我们的方法.     

一类有理Bézier曲线的等距线算法及MATLAB实现

构造一类正则有理Bézier曲线,利用改进的有理de casteljau算法求得这类正则有理n次Bézier曲线各点处的切矢,由此得出各点的单位法矢量,应用于原始曲线等距线的计算.该方法几何意义明显,算法简洁,实践效果比较好.同时给出了用Matlab绘制有理Bézier曲线及其等距线的程序,准确快捷,实践效果较好.     

基于参数连续HC Bézier-like曲线的过渡曲线的构造

在形状调配过程中,过渡曲线的连续性往往是很难保证的.给出HC Bézier-like曲线的定义,然后从过渡曲线满足一定连续性的角度出发,利用HC Bézier-like曲线的端点性质,研究形状参数曲线的参数连续特征保持问题.给出线性混合过程中,一阶和二阶参数连续保持条件,从而得出一般的HC Bézier-like曲线在...     

C~3连续的三角样条函数与曲线

本文构造了一种三次三角样条函数 ,函数的每一段由三个函数值生成 ,具有C3连续性和较好的逼近性 ,可方便地进行插值 .基于同样的方法得出了一种C3连续的三角样条曲线 ,曲线也有较好的逼近性 ,而且具有局部性、保凸性等特性 .     

任意阶F-Bézier基的一些基本性质

任意次的F-Bézier基统一了三角多项式空间上的C-Bézier基和双曲多项式空间上的H-Bézier基,我们证明这种基函数具有类似于基函数的优良性质,包括端点性质、对称性、升阶性质、线性无关性等,并且证明当形状参数趋于零时F-Bézier基收敛Bernstein基.     

Piecewise quadratic trigonometric polynomial curves

Analogous to the quadratic B-spline curve, a piecewise quadratic trigonometric polynomial curve is presented in this paper. The quadratic trigonometric polynomial curve has continuity, while the quadratic B-spline curve has continuity. The quadratic trigonometric polynomial curve is closer to the given control polygon than the quadratic B-spline curve.