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.     

Generating Bézier points for curves and surfaces from boundary information

This paper presents efficient methods for directly generating Bézier points of curves and surfaces explicitly from the given compatible arbitrary order boundary information of Hermite curves, Coons-Hermite Cartesian sum patches and Coons-Boolean sum patches. The explicit expressions for the generalized Hermite functions are also developed. Furthermore, a method for determining the twist control points and higher level sets of interior control points from their boundary and lower level sets of control points by using the Coons-Boolean sum schema presented. Many interesting and useful examples are also given in this paper.     

Dual Bézier curves and convexity preserving interpolation

A convexity preserving interpolation problem is analyzed from a geometrical point of view. A dualization of the usual Bézier techniques allows us to define a subdivision algorithm which generates certain conic sections. This algorithm can be used to define a rational convexity preserving interpolant. We also describe some particular dual Bézier curves which are particularly suitable for the design of convex functions.     

Three-dimensional Bézier interpolation in solid modeling and grid generation

Bézier and B-spline patches are popular tools in surface modeling. With these methods, a surface is represented by the product of univariate approximations. The extension of this concept to three-dimensions is straightforward and can be applied to the problem of grid generation. This report will demonstrate how rational Bézier basis functions can be used to generate three-dimensional solids that can be used in solid modeling and in the generation of grids. Examples will be given demonstrating the ability to generated three-dimensional grids directly from a set of data points without having to first set up the boundary surfaces. Many geometric grid properties can be imposed by the proper choice of the control net, the weights, and the twist models.     

Curvature of singular Bézier curves and surfaces

This paper presents a general approach for finding the limit curvature at a singular endpoint of a rational Bézier curve and the singular corner of a rational Bézier surface patch. Conditions for finite Gaussian and mean limit curvatures are expressed in terms of the rank of a matrix.     

Matrix formulations of Bézier technique

At the start of this paper, a recurrence formula for calculating the Bézier functions of any order is proved. Based on this formula, the Bézier functions are written in matrix form which enables us to develop the Bézier technique in a concise way. There are some new identities of the Bézier functions which are potentially useful in CAGD. Surprisingly a matrix involving the Bézier functions has been shown to be a doubly-stochastic matrix and the convergence behaviour of its powers is determined. Based upon these facts, the Kelisky — Rivlin theorem for the Bernstein polynomials has been extended to the Bézier curves.     

The bound on derivatives of rational Bézier curves

The estimation of bounds on derivatives of rational Bézier curves has important application in CAGD. This paper derives some new bounds according to the properties of derivation and recursion of Bernstein basis, and shows that the new bounds are better than existing ones and more effective.     

Computer-aided geometric design and panel generation for hull forms based on rational cubic Bézier curves

For generation of hull forms, a method using rational cubic Bézier curves is chosen because of their superior segmentwise local-weighted behavior. A hull form is defined by two sets of grid lines—transverse grid lines arranged in length direction and longitudinal grid lines arranged in depth direction. Transverse lines are first defined, the points on the transverse lines with the same curve parameter values are then fitted to define longitudinal lines. Thereby, each curve is described by a rational cubic Bézier curve in space. The bilge, flat side and flat bottom can be defined precisely and more flexibilities are provided for defining bow and stern regions. By the way, a hull surface can be generated which is useful to produce desired data for hydrostatic or panel generations.     

Link between Bézier and Lagrange curve and surface schemes

A class of curves that vary continuously between polynomial Lagrange interpolants and polynomial Bézier curves is discussed. An element in this class is specified by a real number which could be used as a shape parameter for Bézier curves. A geometric derivation of this scheme is given, and the connection to Pólya curves is pointed out. A generalization to the case of tensor product and triangular surface patches is also described.     

Boundary evaluation for interval Bézier curve

The objective of this paper is to provide an efficient and reliable algorithm for representing and evaluating the boundary of the interval Bézier curve in 2- and 3-D. The boundary of the planar Bézier curve is represented by a sequence of Bézier curve segments with same degree and line segments in the order they are encountered when marching counter-clockwise along its boundary. The boundary can also be represented as a single B-spline curve having the same degree with the interval Bézier curve. The boundary of the 3-D interval Bézier curve is made up of trimmed Bézier surface patches and rectangular patches. Some examples illustrate our algorithms.     

带形状参数的QT-Bézier曲线曲面的构造和应用

为了更加方便地表示和修改曲线曲面,提出了带形状参数的四次三角Bézier曲线曲面QTBézier的构造方法和应用。首先仿照Bézier曲线性质,构造了带形状参数的基函数,定义了带形状参数的QT-Bézier曲线曲面并研究了他们的一些主要性质,并就参数的选取做了一些分析。这种带形状参数的QT-Bézier曲线曲面是已有的一些曲线曲面的一般表达方法,如果选取一些特殊的参数,可以表示特殊的和已知的曲线曲面,还可以构造不同形状的旋转面。带形状参数的QT-Bézier曲线曲面可以很好地通过形状参数来调整曲线曲面的外形,而且能构造不同的旋转面,由于有额外的形状参数,更便于交互。     

带两个形状参数的四次Bézier 曲线的扩展

给出了两组带两个形状参数λ , μ 的六次多项式基函数,它们是四次 Bernstein 基函数的扩展。分析了这两组基函数的性质,基于这两组基分别定义了带形状参数 的两类多项式曲线,两类曲线具有与四次Bézier 曲线类似的性质,且在控制顶点不变的情 况下,可通过改变形状参数的值实现对曲线形状的调整。参数λ, μ 具有明显的几何意义。当 λ =μ = 0 时,均退化为四次Bézier 曲线。实例表明,论文所采用的方法控制灵活,方便有效。     

Bézier representation for cubic surface patches

The paper describes a new method for creating rectangular Bézier surface patches on an implicit cubic surface. Traditional techniques for representing surfaces have relied on parametric representations of surfaces, which, in general, generate surfaces of implicit degree 8 in the case of rectangular Bézier surfaces with rational biquadratic parameterization. The method constructs low-degree algebraic surface patches by reducing the implicit degree from 8 to 3. The construction uses a rectangular biquadratic Bézier control polyhedron that is embedded within a tetrahedron and satisfies a projective constraint. The control polyhedron and the resulting cubic surface patch satisfy all of the standard properties of parametric Bézier surfaces, including interpolation of the corners of the control polyhedron and the convex-hull property.     

Bézier 曲线的多项式重新参数化检测

研究了一种用于精确检测一条Bézier 曲线的次数是否可以通过多项式重新参数化 降低的算法。该算法对任意一条Bézier 曲线,将重新参数化前后的基函数的关系用方程组的形 式表达,但不需要解方程,而是通过系数表示的金字塔算法直接计算,可以精确求出用于重新 参数化的多项式和降低次数后的Bézier 曲线的控制顶点,并且该重新参数化的多项式在相差一 个线性变换的前提下是唯一的。通过实例应用,该算法运算速度较之前的算法快。     

一类双参数类四次三角Bézier曲线及其扩展

给出了一类双参数的类四次三角Bézier曲线及其扩展曲线的定义,得到了该类曲线及其扩展曲线的性质,给出了两段双参数的类四次三角Bézier曲线[G1(C1),G2(C2)]及两段扩展曲线[G1(C1),G2(C2)]光滑拼接的充要条件,并讨论了这两类曲线的应用。算例表明,该类曲线及其扩展曲线在曲线造型,特别是在非对称图形的造型中,具有很强的描述能力。     

Bézier样条曲线改进的近似弧长参数化方法

提出了Bézier样条曲线利用分割技术近似弧长参数化的一种方法，并给出了相应的算法。通过求出曲线上所谓的‘最坏点’并在相应点处进行分割。可得到两条Bézier样条曲线。让这两条Bézier样条曲线具有与它们的近似弧长成比例的权，并对所得到的新的Bézier样条曲线进行同样的工作最终可得到一条由多条Bézier样条曲线所构成的新曲线。将这多条Bézier样条曲线合并成为一条Bézier样条曲线并通过节点插入技术将所得Bézier样条曲线转化为B一样条曲线的形式可得到全局参数域，其中各条Bézier曲线在全局参数域中所占子区间的长度与它们的权成比例，这样使得到了一条近似弧长参数化曲线。     

On the convexity of parametric Bézier triangular surfaces

In this paper, we discuss the convexity of parametric Bézier triangular patches, give some sufficient conditions of it to be convex, which only depend on the edge vectors and twist vectors. All the conditions we obtained can be served as the extension of the convexity preserving conditions of functional Bézier triangular patches.     

A simple algorithm for designing developable Bézier surfaces

An algorithm is presented that generates developable Bézier surfaces through a Bézier curve of arbitrary degree and shape. The algorithm has two important advantages. No (nonlinear) characterizing equations have to be solved and the control of singular points is guaranteed. Further interpolation conditions can be met.     

Algorithms for rational Bézier curves

A recursive algorithm for the evaluation of rational Bézier curves is presented; it consists of a construction that works with a constant cross ratio. This geometric principle is carried over to other algorithms.     

Bézier curves and surfaces with shape parameters

Bézier curves with n shape parameters and triangular Bézier surfaces with 3n(n+1)/2 shape parameters are presented in this paper. The geometric significance of the shape parameters and the geometric properties of these curves and surfaces are discussed. The shapes of the curves and the surfaces can be modified intuitively, foreseeably and precisely by changing the values of the shape parameters.