Periodic Bézier curves

We construct closed trigonometric curves in a Bézier-like fashion. A closed control polygon defines the curves, and the control points exert a push-pull effect on the curve. The representation of circles and derived curves turns out to be surprisingly simple. Fourier and Bézier coefficients of a curve relate via Discrete Fourier Transform (DFT). As a consequence, DFT also applies to several operations, including parameter shift, successive differentiation and degree-elevation. This Bézier model is a particular instance of a general periodic scheme, where radial basis functions are generated as translates of a symmetric function. In addition to Bézier-like approximation, such a periodic scheme subsumes trigonometric Lagrange interpolation. The change of basis between Bézier and Lagrange proceeds via DFT too, which can be applied to sample the curve at regularly spaced parameter values. The Bézier curve defined by certain control points is a low-pass filtered version of the Lagrange curve interpolating the same set of points.     

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.     

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。     

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-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,这样便得到了一条近似弧长参数化曲线。     

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.     

The rational cubic Bézier representation of conics

The rational cubic Bézier curve is a very useful tool in CAGD. It incorporates both conic sections and parametric cubic curves as special cases, so its advantage is that one can deal with curves of these two kinds in one computer procedure. In this paper, the necessary and sufficient conditions for representing conics by the rational cubic Bézier form in proper parametrization are investigated; these conditions can be divided into two parts: one for weights and the other for Bézier vertices.     

Construction of Bézier surface patches with Bézier curves as geodesic boundaries

Given four polynomial or rational Bézier curves defining a curvilinear rectangle, we consider the problem of constructing polynomial or rational tensor-product Bézier patches bounded by these curves, such that they are geodesics of the constructed surface. The existence conditions and interpolation scheme, developed in a general context in earlier studies, are adapted herein to ensure that the geodesic-bounded surface patches are compatible with the usual polynomial/rational representation schemes of CAD systems. Precise conditions for four Bézier curves to constitute geodesic boundaries of a polynomial or rational surface patch are identified, and an interpolation scheme for the construction of such surfaces is presented when these conditions are satisfied. The method is illustrated with several computed examples.     

A shape controled fitting method for Bézier curves

The problem of controling a shape when fitting a curve to a set of digitized data points by proceeding to a least squares approximation is considered. A nonlinear method of solving this problem, dedicated to the obtention of planar curves with a smooth and monotonous variation of curvature is introduced. This method uses particular Bézier curves, called typical curves, whose control polygon is partially constrained in order to provide the desired curve shape. The curve fitting principle is based on variations of the tangent direction at the ends of the curve. These variations are controled by the displacement of a given curve point. An automatic procedure using this method to get a curve close to a set of data points has been implemented. An application to car body shape design and a comparison with the least squares approximation method is presented and discussed.     

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.