Partial derivatives of rational Bézier surfaces

A rational surface is the locus of a rational curve that is moving through space and thereby changing its shape by changing its control points and weights. This intuitive definition can be used to derive hodographs of rational Bézier surfaces and their bounds of magnitude.     

Self-intersections of rational Bézier curves

Rational Bézier curves provide a curve fitting tool and are widely used in Computer Aided Geometric Design, Computer Aided Design and Geometric Modeling. The injectivity (one-to-one property) of rational Bézier curve as a mapping function is equivalent to the curve without self-intersections. We present a geometric condition on the control polygon which is equivalent to the injectivity of rational Bézier curve with this control polygon for all possible choices of weights. The proof is based on the degree elevation and toric degeneration of rational Bézier curve.     

Low-harmonic rational Bézier curves for trajectory generation of high-speed machinery

For a given high-speed machinery, a significant source of the internally induced vibrational excitation is the presence of high frequency harmonics in the trajectories that the system is forced to follow. In this paper a special class of rational Bézier curves is presented that correspond to low-harmonic trajectory patterns. Harmonic Bernstein polynomials and harmonic deCasteljau algorithm are also introduced as two major tools for generating the harmonic rational Bézier curves. These curves can be used to synthesize trajectories within the dynamic response limitations of the actuators while avoiding the excitation of the natural modes of vibration of the system.     

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.     

C 1 NURBS representations of G 1 composite rational Bézier curves

This paper is concerned with the re-representation of a G ¹ composite rational Bézier curve. Although the rational Bézier curve segments that form the composite curve are G ¹ continuous at their joint points, their homogeneous representations may not be even C ⁰ continuous in the homogeneous space. In this paper, an algorithm is presented to convert the G ¹ composite rational Bézier curve into a NURBS curve whose nonrational homogeneous representation is C ¹ continuous in the homogeneous space. This re-representation process involves reparameterization using Möbius transformations, smoothing multiplication and parameter scaling transformations. While the previous methods may fail in some situations, the method proposed in this paper always works.     

Multi-degree reduction of tensor product Bézier surfaces with conditions of corners interpolations

This paper studies the multi-degree reduction of tensor product B(?)zier surfaces with any degree interpolation conditions of four corners, which is urgently to be resolved in many CAD/CAM systems. For the given conditions of corners interpolation, this paper presents one intuitive method of degree reduction of parametric surfaces. Another new approximation algorithm of multi-degree reduction is also presented with the degree elevation of surfaces and the Chebyshev polynomial approximation theory. It obtains the good approximate effect and the boundaries of degree reduced surface can preserve the prescribed continuities. The degree reduction error of the latter algorithm is much smaller than that of the first algorithm. The error bounds of degree reduction of two algorithms are also presented .     

Assembly variation analysis of flexible curved surfaces based on Bézier curves

Assembly variation analysis of parts that have flexible curved surfaces is much more difficult than that of solid bodies, because of structural deformations in the assembly process. Most of the current variation analysis methods either neglect the relationships among feature points on part surfaces or regard the distribution of all feature points as the same. In this study, the problem of flexible curved surface assembly is simplified to the matching of side lines. A methodology based on Bézier curves is proposed to represent the side lines of surfaces. It solves the variation analysis problem of flexible curved surface assembly when considering surface continuity through the relations between control points and data points. The deviations of feature points on side lines are obtained through control point distribution and are then regarded as inputs in commercial finite element analysis software to calculate the final product deformations. Finally, the proposed method is illustrated in two cases of antenna surface assembly.     

Approximating a helix segment with a rational Bézier curve

From a differential geometric point of view a helix segment can be considered as a spatial generalization of a circular arc. Thus for problems of shape preservation and geometric modelling an approximate rational representation of a helix segment is of interest. In this paper rational Bézier curves of degree 4, 5 and 6 are presented that approximate a helix segment. The approximants fulfill certain geometric constraints. A generalized degree elevation for rational polynomials in Bézier representation is discussed and used for the construction.     

A basis for the implicit representation of planar rational cubic Bézier curves

We present an approach to finding the implicit equation of a planar rational parametric cubic curve, by defining a new basis for the representation. The basis, which contains only four cubic bivariate polynomials, is defined in terms of the Bézier control points of the curve. An explicit formula for the coefficients of the implicit curve is given. Moreover, these coefficients lead to simple expressions which describe aspects of the geometric behaviour of the curve. In particular, we present an explicit barycentric formula for the position of the double point, in terms of the Bézier control points of the curve. We also give conditions for when an unwanted singularity occurs in the region of interest. Special cases in which the method fails, such as when three of the control points are collinear, or when two points coincide, will be discussed separately.     

Lagrange geometric interpolation by rational spatial cubic Bézier curves

In the paper, the Lagrange geometric interpolation by spatial rational cubic Bézier curves is studied. It is shown that under some natural conditions the solution of the interpolation problem exists and is unique. Furthermore, it is given in a simple closed form which makes it attractive for practical applications. Asymptotic analysis confirms the expected approximation order, i.e., order six. Numerical examples pave the way for a promising nonlinear geometric subdivision scheme.     

Generalized rational Bézier curves for the rigid body motion design

In this paper, we present a new method for the smooth interpolation of the orientations of a rigid body motion. The method is based on the geometrical Hermite interpolation in a hypersphere. However, the non-Euclidean structure of a sphere brings a great challenge to the interpolation problem. For this consideration and the requirements for practical application, we construct the spherical analogue of classical rational Bézier curves, called generalized rational Bézier curves. The new spherical curves are obtained using the generalized rational de Casteljau algorithm, which is a generalization of the classical rational de Casteljau algorithm to a hypersphere. Then, \(G^2\) Hermite interpolation problem in hypersphere is solved analytically using the generalized rational Bézier curve of degree 5. The new method offers residual free parameters including shape parameters and weights, which guarantee the existence of the interpolant to arbitrary motion data and offer great flexibility for the shape design of the motion. Numerical examples show that our method is far better behaved according to the energy functional which is regarded as a measure of the motion shape.     

G 2 curve design with planar quadratic rational Bézier spiral segments

Spiral segments are useful in the design of fair curves. They are important in computer-aided design (CAD) and manufacturing applications, the design of highway and railway routes, trajectories of mobile robots, and other similar applications. Quadratic rational Bézier curves are often used in CAD applications because they can be used to draw conic sections exactly. This paper shows how curvature continuous curves can be designed using quadratic rational Bézier curve segments of monotone curvature.     

Bézier representation of geometrically continuous splines

As an intrinsic measure of smoothness,geometric continuity is an important problem in the fields of computer aided geometric design.It can afford more degrees of freedom for manipulating the shape of curve.However,piecewise polynomial functions of geometrically continuous splines are difficult to be constructed.In this paper,the conversion matrix between geometrically continuous spline basis functions and Bézier representation is analyzed.Based on this,construction of arbitrary degree geometrically continuous spline basis functions can be translated into a solution of linear system of equations.The original construction of geometrically continuous spline is simplified.     

Geometric construction of energy-minimizing Bézier curves

Modeling energy-minimizing curves have many applications and are a basic problem of Geometric Modeling.In this paper,we propose the method for geometric design of energy-minimizing B′ezier curves.Firstly,the necessary and sufficient condition on the control points for B′ezier curves to have minimal internal energy is derived.Based on this condition,we propose the geometric constructions of three kinds of B′ezier curves with minimal internal energy including stretch energy,strain energy and jerk energy.Given...     

广义Bézier曲线

为了有效地改进Bézier曲线的形状,给出了带局部形状参数的广义Bézier曲线,该曲线的表示式以一种函数的高阶逼近式为依据.通过对目标导矢和目标二阶导矢的系数的调整,生成满意的多项式曲线.所给曲线以Bézier曲线为特殊情形,能对较高次的B啨zier曲线进行有效地修改,也能方便地进行曲线段的拼接.     

插值有理Bézier渐近四边形的有理Bézier曲面

为构造封闭的曲线为有理Bézier曲面的边界渐近线,给出封闭四边曲线为渐近四边形的条件,并提出插值该四边形的曲面构造方法.首先在给定角点数据的前提下构造优化的n次有理Bézier渐近四边形;然后利用该四边形和曲面在四边形上的切矢确定曲面沿边界的两排控制顶点和权;最后极小化曲面薄板能量函数确定剩余自由的控制顶点,进而构造出光滑的双5n–7次有理Bézier插值曲面.实例展示边界曲线为有理3,4,5次时曲面的构造结果,以及边界曲线含有直线或者拐点的情况,表明该方法是可行的.