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.     

An approximation approach of the clothoid curve defined in the interval [0, π/2] and its offset by free-form curves

We propose an algorithm to approximate the clothoid curve defined in the interval [0, π/2] and its offset curves with Bézier curves and the approximation errors converge to zero as the degree of the Bézier curves is increased. Secondly, we discuss how to approximate the clothoid curve by B-spline curves of low degrees. By employing our method, the clothoid curve and its offset can be efficiently incorporated into CAD/CAM systems, which are important for the development of 3D civil engineering CAD systems, especially for 3D highway road design systems. The proposed method has been implemented on AutoCAD R14.     

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.     

空间曲线几何Hermite插值的B样条方法

在给定的GC2插值条件,利用de Boor的构造平面曲线的GC2-Hermite插值方法,构造了一条具有两个自由度的三次B样条插值曲线,并证明插值曲线是局部存在的且具有4阶精度.     

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.     

Projectively invariant classes of geometric continuity for CAGD

A new classification of geometric continuity is presented. It is based on the simultaneous application of the concept of ‘contact of order r” to curves (or surfaces) and to their tangent surfaces and generalized tangent surfaces; this implies projective invariance. The developments are illustrated by algorithms for geometrically continuous Bézier curves. Moreover, we obtain new properties of certain splines derived by variational formulations.     

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.     

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.     

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.     

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.     

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.     

Improved bounds on partial derivatives of rational triangular Bézier surfaces

This paper applies inequality skill, degree elevation of triangular Bézier surfaces and difference operators to deduce the bounds on first and second partial derivatives of rational triangular Bézier surfaces. Further more, we prove that the new bounds are tighter and more effective than the known ones. All the results are obviously helpful for further optimization of geometric design systems.     

Tool path generation for free form surfaces using Bézier curves/surfaces

Presented in this paper is a tool path generation method for multi-axis machining of free-form surfaces using Bézier curves and surfaces. The tool path generation includes two core steps. First is the forward-step function that determines the maximum distance, called forward step, between two cutter contact (CC) points with a given tolerance. The second component is the side step function which determines the maximum distance, called side step, between two adjacent tool paths with a given scallop height. Using the Bézier curves and surfaces, we generate cutter contact (CC) points for free-form surfaces and cutter location (CL) data files for post processing. Several parts are machined using a multi-axis milling machine. As part of the validation process, the tool paths generated from Bézier curves and surfaces are analyzed to compare the machined part and the desired part.     

Gregory-type patches bounded by low degree intergral curves for G continuity

G² continuity of free-form surfaces is sometimes very important in engineering applications. The conditions for G² continuity to connect two Bézier patches were studied and methods have been developed to ensure it. However, they have some restrictions on the shapes of patches of the Bézierpatch formulation. Gregory patch is a kind of free-form surface patch which is extended from Bézier patch so that four first derivatives on its boundary curves can be specified without restrictions of the compatibility condition. Several types of Gregory patches have been developed for intergral, rational, and NURBS boundary curves. In this paper, we propose some intergral boundary Gregorytype patches bounded by cubic and quartic curves for G² continuity.     

Local surface interpolation with Bézier patches

A surface interpolation method for meshes of cubic curves is described. A mesh of cubic curve is constructed between the given vertices. This mesh is filled with Bézier patches, so that the surface is represented as a union of geometrically continuous bicubic quadrilateral and/or quartic triangular Bézier patches. The method is local and uses Farin's [Farin '83] conditions of G¹ continuity between patches. The procedure for finding the needed control points of the Bézier patches is simple and efficient.     

广义三阶Bezier曲线的几何构造

针对一类含有3个形状参数的广义三阶Bézier（GCB）曲线,推导出GCB曲线的基函数与四次Bernstein基函数的转换公式。利用升阶公式,建立了它与四次Bézier曲线的关系,给出了几何结构和矩阵表示形式。GCB曲线不仅具有三次Bézier曲线的特征,而且在控制多边形保持不变的条件下,具有形状可调性和对控制多边形更好的逼近性。实例表明：构造的GCB曲线为曲线曲面设计提供了有效的新方法。     

Linear and angular feedrate interpolation for planar implicit curves

In multi-axis CNC machining, a cutting tool combines translation and rotational movements with respect to a workpiece. This entails CNC interpolation to take angular feedrates defining the speed of the rotational movement into account, while current interpolations are overwhelmingly based on linear feedrates defining the speed of the translation movement. This paper considers linear and angular feedrate interpolations of 2D tool paths for multi-axis machining. A 2D tool path is identified by a position curve assumed to be a 2D implicit curve and a tool orientation curve based on the implicit curve. The paper will relate the angular feedrate to the linear feedrate along the tool paths. The result is then used to develop the angular feedrate interpolation from the linear feedrate interpolation. Detailed interpolation schemes for such hybrid feedrates as a constant linear feedrate with constrained angular feedrate (CLCA) and a constant angular feedrate with constrained linear feedrate (CACL) have further been constructed. A corrected interpolation scheme is further present to augment the initial interpolation. Example interpolations including a quadric curve, the nodal cubic and a quartic curve are carried out to illustrate the feasibility and effectiveness of the proposed approach.     

Geometric modeling of curves using weighted linear and circular segments

This paper presents a method for geometric modeling of curves. Linear and circular segments with weight functions are used for curve design. Similar weight functions are used for smoothing and interpolation. The curve can be composed of consecutive linear and circular segments with smoothing at knots. If an interpolating curve is necessary, then weighted linear and circular segments are used for blending. Smoothness of the designed curve depends on the chosen weight function. It is possible to design the curve of variable smoothness using different weight functions.     

A class of general quartic spline curves with shape parameters

With a support on four consecutive subintervals, a class of general quartic splines are presented for a non-uniform knot vector. The splines have C² continuity at simple knots and include the cubic non-uniform B-spline as a special case. Based on the given splines, piecewise quartic spline curves with three local shape parameters are given. The given spline curves can be C²∩G³ continuous by fixing some values of the curve?s parameters. Without solving a linear system, the spline curves can also be used to interpolate sets of points with C² continuity. The effects of varying the three shape parameters on the shape of the quartic spline curves are determined and illustrated.