单圆弧样条保形插值算法

该文以插值具有偶数个点的闭多边形为例提出了一种新的圆弧样条插值算法。这种算法具有以下3个特点：（1）生成的圆弧样条曲线具有保形的特点；（2）圆弧样条中圆弧的段数与型值点个数相同。（3）圆弧段之间的连接点不一定在插值的型值点上，这样就能用更多的自由度来控制拟合曲线的形状。同此文中还提出了一个优化的算法来得到光顺的插值曲线，同时还给出了几个例子加以说明。     

Error-bounded biarc approximation of planar curves

Presented in this paper is an error-bounded method for approximating a planar parametric curve with a G¹ arc spline made of biarcs. The approximated curve is not restricted in specially bounded shapes of confined degrees, and it does not have to be compatible with non-uniform rational B-splines (NURBS). The main idea of the method is to divide the curve of interest into smaller segments so that each segment can be approximated with a biarc within a specified tolerance. The biarc is obtained by polygonal approximation to the curve segment and single biarc fitting to the polygon. In this process, the Hausdorff distance is used as a criterion for approximation quality. An iterative approach is proposed for fitting an optimized biarc to a given polygon and its two end tangents. The approach is robust and acceptable in computation since the Hausdorff distance between a polygon and its fitted biarc can be computed directly and precisely. The method is simple in concept, provides reasonable accuracy control, and produces the smaller number of biarcs in the resulting arc spline. Some experimental results demonstrate its usefulness and quality.     

A practicable approach to G biarc approximations for making accurate, smooth and non-gouged profile features in CNC contouring

Extensive research on G¹ biarc approximations to free-form curves has been conducted for the production of accurate, smooth and non-gouged profile features in CNC contouring. However, all the published work has only focused on improving the fitting accuracy between the biarc curve and the nominal free-form curve of a profile and minimizing the biarc number; as a result, the radii of the concave arcs of some biarcs could be less than the pre-determined tool radius, and the tool would overcut these arcs in machining, eventually gouging the profile. In this work, a new, practicable approach is proposed to completely solve this problem. The main feature of this approach is to find the gouging-free parameter interval of a biarc family, among which the radii of all the concave arcs are larger than the tool radius, and then to search in this interval for a best fitting biarc so that its approximation accuracy is within the tolerance. This approach is robust and easy to implement and can substantially promote the use of G¹ biarc curves for CNC machining.     

Data Approximation Using Biarcs

. An algorithm for data approximation with biarcs is presented. The method uses a specific formulation of biarcs appropriate for parametric curves in Bézier or NURBS formulation. A base curve is applied to obtain tangents and anchor points for the individual arcs joining in G 1 continuity. Data sampled from circular arcs or straight line segments is represented precisely by one biarc. The method is most useful in numerical control to drive the cutter along straight line or circular paths.     

开圆弧样条的保形插值算法

提出一种G1圆弧样条插值算法.该算法选取部分满足条件的型值点构造初始圆,然后过剩下的型值点分别构造相邻初始圆的公切圆.在此过程中,让所有型值点均为相应圆弧的内点,且每段圆弧尽量通过2个型值点.在型值点列满足较弱的条件下,曲线具有在事先给定首末切向的情况下圆弧总段数比型值点个数少且保形的特点.     

An approximation of circular arcs by quartic Bézier curves

In this paper we propose an approximation method for circular arcs by quartic Bézier curves. Using an alternative error function, we give the closed form of the Hausdorff distance between the circular arc and the quartic Bézier curve. We also show that the approximation order is eight. By subdivision of circular arcs with equi-length, our method yields the curvature continuous spline approximation of the circular arc. We confirm that the approximation proposed in this paper has a smaller error than previous quartic Bézier approximations.     

变曲率对称圆弧曲线及其在圆弧样条拟合中的应用

针对数控加工的需要，对圆弧样条拟合曲线的形状进行局部修改和优化，提出了一种新的圆弧样条曲线的基本形式－变曲率对称圆弧曲线，并给出了其计算方法和具体应用，该方法可满足不同运算字长数控系统对拟合后圆弧样条曲线最大曲率半径的要求，同时还可满足随动控制加工对拟合曲率变动量的要求。     

Using a biarc filter to compute curvature extremes of NURBS curves

A method to compute curvature minima and maxima of parametric curves (represented in NURBS format) is presented in this paper. Since the curvature changes vary rapidly along the path of (even smooth) curves, a biarc filter is employed to approximate the curvature function with a piecewise constant function. This allows the isolation of curvature extreme values that are found within-engineering tolerances via repeated biarc approximation followed by golden section search. Because the derivative of the curvature is numerically very unstable, only optimization without derivatives is feasible. However, given the excellent isolation property of biarc filters, curvature extremes are found within 10–20 steps even for high accuracy requirements ranging from 10⁻⁴ to 10⁻⁶.     

Approximation by fat arcs and fat biarcs

A general discussion of the including approximation of a curve by a fat arc is given followed by an algorithm for constructing an including fat arc for a parametric Bézier curve. An example of applying the algorithm is given. The results for a fat arc are then used to develop an including approximation for a curve segment using a fat biarc. An algorithm for a fat biarc including approximation is provided followed by examples of Bézier curves being included by a fat biarc.     

A Recursive Subdivision Algorithm for Piecewise Circular Spline

We present an algorithm for generating a piecewise G ¹ circular spline curve from an arbitrary given control polygon. For every corner, a circular biarc is generated with each piece being parameterized by its arc length. This is the first subdivision scheme that produces a piecewise biarc curve that can interpolate an arbitrary set of points. It is easily adopted in a recursive subdivision surface scheme to generate surfaces with circular boundaries with pieces parameterized by arc length, a property not previously available. As an application, a modified version of Doo–Sabin subdivision algorithm is outlined making it possible to blend a subdivision surface with other surfaces having circular boundaries such as cylinders.     

参数曲线近似弧长参数化的插值方法

本文提出了参数曲线近似弧长参数化的一种插值方法。参数曲线的弧长函数的单调增的，近似弧长参数化可以转化为弧长函数的保单调分段有理线性插值。用这种插值得到的近似弧长参数化曲线插值原曲线上的一组点，最后，两个实例表明了近似弧长参数化曲线能很好地逼近原曲线，且没有所不希望的波动。     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.     

Biarc curves

A numerical curve fitting technique is described based on chains of circular arc and straight line segments. For portions of the curve which are not straight, two circular arcs are used between each pair of data points, and for this reason the term biarc has been applied to the resulting curves. This type of curve fitting was designed for the shipbuilding industry in 1970 and provides the basic curve and hull surface definition for the widely-used BRITSHIPS system.     

平面NURBS曲线及其Offset的双圆弧逼近

除直线、圆弧、速端曲线等少数几种曲线外,平面参数曲线的offset曲线通常不能表示成有 理参数形式,因此在实际应用中,为了方便造型系统中数据结构和几何算法的统一表示,offse t曲线通常用低次曲线逼近来表示.通过用双圆弧逼近表示NURBS(non-uniform rational B -spline)曲线及其offset,并利用双圆弧逼近的特有性质,把offset的双圆弧逼近转化为原 曲线的双圆弧逼近,简化了问题的求解.同时考虑了双圆弧逼近算法中分割点的选取、公切点 的确定以及误差估计等主要问题.具体算     

An optimization approach for biarc curve-fitting of B-spline curves

We present an approach to the optimal fitting of a biarc-spline to a given B-spline curve. The objective is to minimize the area between the original B-spline curve and the fitted curve. Such an objective has obvious practical implications. This approach differs from conventional biarc curve-fitting techniques in two main aspects and has some desirable features. Firstly, it exploits the inherent freedom in the choice of the biarc that can be fitted to a given pair of end-points and their tangents. The conventional approach to biarc curve-fitting introduces additional constraints, such as the minimal difference in curvature or others to uniquely determine successive biarcs. In this approach, such constraints are not imposed. Instead, the freedom is exploited in the problem formulation to achieve a better fit. Secondly, the end-points do not lie on the curve so that appropriate tolerance control can be imposed through the use of additional constraints. Almost all previous biarc-fitting methods consider end-points that are on the original curve. As a result of these two aspects, the resulting biarc curve fits closely to the original curve with relatively fewer segments. This has a desirable effect on the surface finish, verification of CNC codes and memory requirement. Numerical results of the application of this approach to several examples are presented.     

二次均匀B样条曲线的双圆弧逼近方法*

提出了一种用双圆弧对二次均匀B样条曲线的分段逼近方法。首先,对一条具有n 1个控制顶点的二次均匀B样条曲线按照相邻两节点界定的区间分成n-1段只有三个控制顶点的二次均匀B样条曲线段;然后对每一曲线段构造一条双圆弧进行逼近。所构造的双圆弧满足端点及端点切向量条件,即双圆弧的两个端点分别是所逼近的曲线段的端点,而且双圆弧在两个端点处的切向量是所逼近的曲线段在端点处的单位切向量。同时,双圆弧的连接点是双圆弧连接点轨迹圆与其所逼近的曲线段的交点。这些新构造出来的双圆弧连接在一起构成了一条圆弧样条曲线,即二次均匀B样条曲线的逼近曲线。另外给出了逼近误差分析和实例说明。     

Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics

This paper compares two techniques for the approximation of the offsets to a given planar curve. The two methods are based on approximate conversion of the planar curve into circular splines and Pythagorean hodograph (PH) splines, respectively. The circular splines are obtained using a novel variant of biarc interpolation, while the PH splines are constructed via Hermite interpolation of C¹ boundary data.We analyze the approximation order of both conversion procedures. As a new result, the C¹ Hermite interpolation with PH quintics is shown to have approximation order 4 with respect to the original curve, and 3 with respect to its offsets. In addition, we study the resulting data volume, both for the original curve and for its offsets. It is shown that PH splines outperform the circular splines for increasing accuracy, due to the higher approximation order.     

平面曲面的曲率表示及其应用

通过平面曲线的曲率函数显式表示,对这种样条曲线及其造型作了研究,并给出了在线性曲率条件下的插值样条曲线生成算法。     

集多种特性的三次三角伪B样条

目的 为了同时解决传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,提出了一类集多种特性的三次三角伪B样条。方法 首先构造了一组带两个参数的三次三角伪B样条基函数,然后在此基础上定义了相应的参数伪B样条曲线,并讨论了该曲线的特性及光顺性问题,最后研究了相应的代数伪B样条,并给出了最优代数伪B样条的确定方法。结果 参数伪B样条曲线不仅满足C²连续,而且无需求解方程系统即可自动插值于给定的型值点。当型值点保持不变时,插值曲线的形状还可通过自带的两个参数进行调控。在适当条件下,该参数伪B样条曲线可精确表示圆弧、椭圆弧、星形线等常见的工程曲线。相应的代数伪B样条具有参数伪B样条曲线类似的性质,利用最优代数伪B样条可获得满意的插值效果。结论 所提出的伪B样条同时解决了传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,是一种实用的曲线造型方法。