Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

Doo-Sabin细分模式的尖锐特征造型

通过推广准均匀二次B样条的节点插入算法,对边界面、折痕面、角点面等特征面给出新的细分规则,从而使Doo-Sabin细分模式可以表示边界、折痕、角点、刺点等尖锐特征,且特征处不受拓扑结构的限制.在特征附近进行了连续性分析,所得到的极限曲面具有分片G1连续性.该算法既可以设计有特征的、任意拓扑的复杂曲面,又可以精确地表示球面、柱面、锥面等工程技术中常用的二次曲面,在CAD/CAM领域具有广泛的应用前景.     

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.     

散乱数据点的细分曲面重建算法及实现

提出一种对海量散乱数据根据给定精度拟合出无需裁剪和拼接的、反映细节特征的、分片光滑的细分曲面算法．该算法的核心是基于细分的局部特性，通过对有特征的细分控制网格极限位置分析，按照拟合曲面与数据点的距离误差最小原则，对细分曲面控制网格循环进行调整、优化、特征识别、白适应细分等过程，使得细分曲面不断地逼近原始数据．实例表明：该算法不仅具有高效性、稳定性，同时构造出的细分曲面还较好地反映了原始数据的细节特征。     

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。     

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。     

A subdivision scheme for Poisson curves and surfaces

The de Casteljau evaluation algorithm applied to a finite sequence of control points defines a Bézier curve. This evaluation procedure also generates a subdivision algorithm and the limit of the subdivision process is this same Bézier curve. Extending the de Casteljau subdivision algorithm to an infinite sequence of control points defines a new family of curves. Here, limits of this stationary non-uniform subdivision process are shown to be equivalent to curves whose control points are the original data points and whose blending functions are given by the Poisson distribution. Thus this approach generalizes standard subdivision techniques from polynomials to arbitrary analytic functions. Extensions of this new subdivision scheme from curves to tensor product surfaces are also discussed.     

A new subdivision based approach for piecewise smooth approximation of 3D polygonal curves

This paper presents an algorithm dealing with the data reduction and the approximation of 3D polygonal curves. Our method is able to approximate efficiently a set of straight 3D segments or points with a piecewise smooth subdivision curve, in a near optimal way in terms of control point number. Our algorithm is a generalization for subdivision rules, including sharp vertex processing, of the Active B-Spline Curve developed by Pottmann et al. We have also developed a theoretically demonstrated approach, analysing curvature properties of B-Splines, which computes a near optimal evaluation of the initial number and positions of control points. Moreover, our original Active Footpoint Parameterization method prevents wrong matching problems occurring particularly for self-intersecting curves. Thus, the stability of the algorithm is highly increased. Our method was tested on different sets of curves and gives satisfying results regarding to approximation error, convergence speed and compression rate. This method is in line with a larger 3D CAD object compression scheme by piecewise subdivision surface approximation. The objective is to fit a subdivision surface on a target patch by first fitting its boundary with a subdivision curve whose control polygon will represent the boundary of the surface control polyhedron.     

Computing the arc length of parametric curves

Specifying constraints on motion is simpler if the curve is parameterized by arc length, but many parametric curves of practical interest cannot be parameterized by arc length. An approximate numerical reparameterization technique that improves on a previous algorithm by using a different numerical integration procedure that recursively subdivides the curve and creates a table of the subdivision points is presented. The use of the table greatly reduces the computation required for subsequent arc length calculations. After table construction, the algorithm takes nearly constant time for each arc length calculation. A linear increase in the number of control points can result in a more than linear increase in computation. Examples of this type of behavior are shown     

Conjugate-Gradient Progressive-Iterative Approximation for Loop and Catmull-Clark Subdivision Surface Interpolation

Loop and Catmull-Clark are the most famous approximation subdivision schemes, but their limit surfaces do not interpolate the vertices of the given mesh. Progressive-iterative approximation (PIA) is an efficient method for data interpolation and has a wide range of applications in many fields such as subdivision surface fitting, parametric curve and surface fitting among others. However, the convergence rate of classical PIA is slow. In this paper, we present a new and fast PIA format for constructing interpolation subdivision surface that interpolates the vertices of a mesh with arbitrary topology. The proposed method, named Conjugate-Gradient Progressive-Iterative Approximation (CG-PIA), is based on the Conjugate-Gradient Iterative algorithm and the Progressive Iterative Approximation (PIA) algorithm. The method is presented using Loop and Catmull-Clark subdivision surfaces. CG-PIA preserves the features of the classical PIA method, such as the advantages of both the local and global scheme and resemblance with the given mesh. Moreover, CG-PIA has the following features. 1) It has a faster convergence rate compared with the classical PIA and W-PIA. 2) CG-PIA avoids the selection of weights compared with W-PIA. 3) CG-PIA does not need to modify the subdivision schemes compared with other methods with fairness measure. Numerous examples for Loop and Catmull-Clark subdivision surfaces are provided in this paper to demonstrate the efficiency and effectiveness of CG-PIA.     

用细分螺线插值容许G2Hermite 数据

为使几何细分方法生成的平面螺线段插值平面容许G2Hermite 数据,基于 平面双圆弧插值理论提出了该方法首末端点处新的细分规则。理论分析表明,修改后的细分 方法所得极限曲线是曲率单调、不变号的螺线段,且插值首末端点处的点、切向、曲率。数 值算例表明,修改后的细分方法收敛速度较快,极限曲线具有较好的形状。     

Curve interpolation in recursively generated B-spline surfaces over arbitrary topology

Recursive subdivision is receiving a great deal of attention in the definition of B-spline surfaces over arbitrary topology. The technique has recently been extended to generate interpolating surfaces with given normal vectors at the interpolated vertices. This paper describes an algorithm to generate recursive subdivision surfaces that interpolate B-spline curves. The control polygon of each curve is defined by a path of vertices of the polyhedral network describing the surface. The method consists of applying a one-step subdivision of the initial network and modifying the topology in the neighborhood of the vertices generated from the control polygons. Subsequent subdivisions of the modified network generate sequences of polygons each of which converges to a curve interpolated by the limit surface. In the case of regular networks, the method can be reduced to a knot insertion process.     

曲线插值的一种保凸细分方法

为了弥补以四点插值细分方法为代表的线性细分方法在形状控制方面的缺陷,提出一种基于几何的插值型保凸细分方法.细分过程每一步中,每条边所对应的新控制顶点由原控制顶点及其切向共同确定;每点处的切向由其邻近的点所确定,并且随细分过程逐步调整.理论分析表明,该方法的极限曲线是G1连续的保凸曲线.如果所有的初始点取自圆弧段,则极限曲线就是该圆弧段.数值实例表明,采用文中方法得到的曲线较为光顺.     

Construction of minimal subdivision surface with a given boundary

The fascinating characters of minimal surface make it to be widely used in shape design. While the flexibility and high quality of subdivision surface make it a powerful mathematical tool for shape representation. In this paper, we construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric partial differential equation. This equation is solved by a finite element method where the finite element space is spanned by the limit functions of an extended Loop’s subdivision scheme proposed by Biermann et al. Using this extended Loop’s subdivision scheme we can treat a surface with boundary, thereby construct the perfect minimal subdivision surfaces with any topology of the control mesh and any shaped boundaries.     

The error analysis for degree reduction of Bézier curves

The error analysis of Farin's and Forrest's algorithms for generating an approximation of degree n − 1 to an n^th degree Bézier curve is presented. Algorithms are based on observations of the geometric properties of the Bézier curve which allow the development of detailed error analysis. By combining subdivision with a degree reduction algorithm, a piecewise approximation can be generated, which is within some preset error tolerance of the original curve. The number of subdivisions required can be determined a priori and a piecewise approximation of degree m can be generated by iterating the scheme.     

插值细分曲面三角剖分研究

把三维参数化曲面的离散化算法应用到三角网格表示的离散曲面上。用一种可生成C1阶连续曲面的插值分割技术——改进蝶形算法,重新构造极限面。用推进波前法在物理空间直接离散化,所以不需要进行参数化。     

On the smooth convergence of subdivision and degree elevation for Bézier curves

Bézier subdivision and degree elevation algorithms generate piecewise linear approximations of Bézier curves that converge to the original Bézier curve. Discrete derivatives of arbitrary order can be associated with these piecewise linear functions via divided differences. Here we establish the convergence of these discrete derivatives to the corresponding continuous derivatives of the initial Bézier curve. Thus, we show that the control polygons generated by subdivision and degree elevation provide not only an approximation to a Bézier curve, but also approximations of its derivatives of arbitrary order.     

Method for intersecting algebraic surfaces with rational polynomial patches

The paper presents a hybrid algorithm for the computation of the intersection of an algebraic surface and a rational polynomial parametric surface patch. This algorithm is based on analytic representation of the intersection as an algebraic curve expressed in the Bernstein basis; automatic computation of the significant points of the curve using numerical techniques, subdivision and convexity properties of the Bernstein basis; partitioning of the intersection domain at these points; and tracing of the resulting monotonic intersection segments using coarse subdivision and faceting methods coupled with Newton techniques. The algorithm described in the paper treats intersections of arbitrary order algebraic surfaces with rational biquadratic and bicubic patches and introduces efficiency enhancements in the partitioning and tracing parts of the solution process. The algorithm has been tested with up to degree four algebraics and bicubic patches.     

基于最佳圆弧样条逼近的快速距离曲面计算

距离曲面是一种常用的隐式曲面,它在几何造型和计算机动画中具有重要的应用价值,但以往往在对距离曲面进行多边形化时速较慢,为了提高点到曲线最近距离计算的效率,提出了一种基于最佳圆弧样条逼近的快速线骨架距离曲面计算方法,该算法对于一条任意的二维NURBS曲线,在用户给定的误差范围内,先用最少量的圆弧样条来逼近给定的曲线,从而把点到NURBS曲线最近距离的计算问题转化为点到圆弧样条最近距离的计算问题,由于在对曲面进行多边形化时,需要大量的点到曲线最近距离的计算,而该处可以将点到圆弧样条最近距离很少的计算量来解析求得,故该算法效率很高,该实验表明,算法简单实用,具有很大的应用价值。     

Loop subdivision surfaces interpolating -spline curves

This paper presents a novel method for defining a Loop subdivision surface interpolating a set of popularly-used cubic B-spline curves. Although any curve on a Loop surface corresponding to a regular edge path is usually a piecewise quartic polynomial curve, it is found that the curve can be reduced to a single cubic B-spline curve under certain constraints of the local control vertices. Given a set of cubic B-spline curves, it is therefore possible to define a Loop surface interpolating the input curves by enforcing the interpolation constraints. In order to produce a surface of local or global fair effect, an energy-based optimization scheme is used to update the control vertices of the Loop surface subjecting to curve interpolation constraints, and the resulting surface will exactly interpolate the given curves. In addition to curve interpolation, other linear constraints can also be conveniently incorporated. Because both Loop subdivision surfaces and cubic B-spline curves are popularly used in engineering applications, the curve interpolation method proposed in this paper offers an attractive and essential modeling tool for computer-aided design.