Circle approximation using integral B-splines

A method to approximate a circular arc of any sweep angle with integral B-spline curves of any degree is presented. The idea is to interpolate end derivatives as well as some internal points with integral B-splines of given degree and continuity. The critical element is the choice of the right end derivative directions and magnitudes. The points and the derivatives at these locations are sampled uniformly using the trigonometric equation of the circle. The method is very general in that any degree and any level of parametric continuity can be specified.     

Curvature continuous path generation for autonomous vehicle using B-spline curves

In this paper we introduce a roadmap algorithm for generating collision-free paths in terms of cubic B-spline curves for unmanned vehicles used in mining operations. The algorithm automatically generates collision-free paths that are curvature continuous with an upper bounded curvature and a small slope discontinuity of curvature at knots, when we are given the locations of the obstacles, the boundary geometry of the working area, positions and directions of the vehicle at the start, loading, and the goal points. Our algorithm also allows us to find a switch back point where the vehicle reverses its direction to enter the loading area. Examples are provided to demonstrate the effectiveness of the proposed algorithms.     

Design of motion along parameterized curves using B-splines

The paper presents a method for design of motion along parameterized curves using B-splines. The presented approach is based on local parameterizations of orientations with respect to the Frenet frame moving along the parameterized curve. Conditions, which ensure C^p continuity of the designed motion, are established. It is shown that the designed motion is invariant under transition between orthonormal reference frames of the Euclidean affine space E³.     

Interactive design of fair hull surfaces using B-splines

Two related techniques for the interactive computer aided design of lines and molded surfaces are presented. The first technique is aimed at lines generation from hull form parameters. The second is aimed at fairing a mathematical surface that is based upon the lines. Both schemes employ B-splines to represent waterlines and stations in the lines drawing. The lines of a modern ship which have been generated by the first method are shown. A new indicator of surface fairness called Euler's net is illustrated. Indications are given of the expected future developments in these continuiing efforts.     

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.     

Exact and approximate computation of B-spline curves on surfaces

Curves on surfaces are important elements in computer aided geometric design. After presenting a method to explicitly compute these curves in three-dimensions, practical algorithmic issues are discussed concerning the efficiency of the implementation. Good approximations are important because of the quite high degree of exact curves on surfaces. We present two approximate solutions to the problem. The first is derived from the exact representation, while the second extends conventional least-squares approximation by incorporating the geometry of the surface as well. The efficiency and behaviour of the algorithms are evaluated by means of examples.     

Adaptive knot placement using a GMM-based continuous optimization algorithm in B-spline curve approximation

One of the key problems in using B-splines successfully to approximate an object contour is to determine good knots. In this paper, the knots of a parametric B-spline curve were treated as variables, and the initial location of every knot was generated using the Monte Carlo method in its solution domain. The best km knot vectors among the initial candidates were searched according to the fitness. Based on the initial parameters estimated by an improved k-means algorithm, the Gaussian Mixture Model (GMM) for every knot was built according to the best km knot vectors. Then, the new generation of the population was generated according to the Gaussian mixture probabilistic models. An iterative procedure repeating these steps was carried out until a termination criterion was met. The GMM-based continuous optimization algorithm could determine the appropriate location of knots automatically. A set of experiments was then implemented to evaluate the performance of the new algorithm. The results show that the proposed method achieves better approximation accuracy than methods based on artificial immune system, genetic algorithm or squared distance minimization (SDM).     

Constructive sculpting of heterogeneous volumetric objects using trivariate B-splines

This paper deals with modeling heterogeneous volumetric objects as point sets with attributes using trivariate B-splines. In contrast to homogeneous volumes with uniform distribution of material and other properties, a heterogeneous volumetric object has a number of variable attributes assigned at each point. An attribute is a mathematical model of an object property of an arbitrary nature (material, photometric, physical, statistical, etc.). In our approach, the function representation (FRep) is used as the basic model for both object geometry and attributes represented independently using real-valued scalar functions of point coordinates. While FRep directly defines object geometry, for an attribute it specifies a space partition used to define the attribute function. We propose a volume sculpting scheme with multiresolution capability based on trivariate B-spline functions to define both object geometry and its attributes. A new trivariate B-spline primitive is proposed that can be used as a leaf in an FRep constructive tree. An interactive volume modeler based on trivariate B-splines and other simple primitives is described, with a real-time repolygonization of the surface during modeling. We illustrate that the space partition obtained in the modeling process can be applied to define attributes for the objects with an arbitrary geometry model such as BRep or homogeneous volume models.     

FPGA-based hardware CNC interpolator of Bezier,splines, B-splines and NURBS curves for industrial applications

Tool path interpolation is an important part of Computerized Numerical Control (CNC) systems because it is related to the machining accuracy, tool-motion smoothness and overall efficiency. The use of parametric curves to generate tool-motion trajectories on a workpiece for high accuracy machining has become a standard data format that is used for CAD/CAM (Computer Aided Design/Computer Aided Manufacturing) and CNC systems. Splines, Bezier, B-splines, and NURBS (Non-Uniform Rational B-splines) curves are the common parametric technique used for tool path design. However, the reported works bring out the high computational load required for this type of interpolation, and then at best only one interpolation algorithm is implemented. The contribution of this paper is the development of a hardware processing unit based on Field Programmable Gate Arrays (FPGA) for industrial CNC machines, which is capable of implementing the four main interpolation techniques. It allows the selection of the required interpolation technique according the application. Two CAD models are designed for test the CNC interpolations; experimental results show the efficiency of the proposed methodology.     

曲线曲面逼近与插值的统一表示

为了用一种模型实现从逼近到插值的转换,在多项式空间上构造了含一个参数的调配函数,由之定义了基于4点分段的曲线,该曲线可以理解为由相同的一组控制顶点定义的逼近曲线和插值曲线的线性组合,其中的逼近曲线为3次均匀B样条曲线,插值曲线经过除首末点以外的所有控制点。在均匀参数分割下,曲线具有C2连续性,取特殊参数时可达C3连续。在参数变化过程中,曲线各段起点、终点的位置发生改变,但这些点处的一阶、二阶导矢始终保持不变,即始终与3次B样条曲线相同。曲线形状与端点条件密切相关,而B样条曲线具有良好的保形性,这些综合因素使得曲线在形状变化的过程中始终可以较好地保持控制多边形的特征。采用张量积方法将曲线推广至曲面,曲线曲面图例显示了该方法在造型设计中的有效性。     

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.     

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.     

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 review of stochastic algorithms with continuous value function approximation and some new approximate policy iteration algorithms for multidimensional continuous applications

We review the literature on approximate dynamic programming, with the goal of better understanding the theory behind practical algorithms for solving dynamic programs with continuous and vector-valued states and actions and complex information processes. We build on the literature that has addressed the well-known problem of multidimensional (and possibly continuous) states, and the extensive literature on model-free dynamic programming, which also assumes that the expectation in Bellman’s equation cannot be computed. However, we point out complications that arise when the actions/controls are vector-valued and possibly continuous. We then describe some recent research by the authors on approximate policy iteration algorithms that offer convergence guarantees (with technical assumptions) for both parametric and nonparametric architectures for the value function.     

The minimization of axiom sets characterizing generalized approximation operators

In the axiomatic approach of rough set theory, rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators. Thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory. In this paper, the independence of axioms of generalized crisp approximation operators is investigated, and their minimal sets of axioms are presented.     

基于二进制具有量子行为的粒子群算法的多边形近似

提出了适合二进制搜索空间的具有量子行为的粒子群优化算法（BQPSO）。在二进制环境中重新定义粒子的位置向量及距离向量，调整了QPSO算法的进化公式。用二进制具有量子行为的粒子群算法求解平面数字曲线的多边形近似，解决了传统BPSO算法中粒子搜索范围受限的问题。用2条通用benchmark曲线进行测试，结果表明，该算法较BPSO加快了收敛速度，在相同的容忍误差和迭代次数下找到了更少顶点的多边形。     

Modelling and effective modification of smooth boundary geometry in boundary problems using B-spline curves

To create curves in computer graphics, we use, among others, B-splines since they make it possible to effectively produce curves in a continuous way using a small number of de Boor’s control points. The properties of these curves have also been used to define and create boundary geometry in boundary problems solving using parametric integral equations system (PIES). PIES was applied for resolution 2D boundary-value problems described by Laplace’s equation. In this PIES, boundary geometry is theoretically defined in its mathematical formalism, hence the numerical solution of the PIES requires no boundary discretization (such as in BEM) and is simply reduced to the approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained were compared with both exact and numerical solutions.     

Optimal polygonal approximation of digitized curves using the sum of square deviations criterion

We address the problem of finding an optimal polygonal approximation of digitized curves. Several optimal algorithms have already been proposed to determine the minimum number of points that define a polygonal approximation of the curve, with respect to a criterion error. We present a new algorithm with reasonable complexity to determine the optimal polygonal approximation using the mean square error norm. The idea is to estimate the remaining number of segments and to integrate the cost in the A^* algorithm. The solution is optimal in the minimum number of segments.