Developable Bézier patches: properties and design

Geometric design of quadratic and cubic developable Bézier patches from two boundary curves is studied in this paper. The conditions for developability are derived geometrically from the de Casteljau algorithm and expressed as a set of equations that must be fulfilled by the Bézier control points. This set of equations allows us to infer important properties of developable Bézier patches that provide useful parameters and simplify the solution process for the patch design. With one boundary curve freely specified, five more degrees of freedom are available for a second boundary curve of the same degree. Various methods are introduced that fully utilize these five degrees of freedom for the design of general quadratic and cubic developable Bézier patches in 3D space. A more restricted generalized conical model or cylindrical model provides simple solutions for higher-order developable patches.     

Unknots with highly knotted control polygons

An example is presented of a cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bézier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its Bézier curve.     

Boundary evaluation for interval Bézier curve

The objective of this paper is to provide an efficient and reliable algorithm for representing and evaluating the boundary of the interval Bézier curve in 2- and 3-D. The boundary of the planar Bézier curve is represented by a sequence of Bézier curve segments with same degree and line segments in the order they are encountered when marching counter-clockwise along its boundary. The boundary can also be represented as a single B-spline curve having the same degree with the interval Bézier curve. The boundary of the 3-D interval Bézier curve is made up of trimmed Bézier surface patches and rectangular patches. Some examples illustrate our algorithms.     

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.     

Approximate convolution with pairs of cubic Bézier LN curves

In this paper we present an approximation method for the convolution of two planar curves using pairs of compatible cubic Bézier curves with linear normals (LN). We characterize the necessary and sufficient conditions for two compatible cubic Bézier LN curves with the same linear normal map to exist. Using this characterization, we obtain the cubic spline approximation of the convolution curve. As illustration, we apply our method to the approximation of a font where the letters are constructed as the Minkowski sum of two planar curves. We also present numerical results using our approximation method for offset curves and compare our method to previous results.     

Constrained approximation of rational Bézier curves based on a matrix expression of its end points continuity condition

For high order interpolations at both end points of two rational Bézier curves, we introduce the concept of C^(v,u)-continuity and give a matrix expression of a necessary and sufficient condition for satisfying it. Then we propose three new algorithms, in a unified approach, for the degree reduction of Bézier curves, approximating rational Bézier curves by Bézier curves and the degree reduction of rational Bézier curves respectively; all are in L₂ norm and C^(v,u)-continuity is satisfied. The algorithms for the first and second problems can get the best approximation results, and for the third one, resorting to the steepest descent method in numerical optimization obtains a series of degree reduced curves iteratively with decreasing approximation errors. Compared to some well-known algorithms for the degree reduction of rational Bézier curves, such as the uniformizing weights algorithm, canceling the best linear common divisor algorithm and shifted Chebyshev polynomials algorithm, the new one presented here can give a better approximation error, do multiple degrees of reduction at a time and preserve high order interpolations at both end points.     

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.     

3个控制顶点的类三次Bézier螺线

目的 为了使得过渡曲线的设计更为简单高效。提出基于3个控制顶点的类三次Bézier螺线。方法 通过对基函数的研究首先构造了3条在一定条件下曲率单调递减的类三次Bézier曲线,并由参数的对称性得另3条曲率单调递增的曲线。它们具有端点性、凸包性、几何不变性等三次Bézier曲线的基本性质,特点是只有3个控制顶点。接着严格地证明了此类曲线曲率单调的充分条件。 结果 有两条曲线比三次Bézier曲线的曲率单调条件范围大,且类三次Bézier螺线与三次Bézier螺线存在一定的位置关系。这6条曲线中有4条曲线的一个端点处曲率为零,可组合成4对类三次Bézier螺线来构造两圆弧间半径比例不受限制的S型和C型G²连续过渡曲线;剩下的两条曲线在两圆弧半径相差较大的情况下都可做不含曲率极值点的过渡曲线。最后用实例表明了此类曲线的有效性。结论 在过渡曲线设计中基于3个控制顶点的类三次Bézier螺线比三次Bézier螺线更为简单高效。     

Finding the best conic approximation to the convolution curve of two compatible conics based on Hausdorff distance

We consider the convolution of two compatible conic segments. First, we find an exact parametric expression for the convolution curve, which is not rational in general, and then we find the conic approximation to the convolution curve with the minimum error. The error is expressed as a Hausdorff distance which measures the square of the maximal collinear normal distance between the approximation and the exact convolution curve. For this purpose, we identify the necessary and sufficient conditions for the conic approximation to have the minimum Haudorff distance from the convolution curve. Then we use an iterative process to generate a sequence of weights for the rational quadratic Bézier curves which we use to represent conic approximations. This sequence converges to the weight of the rational quadratic Bézier curve with the minimum Hausdorff distance, within a given tolerance. We verify our method with several examples.     

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.     

Quadratic approximation to plane parametric curves and its application in approximate implicitization

Expressing complex curves with simple parametric curve segments is widely used in computer graphics, CAD and so on. This paper applies rational quadratic B-spline curves to give a global C ¹ continuous approximation to a large class of plane parametric curves including rational parametric curves. Its application in approximate implicitization is also explored. The approximated parametric curve is first divided into intrinsic triangle convex segments which can be efficiently approximated with rational quadratic Bézier curves. With this approximation, we keep the convexity and the cusp (sharp) points of the approximated curve with simple computations. High accuracy approximation is achieved with a small number of quadratic segments. Experimental results are given to demonstrate the operation and efficiency of the algorithm.     

Identification of actors drawn in Ukiyoe pictures

This paper presents the development of line image keywords for the identification of actors drawn in Japanese traditional painting pictures known as Ukiyoe pictures. The system is based on visual features of the face from the image database files and is organized as a set of classifiers whose outputs are integrated after a normalization step. Line profile from the picture has been extracted in this investigation and has been approximated by Bézier curves. A learning algorithm has been developed to obtain the control points at high accuracy. A new curve matching method has been developed based on the feature points, rather than the corresponding points. This method can automatically fit a set of data points with piecewise geometrically continuous third order Bézier curves. Last of all, a new approach for distance calculation, namely “apple-node distance” has been introduced here for similarity calculation in image retrieval systems. The computation of similarity between curves has been established on the basis of this “apple-node” distance. The effectiveness of our method has been confirmed through computer simulation. The method developed here can be expanded to one of three dimensional shape-analyzing tools.     

Curvature continuous offset approximation based on circle approximation using quadratic Bézier biarcs

We present a method for G² end-point interpolation of offset curves using rational Bézier curves. The method is based on a G² end-point interpolation of circular arcs using quadratic Bézier biarcs. We also prove the invariance of the Hausdorff distance between two compatible curves under convolution. Using this result, we obtain the exact Hausdorff distance between an offset curve and its approximation by our method. We present the approximation algorithm and give numerical examples.     

Periodic Bézier curves

We construct closed trigonometric curves in a Bézier-like fashion. A closed control polygon defines the curves, and the control points exert a push-pull effect on the curve. The representation of circles and derived curves turns out to be surprisingly simple. Fourier and Bézier coefficients of a curve relate via Discrete Fourier Transform (DFT). As a consequence, DFT also applies to several operations, including parameter shift, successive differentiation and degree-elevation. This Bézier model is a particular instance of a general periodic scheme, where radial basis functions are generated as translates of a symmetric function. In addition to Bézier-like approximation, such a periodic scheme subsumes trigonometric Lagrange interpolation. The change of basis between Bézier and Lagrange proceeds via DFT too, which can be applied to sample the curve at regularly spaced parameter values. The Bézier curve defined by certain control points is a low-pass filtered version of the Lagrange curve interpolating the same set of points.     

对可调控Bézier曲线的改进

目的 在用Bézier曲线表示复杂形状时,相邻曲线的控制顶点间必须满足一定的光滑性条件。一般情况下,对光滑度的要求越高,条件越复杂。通过改进文献中的“可调控Bézier曲线”,以构造具有多种优点的自动光滑分段组合曲线。方法 首先给出了两条位置连续的曲线G^l连续的一个充分条件,进而证明了“可调控Bézier曲线”在普通Bézier曲线的G^l光滑拼接条件下可达G^l(l为曲线中的参数)光滑拼接。然后对“可调控Bézier基”进行改进得到了一组新的基函数,利用该基函数按照Bézier曲线的定义方式构造了一种新曲线。分析了该曲线的光滑拼接条件,并根据该条件定义了一种分段组合曲线。结果 对于新曲线而言,只要前一条曲线的最后一条控制边与后一条曲线的第1条控制边重合,两条曲线便自动光滑连接,并且在连接点处的光滑度可以简单地通过改变参数的值来自由调整。由新曲线按照特殊方式构成的分段组合曲线具有类似于B样条曲线的自动光滑性和局部控制性。不同的是,组合曲线的各条曲线段可以由不同数量的控制顶点定义,选择合适的参数,可以使曲线在各个连接点处达到任何期望的光滑度。另外,改变一个控制顶点,至多只会影响两条曲线段的形状,改变一条曲线段中的参数,只会影响当前曲线段的形状,以及至多两个连接点处的光滑度。结论 本文给出了构造易于拼接的曲线的通用方法,极大简化了曲线的拼接条件。此基础上,提出的一种新的分段组合曲线定义方法,无需对控制顶点附加任何条件,所得曲线自动光滑,且其形状、光滑度可以或整体或局部地进行调整。本文方法具有一般性,为复杂曲线的设计创造了条件。     

Tunnel-free voxelisation of rational Bézier surfaces

To synthesize natural and artificial objects into a hybrid graphics scene represented by a set of voxels, voxelisation of geometric models is necessary. Rational parametric surfaces have been widely used in the representation of free-form surfaces. Voxelisation of these surfaces is therefore of great importance in the development of a voxel-based modeling system. A key issue is to develop a tunnel-free voxelisation algorithm for these continuous surfaces. In this paper, we propose such an algorithm for a rational Bézier surface. We derive the bound of the parametric steps to ensure that the voxelised rational Bézier surface is, by our algorithm, 6-tunnel-free, and we give the mathematical proof of this property. For efficient computation, we employ the forward difference technique in homogeneous form in the implementation of the algorithm. For more general applications, we show that voxelisation of a NURBS surface can be realised by first converting it into a piecewise rational Bézier surface and then voxelising each of the rational Bézier surfaces. We indicate the advantages carrying out this procedure.     

Bézier曲线降阶的迭代算法

为提高Bézier曲线降阶的稳定性,提出以基于L_2范数的逼近误差为指导的一种迭代算法. 该算法从一条初始Bézier曲线开始逐渐地对其控制顶点进行偏移,得到具有误差最小的逼近曲线; 同时,应用线性搜索方法来优化控制顶点的偏移,使得在每次迭代后逼近误差可以达到局部最小. 实例结果表明了该算法的快速收敛性.     

Interpolating G Bézier surfaces over irregular curve networks for ship hull design

We propose a local method of constructing piecewise G¹ Bézier patches to span an irregular curve network, without modifying the given curves at odd- and 4-valent node points. Topologically irregular regions of the network are approximated by implicit surfaces, which are used to generate split curves, which subdivide the regions into triangular and/or rectangular sub-regions. The subdivided regions are then interpolated with Bézier patches. We analyze various singular cases of the G¹ condition that is to be met by the interpolation and propose a new G¹ continuity condition using linear and quartic scalar weight functions. Using this condition, a curve network can be interpolated without modification at 4-valent nodes with two collinear tangent vectors, even in the presence of singularities. We demonstrate our approach in a ship hull.     

Progressive iterative approximation for triangular Bézier surfaces

Recently, for the sake of fitting scattered data points, an important method based on the PIA (progressive iterative approximation) property of the univariate NTP (normalized totally positive) bases has been effectively adopted. We extend this property to the bivariate Bernstein basis over a triangle domain for constructing triangular Bézier surfaces, and prove that this good property is satisfied with the triangular Bernstein basis in the case of uniform parameters. Due to the particular advantages of triangular Bézier surfaces or rational triangular Bézier surfaces in CAD (computer aided design), it has wide application prospects in reverse engineering.     

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.