Improvement of the termination criterion for subdivision of the rational Bézier curves

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Rational offset approximation of rational Bézier curves

INTRODUCTION Offset curves/surfaces, also called parallel curves/surfaces, are defined as the locus of the points which are at constant distance along the normal from the generator curves/surfaces. As for a planar gen- erator curve Γ:C(t)=(x(t),y(t)), the parametric speed and its norm σ(t) are defined respectively as (Farouki, 1992) C ′( t ) =( x ′( t ), y ′(t )),σ (t ) = x ′ 2 (t ) y ′2(t ). (1) Subsequently the offset curve of the generator curve, which is at constant distanc…     

Constrained multi-degree reduction of rational Bézier curves using reparameterization

Applying homogeneous coordinates, we extend a newly appeared algorithm of best constrained multi-degree reduc- tion for polynomial Bézier curves to the algorithms of constrained multi-degree reduction for rational Bézier curves. The idea is introducing two criteria, variance criterion and ratio criterion, for reparameterization of rational Bézier curves, which are used to make uniform the weights of the rational Bézier curves as accordant as possible, and then do multi-degree reduction for each component in homogeneous coordinates. Compared with the two traditional algorithms of "cancelling the best linear common divisor" and "shifted Chebyshev polynomial", the two new algorithms presented here using reparameterization have advantages of simplicity and fast computing, being able to preserve high degrees continuity at the end points of the curves, do multi-degree reduction at one time, and have good approximating effect.     

Bézier curves with shape parameter

INTRODUCTION The Bézier curves and surfaces form a basic toofor constructing free form curves and surfaces. Manbasis-like Bézier basis are presented. Said (1989) anGoodman and Said (1991) constructed the Ball basisMainar et al.(2001) found some bases for the space{1, t, cost, sint, cos2t, sin2t}, {1, t, t2, cost, sint}, an{1, t, cost, sint, tcost, tsint}. Chen and Wang (2003gave the C-Bézier basis in the space {1, t, t2, …, tn?2sint, cost}. Wang and Wang (2004) put forwarUniform…     

Visible region extraction from a sequence of rational Bézier surfaces

INTRODUCTION Computing the visible portions of parameteriz surfaces is a fundamental problem for Comput Graphics, Computer Aided Design, Computer Visio and so on. Generally, a common solution to getti the visible region of the parameterized surfaces is use discrete methods including Z-Buffer (Greene al., 1993), Hierarchical Occlusion Map (HOM (Zhang et al., 1997), Ray-Tracing (Toth, 1985), an tessellating the surface with its u and v isoparametr curves (Maghrabi and Griffiths, 198…     

A quadratic programming method for optimal degree reduction of Bézier curves with G~1-continuity

This paper presents a quadratic programming method for optimal multi-degree reduction of Bézier curves with G1-continuity. The L2 and l2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applica- tions of degree reduction of Bézier curves with their parameterizations close to arc-length parameterizations are also discussed.     

Optimal approximate merging of a pair of Bézier curves with G2-continuity

We present a novel approach for dealing with optimal approximate merging of two adjacent Bezier eurves with G^2-continuity. Instead of moving the control points, we minimize the distance between the original curves and the merged curve by taking advantage of matrix representation of Bezier curve＇s discrete structure, where the approximation error is measured by L2-norm. We use geometric information about the curves to generate the merged curve, and the approximation error is smaller. We can obtain control points of the merged curve regardless of the degrees of the two original curves. We also discuss the merged curve with point constraints. Numerical examples are provided to demonstrate the effectiveness of our algorithms.     

Optimal approximate merging of a pair of Bézier curves with G2-continuity

We present a novel approach for dealing with optimal approximate merging of two adjacent Bezier curves with G2-continuity. Instead of moving the control points, we minimize the distance between the original curves and the merged curve by taking advantage of matrix representation of Bezier curve's discrete structure, where the approximation error is measured by L2-norm. We use geometric information about the curves to generate the merged curve, and the approximation error is smaller. We can obtain control points of the merged curve regardless of the degrees of the two original curves. We also discuss the merged curve with point constraints. Numerical examples are provided to demonstrate the effectiveness of our algorithms.     

Designing Bézier surfaces minimizing the L2-norm of the Gaussian curvature

In freeform surface modelling, developable surfaces have much application value. But, in 3D space, there is not always a regular developable surface which interpolates the given boundary of an arbitrary piecewise smooth closed curve. In this paper, tensor product Bézier surfaces interpolating the closed curves are determined and the resulting surface is a minimum of the functional defined by the L2-integral norm of the Gaussian curvature. The Gaussian curvature of the surfaces is minimized by the method of solving nonlinear optimization problems. An improved approach trust-region form method is proposed. A simple application example is also given.     

Optimal constrained multi-degree reduction of Bézier curves with explicit expressions based on divide and conquer

We decompose the problem of the optimal multi-degree reduction of Bezier curves with comers constraint into two simpler subproblems, namely making high order interpolations at the two endpoints without degree reduction, and doing optimal degree reduction without making high order interpolations at the two endpoints. Further, we convert the second subproblem into multi-degree reduction of Jacobi polynomials. Then, we can easily derive the optimal solution using orthonormality of Jacobi polynomials and the least square method of unequally accurate measurement. This method of ＇divide and conquer＇ has several advantages including maintaining high continuity at the two endpoints of the curve, doing multi-degree reduction only once, using explicit approximation expressions, estimating error in advance, low time cost, and high precision. More importantly, it is not only deduced simply and directly, but also can be easily extended to the degree reduction of surfaces. Finally, we present two examples to demonstrate the effectiveness of our algorithm.     

Designing Bézier surfaces minimizing the L~2-norm of the Gaussian curvature

INTRODUCTION Struik (1988) stated that the Gaussian curvatureof a developable surface is zero, and vice versa. If apiecewise smooth closed curve is given as a boundarycurve, there is not always a regular developable sur-face interpolating the boundary in theory, as DoCarmo (1976) proposed. There is still not a simpleand effective method for constructing a developablesurface in Computer Graphics (CG) and ComputerAided Geometric Design (CAGD). But, a Béziersurface which has a minimu…     

A new algorithm for designing developable Bézier surfaces

INTRODUCTION A ruled surface is a curved surface which can be generated by the continuous motion of a straight line in space along a space curve called a directrix (Chen et al., 2001; Zheng and Sederberg, 2001). This straight line is called a generator, or ruling, of the surface. A developable surface is a special ruled sur- face which has the same tangent plane at all points along a generator, or to which the tangent planes along a ruling coincide. A developable surface is also the enve…     

PH-spline approximation for Bézier curve and rendering offset

INTRODUCTIONOffsetsareusedinmanyindustrialapplica-tions,suchastoolpathsinnumerical-control(NC)machining,planningpathsformobilerobotsandinCAD/CAMfields.TheparametricrepresentationofcurveinCAGDisbasedonemployingpolyno-mialorrationalfunction.Planecurveanditsoff-setsareusuallydefinedbyparametricformssuchasr(t)=(x(t),y(t));offsetsared()()()ttdt=眗rn,n(t)isnormalvectorofr(t),disdistancealongn(t).Butthegenerationofoffsetcurvesisnotasimpletaskbecausen(t)ingeneralhasnorationalexpression.Sofar…     

Constrained multi-degree reduction of rational Bezier curves using reparametenzation

Applying homogeneous coordinates, we extend a newly appeared algorithm of best constrained multi-degree reduction for polynomial Bezier curves to the algorithms of constrained multi-degree reduction for rational Bezier curves. The idea is introducing two criteria, variance criterion and ratio criterion, for reparameterization of rational Bezier curves, which are used to make uniform the weights of the rational Bezier curves as accordant as possible, and then do multi-degree reduction for each component in homogeneous coordinates. Compared with the two traditional algorithms of ＂cancelling the best linear common divisor＂ and ＂shifted Chebyshev polynomial＂, the two new algorithms presented here using reparameterization have advantages of simplicity and fast computing, being able to preserve high degrees continuity at the end points of the curves, do multi-degree reduction at one time, and have good approximating effect.     

Bézier曲线的光滑拼接

Bézier曲线是计算机图形学研究的主要内容.曲线的拼接是曲线曲面造型中的关键技术之一.基于Bézier曲线的拼接原理,在Visual C＋＋6.0环境下开发Bézier曲线的拼接程序,将曲线拼接在界面中动态实现.     

A New Fractional Rational Bézier Curve

1Introduction ArationalB啨ziercurveistypicallydefinedby Q(t)=∑kωkPkBnk(t)∑kωkBnk(t),0≤t≤1,(1)where{Pk}isacollectionofcontrolpointsinRmand{ωk}isasetofscalarweights.RonGoldman[2]definedarationalB啨ziercurveintermsoftherational Bernsteinbasisfunctions,thatis,bysetting R(t)=∑kPkB-nk(t),-∞相似文献   

Bézier曲线的升阶及程序实现

基于Bézier曲线的升阶算法,探讨了程序开发的关键技术,并在Visual C++6.0环境下开发Bézier曲线的升阶程序,曲线升阶在界面中可动态实现.     

椭圆曲线的Bézier多项式逼近

椭圆曲线是计算机辅助几何设计中基本且重要的曲线.本文首先利用Tchebyshev多项式去逼近椭圆,再在此基础上得到插值椭圆首、末端点的n次Bézier多项式逼近.该算法可以逼近整椭圆,而且适合圆的逼近.     

有理Bézier曲线降阶综述

有理Bézier曲线的降阶是样条曲线和曲面造型中的关键技术之一,为了实现不同CAD系统之间的数据交换,都要用到这一技术,因此它已经成为该领域的热点问题.本文结合作者在该领域的研究成果,综述了近年来国内外专家学者关于有理Bezier曲线的降阶逼近研究的方法、理论成果及实际应用情况,对各种不同的方法进行了分析比较.     

一类类Bézier曲线及其扩展

该文研究一类类Bezier曲线及其扩展。主要根据一组三角基函数构造由四个控制顶点组成的类Bezier曲线,随后引入基函数中的形状控制参数得到带形状参数的类Bezier曲线,进而构造了带形状参数的类Bezier插值曲线,这是基于通过混合函数混合参数化的控制多边形及类Bezier曲线得到的一类曲线。它具有局部性、端点性质、几何不变性与仿射不变性、对称性、凸包性、变差缩减性等性质。同时它可以表示Bezier曲线所不能表示的圆锥曲线,最重要的性质是其具有插值性,这样就大大增强了其在实际领域中得到应用的可能性。