Bézier representation of the constrained dual Bernstein polynomials

Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed.     

Approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

An attractive method for approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces is introduced. The main result is that the arbitrary given order derived vectors of a polynomial triangular surface converge uniformly to those of the approximated rational triangular Bézier surface as the elevated degree tends to infinity. The polynomial triangular surface is constructed as follows. Firstly, we elevate the degree of the approximated rational triangular Bézier surface, then a polynomial triangular Bézier surface is produced, which has the same order and new control points of the degree-elevated rational surface. The approximation method has theoretical significance and application value: it solves two shortcomings-fussy expression and uninsured convergence of the approximation-of Hybrid algorithms for rational polynomial curves and surfaces approximation.     

An extension of the Bézier model

In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein-Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.     

PDE triangular Bézier surfaces: Harmonic, biharmonic and isotropic surfaces

We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a fourth-order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular Bézier surfaces.     

Iterative process for G-multi degree reduction of Bézier curves

In this paper, the issue of multi-degree reduction of Bézier curves with C¹ and G²-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to demonstrate the efficiency of this method.     

On the conditions for the coincidence of two cubic Bézier curves

In a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree.     

Approximating tensor product Bézier surfaces with tangent plane continuity

We present a simple method for degree reduction of tensor product Bézier surfaces with tangent plane continuity in L₂-norm. Continuity constraints at the four corners of surfaces are considered, so that the boundary curves preserve endpoints continuity of any order α. We obtain matrix representations for the control points of the degree reduced surfaces by the least-squares method. A simple optimization scheme that minimizes the perturbations of some related control points is proposed, and the surface patches after adjustment are C^∞ continuous in the interior and G¹ continuous at the common boundaries. We show that this scheme is applicable to surface patches defined on chessboard-like domains.     

A third order partial differential equation for isotropic boundary based triangular Bézier surface generation

We approach surface design by solving a linear third order Partial Differential Equation (PDE). We present an explicit polynomial solution method for triangular Bézier PDE surface generation characterized by a boundary configuration. The third order PDE comes from a symmetric operator defined here to overcome the anisotropy drawback of any operator over triangular Bézier surfaces.     

Conditions for coincidence of two cubic Bézier curves

This paper presents a necessary and sufficient condition for judging whether two cubic Bézier curves are coincident: two cubic Bézier curves whose control points are not collinear are coincident if and only if their corresponding control points are coincident or one curve is the reversal of the other curve. However, this is not true for degree higher than 3. This paper provides a set of counterexamples of degree 4.     

Improved bounds on the magnitude of the derivative of rational Bézier curves

In this paper we derive some new derivative bounds of rational Bézier curves according to some existing identities and inequalities. The comparison of the new bounds with some existing ones is also presented.     

Simultaneous approximation for Bézier variant of Szász-Mirakyan-Durrmeyer operators

We study the rate of convergence in simultaneous approximation for the Bézier variant of Szász-Mirakyan-Durrmeyer operators by using the decomposition technique of functions of bounded variation.     

Shape analysis of cubic trigonometric Bézier curves with a shape parameter

For the cubic trigonometric polynomial curves with a shape parameter (TB curves, for short), the effects of the shape parameter on the TB curve are made clear, the shape features of the TB curve are analyzed. The necessary and sufficient conditions are derived for these curves having single or double inflection points, a loop or a cusp, or be locally or globally convex. The results are summarized in a shape diagram of TB curves, which is useful when using TB curves for curve and surface modeling. Furthermore the influences of shape parameter on the shape diagram and the ability for adjusting the shape of the curve are shown by graph examples, respectively.     

Approximating conic sections by constrained Bézier curves of arbitrary degree

In this paper, an algorithm for approximating conic sections by constrained Bézier curves of arbitrary degree is proposed. First, using the eigenvalues of recurrence equations and the method of undetermined coefficients, some exact integral formulas for the product of two Bernstein basis functions and the denominator of rational quadratic form expressing conic section are given. Then, using the least squares method, a matrix-based representation of the control points of the optimal Bézier approximation curve is deduced. This algorithm yields an explicit, arbitrary-degree Bézier approximation of conic sections which has function value and derivatives at the endpoints that match the function value and the derivatives of the conic section up to second order and is optimal in the _L2$L_2$ norm. To reduce error, the method can be combined with a curve subdivision scheme. Computational examples are presented to validate the feasibility and effectiveness of the algorithm for a whole curve or its part generated by a subdivision.     

Approximate dynamic programming with Bézier Curves/Surfaces for Top-percentile Traffic Routing

Multi-homing is used by Internet Service Providers (ISPs) to connect to the Internet via different network providers. This study develops a routing strategy under multi-homing in the case where network providers charge ISPs according to top-percentile pricing (i.e. based on the θth highest volume of traffic shipped). We call this problem the Top-percentile Traffic Routing Problem (TpTRP).Solution approaches based on Stochastic Dynamic Programming require discretization in state space, which introduces a large number of state variables. This is known as the curse of dimensionality in state space. To overcome this, in previous work we have suggested to use approximate dynamic programming (ADP) to construct value function approximations, which allow us to work in continuous state space. The resulting ADP model provides well performing routing policies for medium sized instances of the TpTRP. In this work we extend the ADP model, by using Bézier Curves/Surfaces to obtain continuous-time approximations of the time-dependent ADP parameters. This modification reduces the number of regression parameters to estimate, and thus accelerates the efficiency of parameter training in the solution of the ADP model, which makes realistically sized TpTRP instances tractable. We argue that our routing strategy is near optimal by giving bounds.     

A note on iterative process for G-multi degree reduction of Bézier curves

In the paper [A. Rababah, S. Mann, Iterative process for G²-multi degree reduction of Bézier curves, Applied Mathematics and Computation 217 (2011) 8126-8133], Rababah and Mann proposed an iterative method for multi-degree reduction of Bézier curves with C¹ and G²-continuity at the endpoints. In this paper, we provide a theoretical proof for the existence of the unique solution in the first step of the iterative process, while the proof in their paper applies only in some special cases. Also, we give a complete convergence proof for the iterative method. We solve the problem by using convex quadratic optimization.     

Circle approximation using LN Bézier curves of even degree and its application

We present an approximation method of circular arcs using linear-normal (LN) Bézier curves of even degree, four and higher. Our method achieves G^m$G^m$ continuity for endpoint interpolation of a circular arc by a LN Bézier curve of degree 2m  , for m=2,3$m=2,3$. We also present the exact Hausdorff distance between the circular arc and the approximating LN Bézier curve. We show that the LN curve has an approximation order of 2m+2$2m+2$, for m=2,3$m=2,3$. Our approximation method can be applied to offset approximation, so obtaining a rational Bézier curve as an offset approximant. We derive an algorithm for offset approximation based on the LN circle approximation and illustrate our method with some numerical examples.     

On the rates of convergence of BBH-Kantorovich operators and their Bézier variant

In the present paper we consider the Bézier variant of BBH-Kantorovich operators J_n,αf for functions f measurable and locally bounded on the interval [0, ∞) with α ? 1. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of J_n,αf(x) at those x > 0 at which the one-sided limits f(x+), f(x−) exist. The very recent result of Chen and Zeng (2009) [L. Chen, X.M. Zeng, Rate of convergence of a new type Kantorovich variant of Bleimann-Butzer-Hahn Operators, J. Inequal. Appl. 2009 (2009) 10. Article ID 852897] is extended to more general classes of functions.     

三角域上带两个形状参数的Bézier曲面的扩展

给出了三角域上带双参数λ1,λ2的类三次Bernstein基函数,它是三角域上三次Bernstein基函数的扩展.分析了该组基的性质并定义了三角域上带有两个形状参数λ1,λ2的类三次Bernstein-Bézier(B-B)参数曲面.该基函数及参数曲面分别具有与三次Bernstein基函数及三次B-B参数曲面类似的性质.当λ1,λ2取特殊的值时,可分别得到三次Bernstein基函数及三次B-B参数曲面以及参考文献中所定义的类三次Bernstein基函数及类三次B-B参数曲面.由实例可知,通过改变形状参数的取值,可以调整曲面的形状.     

A novel generalization of Bézier curve and surface

A new formulation for the representation and designing of curves and surfaces is presented. It is a novel generalization of Bézier curves and surfaces. Firstly, a class of polynomial basis functions with n$n$ adjustable shape parameters is present. It is a natural extension to classical Bernstein basis functions. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n$n$ shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon. It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. Some examples are given by figures.