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1.
Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than π. However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as being able to represent conics up to 2π (but not including 2π) without resorting to negative weights and possessing better parameterization. Therefore, it is actually worth studying the necessary and sufficient conditions for the rational quartic Bézier representation of conics. This paper attributes the rational quartic conic sections to two special kinds, that is, degree-reducible and improperly parameterized; on this basis, the necessary and sufficient conditions for the rational quartic Bézier representation of conics are derived. They are divided into two parts: Bézier control points and weights. These conditions can be used to judge whether a rational quartic Bézier curve is a conic section; or for a given conic section, present positions of the control points and values of the weights of the conic section in form of a rational quartic Bézier curve. Many examples are given to show the use of our results. 相似文献
2.
A new method for approximation of conic section by quartic B′ezier curve is presented, based on the quartic B′ezier approximation of circular arcs. Here we give an upper bound of the Hausdorff distance between the conic section and the approximation curve, and show that the error bounds have the approximation order of eight. Furthermore, our method yields quartic G2 continuous spline approximation of conic section when using the subdivision scheme,and the effectiveness of this method is demonstrated by some numerical examples. 相似文献
3.
HU Qianqian & WANG Guojin Department of Mathematics Zhejiang University Hangzhou China State Key Laboratory of CAD&CG Zhejiang University Hangzhou China 《中国科学A辑(英文版)》2005,48(9):1209-1222
Using algebraic and geometric methods,functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve.That is,the inverse mappings of the mappings represented by the expressions of rational conic segments are given.These formulae relate some triangular areas or some angles,determined by the selected point on the curve and the control points of the curve,as well as by the weights of the rational Bézier curve.Also,the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment,as well as by the weights of the rational Bézier curve.These results are greatly useful for optimal parametrization,reparametrization,etc.,of rational Bézier curves and surfaces. 相似文献
4.
An algorithmic approach to degree reduction of rational Bézier curves is presented. The algorithms are based on the degree reduction of polynomial Bézier curves. The method is introduced with the following steps: (a) convert the rational Bézier curve to polynomial Bézier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bézier curve, (c) determine weights of degree reduced curve, (d) convert the Bézier curve obtained through step (b) to rational Bézier curve with weights in step (c). 相似文献
5.
This paper is concerned with a generalization of Bernstein–Bézier curves. A one parameter family of rational Bernstein–Bézier
curves is introduced based on a de Casteljau type algorithm. A subdivision procedure is discussed, and matrix representation
and degree elevation formulas are obtained. We also represent conic sections using rational q-Bernstein–Bézier curves.
AMS subject classification (2000) 65D17 相似文献
6.
We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier
curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits
their recursive properties. Examples are given, showing the effectiveness of the algorithm. 相似文献
7.
An algorithmic approach to degree elevation of NURBS curves is presented. The new algorithms are based on the weighted blossoming process and its matrix representation. The elevation method is introduced that consists of the following steps: (a) decompose the NURBS curve into piecewise rational Bézier curves, (b) elevate the degree of each rational Bézier piece, and (c) compose the piecewise rational Bézier curves into NURBS curve. 相似文献
8.
Huang youdu 《大学数学》1998,(4)
本文给出了一种三次Bézier曲线的生成算法,在曲线的逐点生成过程中,只用到加减法,故效率极高.而且,此方法可推广到一般多项式或有理参数曲线 相似文献
9.
Xinqiang Qin Gang Hu Yang Yang Guo Wei 《Mathematical Methods in the Applied Sciences》2016,39(6):1336-1348
Quartic C‐Bézier curves possess similar properties with the traditional Bézier curves including terminal property, convex hull property, affine invariance, and approaching the shape of their control polygons as the shape parameter α decreases. In this paper, by adjusting the shape parameter α on the basis of the utilization of the least square approximation and nonlinear functional minimization together with fairing of a quartic C‐Bézier curve with G1 continuity of quartic C‐Bézier curve segments, we develop a fairing and G1 continuity algorithm for any given stitching coefficients λk(k = 1,2,…,n ? 1). The shape parameters αi(i=1, 2, …, n) can be adjusted by the value of control points. The curvature of the resulting quartic C‐Bézier curve segments after fairing is more uniform than before. Moreover, six examples are provided in the paper to demonstrate the efficacy of the algorithm and illustrate how to apply this algorithm to the computer‐aided design/computer‐aided manufacturing modeling systems. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
10.
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 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. 相似文献
11.
This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis. 相似文献
12.
Stanisław Lewanowicz Paweł Keller Paweł Woźny 《Advances in Computational Mathematics》2017,43(4):777-793
Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the L 2-solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed. 相似文献
13.
The existing results of curve degree elevation mainly focus on the degree of algebraic polynomials. The paper considers the elevation of degree of the trigonometric polynomial, from a Bzier curve on the algebraic polynomial space, to a C-B′ezier curve on the algebraic and trigonometric polynomial space. The matrix of degree elevation is obtained by an operator presentation and a derivation pyramid. It possesses not a recursive presentation but a direct expression. The degree elevation process can also be represented as a corner cutting form. 相似文献
14.
两类新的广义Ball曲线 总被引:14,自引:1,他引:13
本文提出两类新的广义Ball曲线,一类介于Wang(王国瑾)和Said广义Ball曲线之间,另一类介于Bezier和Said曲线之间,同时给出有关的升阶公式、递推算法及转化成Bezier形式的系数公式。 相似文献
15.
目前多项式 Bézier曲线的逼近合并问题已研究得比较深入 ,而有理 Bézier情形主要还是通过两类多项式 h和 H来降阶逼近 ,但是在工业制造中有重要意义的有理 Bézier曲线的合并问题一直缺乏研究 .本文通过控制点的优化扰动将两连续的满足权约束条件的有理 Bézier曲线转化成新的两有理Bézier曲线 ,使它们符合精确合并条件 ;并将合并得到的同阶有理 Bézier曲线看成是原两曲线的有理逼近 相似文献
16.
17.
Hans-Peter Seidel 《Constructive Approximation》1991,7(1):257-279
Using the concept of a symmetric recursive algorithm, we construct a new patch representation for bivariate polynomials: the B-patch. B-patches share many properties with B-spline segments: they are characterized by their control points and by a three-parameter family of knots. If the knots in each family coincide, we obtain the Bézier representation of a bivariate polynomial over a triangle. Therefore B-patches are a generalization of Bézier patches. B-patches have a de Boor-like evaluation algorithm, and, as in the case of B-spline curves, the control points of a B-patch can be expressed by simply inserting a sequence of knots into the corresponding polar form. In particular, this implies linear independence of the blending functions. B-patches can be joined smoothly and they have an algorithm for knot insertion that is completely similar to Boehm's algorithm for curves. 相似文献
18.
This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain. 相似文献
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20.
《Journal of Computational and Applied Mathematics》2012,236(6):1675-1677
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. 相似文献