Construction of Generalized Hybrid Trigonometric Bézier Surfaces with Shape Parameters and Their Applications

Journal of Mathematical Imaging and Vision - To tackle the problem of modeling and shape designing of complex engineering surfaces, the continuity constraints between generalized hybrid...     

A Subdivision Scheme for Rational Triangular Bézier Surfaces

1IntroductionSubdivisionalgorithmsforB6ziersurfacesareimportantinCADandcomputergraPhics.ThiangularB6ziersurfaceshavebeenstudiedextensivelyI1'3],andsomesubdivisionschemeshavebeendevelopedtosplittheoriginaltriangularpatchintoanumberofsmalltriangularpatchesI3-5].Asstatedby[7],itisusuaJlyavoidedtoincludetriangularpatchesinexistingornewCADsystems,whichuserectangularpatches.ItiswelLknownthattrian-gularpatcheshaveafundamentallydifferentgeometricstructurefromrectangularpatchesl1].Tliereforetheyr…     

A Subdivision Scheme for Rational Triangular Bézier Surfaces

An explicit formula is developed to decompose a rational triangular Bezier patch into three non-degenerate rational rectangular Bezier patches of the same degree.This formula yields a stable algorithm to compute the control vertices of those three rectangular subpatches.Some properties of the subdivision are discussed and the formula is illustrated with an example.     

Some remarks on Bézier curves

Parametric polynomial curves in Bézier-Bernstein representation are considered as prohections of rational norm curves of degree n in n-space; from this point of view the singularities of a planar Bézier cubic are determined and expressed by its affine invariants. Secondly, for an arbitrary pair of adjacent parametric curves in homogeneous coordinates, the general conditions for geometric continuity of any order k, G^k, are established.This result generalizes the corresponding conditions in the non-homogeneous (affine) case, recently obtained by [Goodman '84]. Some applications are given for Bézier curves. In particular, for γ-splines [Boehm '85], the existence of a global rational parameter that makes it to a C² parametric curve is shown. Furthermore, for two adjacent rational Bézier curves the complete conditions for G³ are stated using the projective properties of the control points only.     

Parallel Computation of Gröbner Bases on Distributed Memory Machines

This paper reports our work on parallelizing an algorithm computing Gröbner bases on a distributed memory parallel machine. When computing Gröbner bases, the efficiency of computation is dominated by the total number of S-polynomials. To decrease the total number of S-polynomials it is necessary to apply a selection strategy that selects the minimum polynomial as a new element of an intermediate base.On a distributed memory parallel machine, as opposed to a shared memory parallel machine, we have to take into account non-trivial communication costs between processors. To reduce such communication costs, it is better to employ coarse grained parallelism rather than fine grained parallelism.We adopt a manager-worker model. S-polynomials are reduced in worker processes in parallel, and the minimum polynomial is selected in the manager process. To implement the selection strategy in this parallel model, synchronization between worker processes is required for every selection of a new element of the intermediate base. However, in spite of synchronization, introducing the selection strategy produces not only a better absolute computation speed but also better speedup with multi-processors. We achieved about 8 times speedup with 64 processors for large problems, T-6 and Ex-17.     

Self-intersections of rational Bézier curves

Rational Bézier curves provide a curve fitting tool and are widely used in Computer Aided Geometric Design, Computer Aided Design and Geometric Modeling. The injectivity (one-to-one property) of rational Bézier curve as a mapping function is equivalent to the curve without self-intersections. We present a geometric condition on the control polygon which is equivalent to the injectivity of rational Bézier curve with this control polygon for all possible choices of weights. The proof is based on the degree elevation and toric degeneration of rational Bézier curve.     

Orthogonalization of unified and extended Bézier basis and its transformation matrix

UE-Bézier (unified and extended Bézier) basis is the unified form of Bézier-like bases,including polynomial Bézier basis,trigonometric polynomial and hyperbolic polynomial Bézier basis.Similar to the original Bézier-like bases,UE-Bézier basis functions are not orthogonal.In this paper,a group of orthogonal basis is constructed based on UE-Bézier basis.The transformation matrices between UE-Bézier basis and the proposed orthogonal basis are also solved.     

A note on degenerate triangular Bézier patches

In (Neamtu and Pfluger, 1994) degenerate Bézier patches were used to interpolate irregularly structured spatial data. This very interesting approach is based on the observation that singular parametrizations are no fundamental obstacle to modeling smooth spline surfaces. However, the smoothness conditions on degenerate Bézier patches provided in (Neamtu and Pfluger, 1994) are inaccurate. Moreover, proving tangent plane continuity is not sufficient to guarantee smoothness in the sense of differential geometry. What has to be shown is regularity, i.e. the existence of a regular smooth re-parametrization representing a degenerate patch locally near the singular point. Both correct sufficient smoothness conditions and the proof of regularity are presented in this paper.     

Bézier representation of geometrically continuous splines

As an intrinsic measure of smoothness,geometric continuity is an important problem in the fields of computer aided geometric design.It can afford more degrees of freedom for manipulating the shape of curve.However,piecewise polynomial functions of geometrically continuous splines are difficult to be constructed.In this paper,the conversion matrix between geometrically continuous spline basis functions and Bézier representation is analyzed.Based on this,construction of arbitrary degree geometrically continuous spline basis functions can be translated into a solution of linear system of equations.The original construction of geometrically continuous spline is simplified.     

Bilinear precision of rational Bézier surfaces

This short note proves the bilinear precision property of rational Bézier surfaces and gives a simple algorithm to compute the weights. Linear precision of rational Bézier surfaces is also discussed.     

Partial derivatives of rational Bézier surfaces

A rational surface is the locus of a rational curve that is moving through space and thereby changing its shape by changing its control points and weights. This intuitive definition can be used to derive hodographs of rational Bézier surfaces and their bounds of magnitude.     

Construction of Bézier rectangles and triangles on the symmetric Dupin horn cyclide by means of inversion

Dupin cyclides may be obtained as offsets of a special Dupin cyclide, the so-called symmetric Dupin horn cyclide. A novel approach based on the concept of inversion is presented for generating rational Bézier patches on the symmetric Dupin horn cyclide. This leads to a new formulation for rational rectangular biquadratic cyclide Bézier patches, and to a rational Bézier representation of triangular patches of degree 4 on the symmetric Dupin horn cyclide.     

Geometric construction of energy-minimizing Bézier curves

Modeling energy-minimizing curves have many applications and are a basic problem of Geometric Modeling.In this paper,we propose the method for geometric design of energy-minimizing B′ezier curves.Firstly,the necessary and sufficient condition on the control points for B′ezier curves to have minimal internal energy is derived.Based on this condition,we propose the geometric constructions of three kinds of B′ezier curves with minimal internal energy including stretch energy,strain energy and jerk energy.Given...     

广义Bézier曲线

为了有效地改进Bézier曲线的形状,给出了带局部形状参数的广义Bézier曲线,该曲线的表示式以一种函数的高阶逼近式为依据.通过对目标导矢和目标二阶导矢的系数的调整,生成满意的多项式曲线.所给曲线以Bézier曲线为特殊情形,能对较高次的B啨zier曲线进行有效地修改,也能方便地进行曲线段的拼接.     

Assembly variation analysis of flexible curved surfaces based on Bézier curves

Assembly variation analysis of parts that have flexible curved surfaces is much more difficult than that of solid bodies, because of structural deformations in the assembly process. Most of the current variation analysis methods either neglect the relationships among feature points on part surfaces or regard the distribution of all feature points as the same. In this study, the problem of flexible curved surface assembly is simplified to the matching of side lines. A methodology based on Bézier curves is proposed to represent the side lines of surfaces. It solves the variation analysis problem of flexible curved surface assembly when considering surface continuity through the relations between control points and data points. The deviations of feature points on side lines are obtained through control point distribution and are then regarded as inputs in commercial finite element analysis software to calculate the final product deformations. Finally, the proposed method is illustrated in two cases of antenna surface assembly.     

插值有理Bézier渐近四边形的有理Bézier曲面

为构造封闭的曲线为有理Bézier曲面的边界渐近线,给出封闭四边曲线为渐近四边形的条件,并提出插值该四边形的曲面构造方法.首先在给定角点数据的前提下构造优化的n次有理Bézier渐近四边形;然后利用该四边形和曲面在四边形上的切矢确定曲面沿边界的两排控制顶点和权;最后极小化曲面薄板能量函数确定剩余自由的控制顶点,进而构造出光滑的双5n–7次有理Bézier插值曲面.实例展示边界曲线为有理3,4,5次时曲面的构造结果,以及边界曲线含有直线或者拐点的情况,表明该方法是可行的.     

Corner cutting curves and a new characterization of Bézier and B-spline curves

We call a curve c corner cutting, if each single point of c can be received from a given polygon by a finite corner cutting process. We prove some results about these curves, especially about interpolation and contact. The obtained results will give a new geometric characterization of Bézier and B-spline curves.     

AHT Bézier Curves and NUAHT B-Spline Curves

In this paper,we present two new unified mathematics models of conics and polynomial curves,called algebraic hyperbolic trigonometric(AHT)Bézier curves and non-uniform algebraic hyperbolic trigonometric(NUAHT)B-spline curves of order n,which are generated over the space span{sin t,cos t,sinh t,cosh t,1,t,...,t~(n-5)},n≥5.The two kinds of curves share most of the properties as those of the Bézier curves and B-spline curves in polynomial space.In particular,they can represent exactly some remarkable transcendental curves such as the helix,the cycloid and the catenary.The subdivision formulae of these new kinds of curves are also given.The generations of the tensor product surfaces are straightforward. Using the new mathematics models,we present the control mesh representations of two classes of minimal surfaces.