Convexity and Positivity of Bivariate Quadratic Bézier Surfaces over a Rectangle

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Visible region extraction from a sequence of rational Bézier surfaces

A method for computing the visible regions of free-form surfaces is proposed in this paper. Our work is focused on accurately calculating the visible regions of the sequenced rational Bézier surfaces forming a solid model and having coincident edges but no inner-intersection among them. The proposed method calculates the silhouettes of the surfaces without tessellating them into triangle meshes commonly used in previous methods so that arbitrary precision can be obtained. The computed silhouettes of visible surfaces are projected onto a plane orthogonal to the parallel light. Then their spatial relationship is applied to calculate the boundaries of mutual-occlusion regions. As the connectivity of the surfaces on the solid model is taken into account, a surface clustering technique is also employed and the mutual-occlusion calculation is accelerated. Experimental results showed that our method is efficient and robust, and can also handle complex shapes with arbitrary precision. Project supported by the National Basic Research Program (973) of China (No. 2002CB312106) and the National Natural Science Foundation of China (Nos. 60533070, and 60403047). The third author was supported by the project sponsored by a Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 200342) and a Program for New Century Excellent Talents in University (No. NCET-04-0088), China     

Bézier curves with shape parameter

In this paper, Bézier basis with shape parameter is constructed by an integral approach. Based on this basis, we define the Bézier curves with shape parameter. The Bézier basis curves with shape parameter have most properties of Bernstein basis and the Bézier curves. Moreover the shape parameter can adjust the curves' shape with the same control polygon. As the increase of the shape parameter, the Bézier curves with shape parameter approximate to the control polygon. In the last, the Bézier surface with shape parameter is also constructed and it has most properties of Bézier surface. Project supported by the National Natural Science Foundation of China (No. 10371110) and the National Basic Research Program (973) of China (No. G2002CB12101)     

A new algorithm for designing developable Bézier surfaces

A new algorithm is presented that generates developable Bézier surfaces through a Bézier curve called a directrix. The algorithm is based on differential geometry theory on necessary and sufficient conditions for a surface which is developable, and on degree evaluation formula for parameter curves and linear independence for Bernstein basis. No nonlinear characteristic equations have to be solved. Moreover the vertex for a cone and the edge of regression for a tangent surface can be obtained easily. Aumann’s algorithm for developable surfaces is a special case of this paper. Project supported by the National Basic Research Program (973) of China (No. 2004CB719400), the National Natural Science Foundation of China (Nos. 60373033 and 60333010) and the National Natural Science Foundation for Innovative Research Groups (No. 60021201), China     

PH-spline approximation for Bézier curve and rendering offset

In this paper, a G¹, C¹, C² PH-spline is employed as an approximation for a give Bézier curve within error bound and further renders offset which can be regarded as an approximate offset to the Bézier curve. The errors between PH-spline and the Bézier curve, the offset to PH-spline and the offset to the given Bézier curve are also estimated. A new algorithm for constructing offset to the Bézier curve is proposed. Project supported by the National Natural Foundation of China (No. 60073023) and the National Basic Research Program (973) of China (No. 2002CB312101)     

A new fractional rational Bézier curve

Based on rational Bézier curves given by Ron Goldman, a new fractional rational Bézier curve was first defined in terms of fractional Bernstein bases. Moreover, some basic properties were dicussed and a theorem connected to Poisson curves was obtained. Some examples in this paper were given by the visual results. Project supported by the National Natural Science Foundation of China (Grant No. 10271074)     

Rational offset approximation of rational Bézier curves

The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications. As for the reason that in most cases the offset curves do not preserve the same polynomial or rational polynomial representations, it arouses difficulty in applications. Thus approximation methods have been introduced to solve this problem. In this paper, it has been pointed out that the crux of offset curve approximation lies in the approximation of parametric speed. Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented. Project supported by the National Basic Research Program (973) of China (No. 2002CB312101) and the National Natural Science Foundation of China (Nos. 60373033 and 60333010)     

Bézier曲线等分作图算法的包络性分析

Bézier曲线是一种最重要且最简便的构造控制参数曲线方法,是计算机图形学的重要内容。Bézier曲线的等分作图算法是一种简便、计算量小的算法;同时Bézier曲线在一些实际应用中可由包络形成因而具有包络性。本文给出了二次三次及n次Bézier曲线等分图算法的包络性证明,证明了Bézier曲线的包络性,为理解Bézier曲线的等分图算法提供了新的方式。     

Computation of lower derivatives of rational triangular Bézier surfaces and their bounds estimation

By introducing the homogenous coordinates, degree elevation formulas and combinatorial identities, also by using multiplication of Bernstein polynomials and identity transformation on equations, this paper presents some explicit formulas of the derives some estimations of bound both on the direction and magnitude of the corresponding derivatives. All the results above have value not only in surface theory but also in practice.     

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.