三角域上Said-Bézier型广义Ball基的对偶泛函及其应用

A family of Said-Bézier type generalized Ball(SBGB) bases and surfaces with a parameter H over triangular domain is introducermediate surfaces.To convert different bases and surfaces,the dual functionals of bases are presented.As an application of dual functionals,the subdivision formulas for surfaces are established.     

Minimal quasi-Bézier surface

The quasi-Bézier surface is a kind of commonly used surfaces in CAGD/CAD systems. In this paper, we study how to find the quasi-Bézier surface of minimal area among all the quasi-Bézier surfaces with prescribed borders, i.e., the Plateau-quasi-Bézier problem. The prescribed borders can not only be polynomial curves, but also catenaries and circular arcs. Moreover, the harmonic and biharmonic quasi-Bézier surfaces are investigated.     

The weighted dual functions for Wang-Bézier type generalized Ball bases and their applications

By introducing the inner-product matrix of two vector functions and using conversion matrix, explicit formulas for the dual basis functions of Wang-Bézier type generalized Ball bases (WBGB) with respect to the Jacobi weight function are given. The dual basis functions with and without boundary constraints are also considered. As a result, the paper includes the weighted dual basis functions of Bernstein basis, Wang-Ball basis and some intermediate bases. Dual functionals for WBGB and the least square approximation polynomials are also obtained.     

Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G=G(t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual functionals for the G-Bernstein basis functions and we use this dual functional property to prove that G-Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G-Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G-Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G-Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G-Bézier curve, and we derive the geometric rate of convergence of this algorithm.     

Degree reduction of SG-Bézier surfaces based on grey wolf optimizer

Aiming at the problem of approximate degree reduction of SG-Bézier surfaces, a method is proposed to achieve the degree reduction from (n × n) to (m × m) (m < n). Starting from the idea of grey wolf optimizer (GWO) algorithm and combining the geometric properties of SG-Bézier surfaces, this method transforms the degree reduction problem of SG-Bézier surfaces into an optimization problem. By choosing the fitness function, the degree reduction approximation of shape-adjustable SG-Bézier surfaces under unconstrained and angular interpolation constraints is realized. At the same time, some concrete examples of degree reduction and its errors are given. The results show that this method not only achieves good degree reduction effect but also is easy to implement and has high precision.     

Bézier form of dual bivariate Bernstein polynomials

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.     

Degenerations of rational Bézier surface with weights in the form of exponential function

Rational Bézier surface is a widely used surface fitting tool in CAD. When all the weights of a rational Bézier surface go to infinity in the form of power function, the limit of surface is the regular control surface induced by some lifting function, which is called toric degenerations of rational Bézier surfaces. In this paper, we study on the degenerations of the rational Bézier surface with weights in the exponential function and indicate the difference of our result and the work of Garc′?a-Puente et al. Through the transformation of weights in the form of exponential function and power function, the regular control surface of rational Bézier surface with weights in the exponential function is defined, which is just the limit of the surface.Compared with the power function, the exponential function approaches infinity faster, which leads to surface with the weights in the form of exponential function degenerates faster.     

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.     

A weak condition for the convexity of tensor-product Bézier and B-spline surfaces

A sufficient condition for a tensor-product Bézier surface to be convex is presented. The condition does not require that the control surface itself is convex, which is known to be a very restrictive property anyway. The convexity condition is generalised toC ¹ tensor-product B-spline surfaces.     

Dual bases for Wang-Bézier basis and their applications

This paper presents the dual bases for Wang-Bézier curves with a position parameter L, which include Bézier curve, Wang-Ball curve and some intermediate curves. The Marsden identity and the transformation formulas from Bézier curve to Wang-Bézier curve are also given. These results are useful for the application of Wang-Bézier curve and their popularization in Computer Aided Geometric Design.