Tensor Product q −Bernstein Bézier Patches

Computer Aided Geometric Design, 2009

A generalisation of Patterson's work, on the invariants of the rational Bézier paths, to the case of surface patches is presented. An equation for the determination of the invariants for patches of degree (n, n) is derived and solved for the bi-quadratics -for which it is shown that seven independent, invariant functions exist. Explicit forms of the invariants are derived and a number of applications are presented.

Applied Mathematics and Computation, 2006

We use the matrices of transformations between Chebyshev and Bernstein basis and the matrices of degree elevation and reduction of Chebyshev polynomials to present a simple and efficient method for r times degree elevation and optimal r times degree reduction of Bézier curves with respect to the weighted L 2 -norm for the interval [0, 1], using the weight function wðxÞ ¼ 1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4x À 4x 2 p . The error of the degree reduction scheme is given, and the degree reduction with continuity conditions is also considered.

Computer Graphics …, 2005

This paper introduces the rational forward difference operator for differential computation on a rational Bézier patch based on its control mesh. With this rational version of the forward difference operator, and by ignoring the appropriate dependent (on lower order derivatives) terms of various derivatives, the derivatives themselves have the same expressions as their polynomial counterparts; the curvature and torsion, and the first and second fundamental forms, all have very similar expressions as those equations from classical differential geometry. This new approach also allows straightforward generalization to higher dimensional rational Bézier patches, the special co-dimension 1 case of which is treated with the result of the first and second fundamental forms.

Journal of Mathematics

A computational model is presented to find the q -Bernstein quasi-minimal Bézier surfaces as the extremal of Dirichlet functional, and the Bézier surfaces are used quite frequently in the literature of computer science for computer graphics and the related disciplines. The recent work [1–5] on q -Bernstein–Bézier surfaces leads the way to the new generalizations of q -Bernstein polynomial Bézier surfaces for the related Plateau–Bézier problem. The q -Bernstein polynomial-based Plateau–Bézier problem is the minimal area surface amongst all the q -Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal condi...

Computer Aided Geometric Design, 2009

We present a novel approach to the problem of multi-degree reduction of Bézier curves with constraints, using the dual constrained Bernstein basis polynomials, associated with the Jacobi scalar product. We give properties of these polynomials, including the explicit orthogonal representations, and the degree elevation formula. We show that the coefficients of the latter formula can be expressed in terms of dual discrete Bernstein polynomials. This result plays a crucial role in the presented algorithm for multi-degree reduction of Bézier curves with constraints. If the input and output curves are of degree n and m, respectively, the complexity of the method is O (nm), which seems to be significantly less than complexity of most known algorithms. Examples are given, showing the effectiveness of the algorithm.

Journal of Computational and Applied Mathematics, 2008

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 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 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.

Numerical Algorithms, 2016

We propose a novel approach to the problem of polynomial approximation of rational Bézier triangular patches with prescribed boundary control points. The method is very efficient thanks to using recursive properties of the bivariate dual Bernstein polynomials and applying a smart algorithm for evaluating a collection of two-dimensional integrals. Some illustrative examples are given.

2010

We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is O(n 2 m 2), n and m being the degrees of the input and output Bézier surfaces, respectively. If the approximation-with appropriate boundary constraints-is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global C r continuity with a prescribed order r. Some illustrative examples are given.

Computer Aided Geometric Design, 2003

This note derives formulae for Gaussian and mean curvatures for tensor-product and triangular rational Bézier patches in terms of the respective control meshes. These formulae provide more geometric intuition than the generic formulae from differential geometry.

Applied Mathematics and Computation, 2013

This paper presents a new method of computation of Bézier curves of any order. This method is based on the Bernstein-Fourier representation of a Bézier curve and utilizes Fast Fourier Transforms to change from the Bernstein basis to a new one that provides efficient computation. For 2 6 n 6 8, where n is the number of control points, this method is still more rapid than the VS method, which is already very fast.