Conversion formulas for simploidal Bernstein polynomials

Journal of Computational and Applied Mathematics, 1994

In this paper a new algorithm for generating values of Bernstein-Bezier polynomials is proposed and investigated. Opposite to the known algorithms, its computational complexity does not depend on the degree of a calculated polynomial. However, it allows to calculate only approximate values of the polynomial. These features allow to recommend the proposed algorithm for solving curve fitting and image processing problems in which processing a large number of data is necessary. The proposed method is based on a probabilistic interpretation of Bernstein-Bezier (BB-) polynomials and its properties are investigated in the statistical language. As a byproduct, a local nature of BB-polynomial approximation is displayed.

We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n. In addition, we create the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis and Bernstein polynomial basis

Graphics Gems, 1994

Journal of Approximation Theory, 2006

The evaluation of multivariate polynomials of n variables in Bernstein-Bézier form is considered. A forward error analysis for the corresponding de Casteljau algorithm and the VS algorithm is performed. We also include algorithms that simultaneously evaluate the polynomial and provide "a posteriori" error bounds, without increasing significantly the computational cost. The sharpness of our running error bounds is shown in the case of trivariate polynomials.

Computer Aided Geometric Design

A new variant of the blossom, the h-blossom, is introduced by altering the diagonal property of the standard blossom. The significance of the h-blossom is that the h-blossom satisfies a dual functional property for h-Bézier curves over arbitrary intervals. Using the h-blossom, several new identities involving the h-Bernstein bases are developed including an h-variant of Marsden's identity. In addition, for each h-Bézier curve of degree n, a collection of n! new, affine invariant, recursive evaluation algorithms are derived. Using two of these recursive evaluation algorithms, a recursive subdivision procedure for h-Bézier curves is constructed. Starting from the original control polygon of an h-Bézier curve, this subdivision procedure generates a sequence of control polygons that converges rapidly to the original h-Bézier curve.

arXiv (Cornell University), 2015

We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n. In addition, we create the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis and Bernstein polynomial basis.

Journal of Mathematical Analysis and Applications, 2007

Explicit formulae, in terms of Bernstein-Bézier coefficients, of the Fourier transform of bivariate polynomials on a triangle and univariate polynomials on an interval are derived in this paper. Examples are given and discussed to illustrate the general theory. Finally, this consideration is related to the study of refinement masks of spline function vectors.

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.

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.

Journal of Approximation Theory

We introduce a new variant of the blossom, the q-blossom, by altering the diagonal property of the standard blossom. This q-blossom is specifically adapted to developing identities and algorithms for q-Bernstein bases and q-Bézier curves over arbitrary intervals. By applying the q-blossom, we generate several new identities including an explicit formula representing the monomials in terms of the q-Bernstein basis functions and a q-variant of Marsden's identity. We also derive for each q-Bézier curve of degree n, a collection of n! new, affine invariant, recursive evaluation algorithms. Using two of these new recursive evaluation algorithms, we construct a recursive subdivision algorithm for q-Bézier curves.