Abedallah Rababah

TRANSFORMATION OF CHEBYSHEV-BERNSTEIN POLYNOMIAL BASIS

Abstract In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is remarkably well-conditioned, allowing one to combine the superior least-squares performance of Chebyshev polynomials with the geometrical insight of the Bernstein form. We also compare it to other basis transformations such as Bernstein-Hermite, power-Hermite, and Bernstein–Legendre basis transformations.

A simple matrix form for degree reduction of Bezier curves using Chebyshev-Bernstein basis transformations

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 Bezier curves with respect to the weighted L2-norm for the interval [0, 1], using the weight function w(x)=1/4x-4x2. The error of the degree reduction scheme is given, and the degree reduction with continuity conditions is also considered.

JACOBI-BERNSTEIN BASIS TRANSFORMATION

Abstract In this paper we derive the matrix of transformation of the Jacobi polynomial basis form into the Bernstein polynomial basis of the same degree n and vice versa. This enables us to combine the superior least-squares performance of the Jacobi polynomials with the geometrical insight of the Bernstein form. Application to the inversion of the Bézier curves is given.

Distance for degree raising and reduction of triangular Bézier surfaces

The problem of degree reduction and degree raising of triangular Bezier surfaces is considered. The L 2 and l 2 measures of distance combined with the least-squares method are used to get a formula for the Bezier points. The methods use the matrix representations of the degree reduction and degree raising.

High order approximation method for curves

Abstract In this paper, an approximation procedure for space curves is described, which significantly improves the standard approximation rate via parametric Taylors approximations. The method takes advantages of the freedom in the choice of the parametrization and yields the order (m + 1) + [ (m + 1) (2d - 1) ] for a curve in R d , where m is the degree of the approximating polynomial parametrization. Moreover, the optimal rate (m + 1)+[ (m - 1) (d - 1) ] , for a curve in R d , is also achieved for a particular set of curves. The cubic case is studied with examples which shows that our approximation method is an interesting quantity as well as quality improvement over standard methods.

Multiple Degree Reduction and Elevation of Bézier Curves Using Jacobi–Bernstein Basis Transformations

In this article, we find the optimal r times degree reduction of Bézier curves with respect to the Jacobi-weighted L 2-norm on the interval [0, 1]. This method describes a simple and efficient algorithm based on matrix computations. Also, our method includes many previous results for the best approximation with L 1, L 2, and L ∞-norms. We give some examples and figures to demonstrate these methods.

Taylor theorem for planar curves

We describe an approximation method for planar curves that significantly improves the standard rate obtained by local Taylor approximations. The method exploits the freedom in the choice of the parametrization and achieves the order 4m/3 where m is the degree of the approximating polynomial parametrization. Moreover, we show for a particular set of curves that the optimal rate 2m is possible

Weighted dual functions for Bernstein basis satisfying boundary constraints

In this paper, we consider the issue of dual functions for the Bernstein basis which satisfy boundary conditions. The Jacobi weight function with the usual inner product in the Hilbert space are used. Some examples of the transformation matrices are given. Some figures for the weighted dual functions of the Bernstein basis with respect to the Jacobi weight function satisfying boundary conditions are plotted. We discuss special cases of the Jacobi weight function as the Legendre weight function and the Chebyshev weight functions of the first, second, and third kinds.

Computing Derivatives of Jacobi Polynomials Using Bernstein Transformation and Differentiation Matrix

In this paper, we give a new, simple, and efficient method for evaluating the pth derivative of the Jacobi polynomial of degree n. The Jacobi polynomial is written in terms of the Bernstein basis, and then the pth derivative is obtained. The results are given in terms of both Bernstein basis of degree n − p and Jacobi basis form of degree n − p and presented in a matrix form. Numerical examples and comparisons with other well-known methods are presented.

The best uniform quadratic approximation of circular arcs with high accuracy

Abstract In this article, the issue of the best uniform approximation of circular arcs with parametrically defined polynomial curves is considered. The best uniform approximation of degree 2 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 4; the error function equioscillates five times; the approximation order is four. For θ = π/4 arcs (quarter of a circle), the uniform error is 5.5 × 10−3. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfy the properties of the best uniform approximation and yield the highest possible accuracy.