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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. [ABSTRACT FROM AUTHOR]

We present a new approach to the problem of G-constrained ( k, l ≤ 3) multi-degree reduction of Bézier curves with respect to the least squares norm. First, to minimize the least squares error, we consider two methods of determining the values of geometric continuity parameters. One of them is based on quadratic and nonlinear programming, while the other uses some simplifying assumptions and solves a system of linear equations. Next, for prescribed values of these parameters, we obtain control points of the multi-degree reduced curve, using the properties of constrained dual Bernstein basis polynomials. Assuming that the input and output curves are of degree n and m, respectively, we determine these points with the complexity O( m n), which is significantly less than the cost of other known methods. Finally, we give several examples to demonstrate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the polynomial Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Abstract: 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 , which seems to be significantly less than complexity of most known algorithms. Examples are given, showing the effectiveness of the algorithm. [Copyright &y& Elsevier]

This paper deals with the merging problem of segments of a composite Bézier curve, with the endpoints continuity constraints. We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (Woźny and Lewanowicz, 2009) [12]to compute the control points of the merged curve. Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging methods. [ABSTRACT FROM AUTHOR]