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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]

Abstract: 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 , and 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 continuity with a prescribed order . Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

Abstract: A four-parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type. The polynomials, which are expressed in terms of univariate big q-Jacobi polynomials, form an extension of Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, Orthogonal polynomials in two variables of -Hahn and -Jacobi type, SIAM J. Algebr. Discrete Methods 1 (1980) 137–151]. We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial q-difference equation. [Copyright &y& Elsevier]

Abstract: Connection coefficients between the two-variable Bernstein and Jacobi polynomial families on the triangle are given explicitly as evaluations of two-variable Hahn polynomials. Dual two-variable Bernstein polynomials are introduced. Explicit formula in terms of two-variable Jacobi polynomials and a recurrence relation are given. [Copyright &y& Elsevier]