Search

This paper deals with the problem of multi-degree reduction of a composite Bézier curve with the parametric continuity constraints at the endpoints of the segments. We present a novel method which is based on the idea of using constrained dual Bernstein polynomials to compute the control points of the reduced composite curve. In contrast to other methods, ours minimizes the L 2 -error for the whole composite curve instead of minimizing the L 2 -errors for each segment separately. As a result, an additional optimization is possible. Examples show that the new method gives much better results than multiple application of the degree reduction of a single Bézier curve. [ABSTRACT FROM AUTHOR]

Abstract: We give structure relations for the orthogonal polynomials in two variables, defined by Lewanowicz and Woźny [S. Lewanowicz, P. Woźny, J. Comput. Appl. Math. 233 (2010) 1554–1561] where , and are univariate big q-Jacobi polynomials. We discuss in full detail the case of the polynomials , which are closely related to Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, SIAM J. Algebra. Discr. Methods 1 (1980) 137–151]. [Copyright &y& Elsevier]

Abstract: Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed. [Copyright &y& Elsevier]

Abstract: We propose an efficient approach to the problem of multi-degree reduction of rectangular Bézier patches, with prescribed boundary control points. We observe that the solution can be given in terms of constrained bivariate dual Bernstein polynomials. The complexity of the method is with m ≔min(m 1, m 2), where (n 1, n 2) and (m 1, m 2) is the degree of the input and output Bézier surface, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined rectangular Bézier surfaces, the result is a composite surface of global C r continuity with a prescribed r ⩾0. In the detailed discussion, we restrict ourselves to r ∈{0,1}, which is the most important case in practical application. Some illustrative examples are given. [ABSTRACT FROM AUTHOR]

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]