arXiv subjects

Stanisław Lewanowicz

Publications and source records attributed to Stanisław Lewanowicz.

7 recordsLinked to original sources

arXiv2016

Bézier form of dual bivariate Bernstein polynomials

Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the $L^2$-solution of the problem of best polynomial approximation of Bézier curve or surface. In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost. In this paper, a set of recurrence relations satisfied by the Bézier coefficients of dual bivariate Bernstein polynomials is derived and an efficient algorithm for evaluation of these coefficients is proposed. Applications of this result to some approximation problems of Computer Aided Geometric Design (CAGD) are discussed.

math.NA↗

arXiv2016

Degree reduction of composite Bézier curves

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.

cs.GR↗

arXiv2015

Constrained approximation of rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

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.

math.NA↗

arXiv2015

Efficient merging of multiple segments of Bézier curves

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 (P. Woźny, S. Lewanowicz, Comput. Aided Geom. Design 26 (2009), 566--579) 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.

math.NA↗

arXiv2015

$G^{k,l}$-constrained multi-degree reduction of Bézier curves

We present a new approach to the problem of $G^{k,l}$-constrained ($k,l \leq 3$) 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 $O(mn)$, which is significantly less than the cost of other known methods. Finally, we give several examples to demonstrate the effectiveness of our algorithms.

cs.GR↗

arXiv2015

Weighted polynomial approximation of rational Bézier curves

We present an efficient method to solve the problem of the constrained least squares approximation of the rational Bézier curve by the Bézier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials, associated with the Jacobi scalar product, and exploits their recursive properties. Examples are given, showing the effectiveness of the algorithm.

math.NA↗

arXiv2011

Bézier representation of the constrained dual Bernstein polynomials

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.

math.NA↗