Related papers: On computing B\'ezier curves by Pascal matrix meth…
In this paper, we derive a family of fast and stable algorithms for multiplying and inverting $n \times n$ Pascal matrices that run in $O(n log^2 n)$ time and are closely related to De Casteljau's algorithm for B\'ezier curve evaluation.…
New geometric methods for fast evaluation of derivatives of polynomial and rational B\'{e}zier curves are proposed. They apply an algorithm for evaluating polynomial or rational B\'{e}zier curves, which was recently given by the authors.…
In this paper, a new method to compute a B\'ezier curve of degree n = 2m-1 is introduced, here formulated as a set of points whose coordinates are calculated from two Hankel forms in $\C^m$. From Vandermonde factorizations of the two…
It is well-known that a $d$-dimensional polynomial B\'{e}zier curve of degree $n$ can be subdivided into two segments using the famous de Casteljau algorithm in $O(dn^2)$ time. Can this problem be solved more efficiently? In this paper, we…
B\'ezier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of B\'ezier curves to the infinite-dimensional space of images. To this end the space of images is equipped with a…
This paper deals with the merging problem of segments of a composite B\'ezier 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.…
We present an efficient method to solve the problem of the constrained least squares approximation of the rational B\'{e}zier curve by the B\'{e}zier curve. The presented algorithm uses the dual constrained Bernstein basis polynomials,…
A new algorithm for computing a point on a polynomial or rational curve in B\'{e}zier form is proposed. The method has a geometric interpretation and uses only convex combinations of control points. The new algorithm's computational…
In this paper, a de Casteljau algorithm to compute (p,q)-Bernstein Bezier curves based on (p,q)-integers is introduced. We study the nature of degree elevation and degree reduction for (p,q)-Bezier Bernstein functions. The new curves have…
A new differential-recurrence relation for the B-spline functions of the same degree is proved. From this relation, a recursive method of computing the coefficients of B-spline functions of degree $m$ in the Bernstein-B\'{e}zier form is…
The approach to curve implicitization through Sylvester and Bezout resultant matrices and bivariate interpolation in the usual power basis is extended to the case of Bernstein-Bezoutian matrices constructed when the polynomials are given in…
B\'ezier splines are widely available in various systems with the curves and surface designs. In general, the B\'ezier spline can be specified with the B\'ezier curve segments and a B\'ezier curve segment can be fitted to any number of…
In the paper, we show that the transformations between modified Jacobi and Bernstein bases of the constrained space of polynomials of degree at most $n$ can be performed with the complexity $O(n^2)$. As a result, the algorithm of degree…
This paper deals with the problem of multi-degree reduction of a composite B\'ezier 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…
In this paper, we give an efficient algorithm of degree reduction of B\'ezier curves with box constraints. The idea is to combine the previous iterative approach, that has been presented recently in (P. Gospodarczyk, Comput. Aided Des. 62…
B\'ezier curves provide the basic building blocks of graphic design in 2D. In this paper, we port B\'ezier curves to manifolds. We support the interactive drawing and editing of B\'ezier splines on manifold meshes with millions of…
How to quickly and stably realize the degree reduction of the rational Bezier curve is an open problem in CAGD. Based on the weighted least squares method and weighted sum method of multi-objective optimization, this paper transforms the…
In this paper, we propose a linear method for $C^{(r,s)}$ approximation of rational B\'{e}zier curve with arbitrary degree polynomial curve. Based on weighted least-squares, the problem be converted to an approximation between two…
Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining B\'{e}zier form of the $L^2$-solution of the problem of best polynomial approximation of B\'{e}zier curve or surface. In this connection, the…
In this paper, we use the blending functions of Lupa\c{s} type (rational) $(p,q)$-Bernstein operators based on $(p,q)$-integers for construction of Lupa\c{s} $(p,q)$-B$\acute{e}$zier curves (rational curves) and surfaces (rational surfaces)…