In this paper, we use the blending functions of Lupaş type (rational) (p, q)-Bernstein operators based on (p, q)-integers for construction of Lupaş (p, q)-Bézier curves (rational curves) and surfaces (rational surfaces) with shape parameters. We study the nature of degree elevation and degree reduction for Lupaş (p, q)-Bézier Bernstein functions. Parametric curves are represented using Lupaş (p...

Let Pa(Z) = Z n + ∑n j=1 ajZ n−j be a C curve of monic polynomials, ai ∈ C(I,C) where I ⊂ R is an interval. We show that if k = k(n) is sufficiently large then any choice of continuous roots of Pa is locally absolutely continuous, in a uniform way with respect to maxj ‖aj‖Ck on compact subintervals. This solves a problem that was open for more then a decade and shows that some systems of pseudo...

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description...

This paper extends the modified affine arithmetic in matrix form method for bivariate polynomial evaluation and algebraic curve plotting in 2D to modified affine arithmetic in tensor form for trivariate polynomial evaluation and algebraic surface plotting in 3D. Experimental comparison shows that modified affine arithmetic in tensor form is not only more accurate but also much faster than affin...

This paper gives an algorithm for computing proper polynomial parametrizations for a particular class of curves. This class is characterized by the existence of a polynomial parametrization and by the absence of affine singularities. The algorithm requires O(n3 logn) field operations, where n is the degree of the curve.  2002 Elsevier Science B.V. All rights reserved.

Reconstructing smooth curve from 2D noise sampled data is a very important issue in many fields, such as data processing, reverse engineering, and so on. In this paper, we present a novel method to reconstruct smooth curve approximating sampled data in plane, at the same time filtering noise in the data. Our method is constructed on a set of local piecewise polynomial approximations associated ...

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description...

Traditional methods for algebraic manipulation of polynomials in Bernstein form try to obtain an explicit formula for each coefficient of the result of a given procedure, such us multiplication, arbitrarily high degree elevation, composition, or differentiation of rational functions. Whereas this strategy often furnishes involved expressions, these operations become trivial in terms of convolut...

We give a new self-contained proof of Bronshtein’s theorem, that any continuous root of a Cn−1,1-family of monic hyperbolic polynomials of degree n is locally Lipschitz, and obtain explicit bounds for the Lipschitz constant of the root in terms of the coefficients. As a by-product we reprove the recent result of Colombini, Orrú, and Pernazza, that a C-curve of hyperbolic polynomials of degree n...

In this paper, I describe the implementation details of some functional, triangular data fitting schemes. The schemes in question use derivative information to find initial settings of control points, giving a C, piecewise polynomial surface with a high degree of polynomial precision. The interior control points are then modified to increase the continuity between patches without decreasing the...