An upper estimate for the Alexander polynomial of an algebraic curve is obtained, which sharpens Libgober’s estimate in terms of the local polynomials at the singular points of the curve: only those singular points may contribute to the Alexander polynomial of the curve which are in excess of the hypothesis of Nori’s vanishing theorem.

A topic which has been of interest since the XIX century is the study of the parabolic curve of smooth surfaces in real three-dimensional space, as shown in the works of Gauss, Darboux, Salmon 1 , Kergosien and Thom 2 , Arnold 3 , among others. The parabolic curve of the graph of a smooth function, f : R2 → R, is the set { p, f p ∈ R2 × R: Hessf p 0}, where Hess f : fxxfyy − f2 xy. In this case...

Rational curves and splines are one of the building blocks of computer graphics and geometric modeling. Although a rational curve is more exible than its polynomial counterpart , many properties of polynomial curves are not applicable to it. For this reason it is very useful to know if a curve presented as a rational space curve has a polynomial parametrization. In this paper, we present an alg...

Through a geometncal approach of the blossoming pnnciple, we achieve a dimension élévation process for extended Chebyshev spaces Applied to a nested séquence ofsuch spaces included in a polynomial one, this allows to compute the Bézier points from the initial Chebyshev-Bézier points This method leads to interesting shape effects © Elsevier, Paris

The evaluation of points and the computations of in¯ection points or cusps on a curve are often necessary in CAGD applications. When a curve is represented in a B-spline form, such computations can be made easier once it is transformed into a set of piecewise polynomial curves in power form. The usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves i...

We investigate the computational complexity of finding the minimum-area circumscribed rectangle of a given two-dimensional domain. We study this problem in the polynomial-time complexity theory of real functions based on the oracle Turing machine model. We show that a bounded domain S with a polynomial-time computable Jordan curve Γ as the boundary may not have a computable minimum-area circums...

We consider the problem of constructing an approximation of the Pareto curve associated with the multiobjective optimization problem minx∈S{(f1(x), f2(x))}, where f1 and f2 are two conflicting polynomial criteria and S ⊂ Rn is a compact basic semialgebraic set. We provide a systematic numerical scheme to approximate the Pareto curve. We start by reducing the initial problem into a scalarized po...

In this paper, we propose a linear method for C approximation of rational Bézier curve with arbitrary degree polynomial curve. Based on weighted least-squares, the problem be converted to an approximation between two polynomial curves. Then applying Bernstein-Jacobi hybrid polynomials, we obtain the resulting curve. In order to reduce error, degree reduction method for Bézier curve is used. A e...

Although numerous researches on s-curve motion profiles have been carried out, up to date, no systematic investigation on the general model of polynomial s-curve motion profiles is considered. In this paper, the model of polynomial s-curve motion profiles is generalized in a recursive form. Based on that, a general algorithm to design s-curve trajectory with time-optimal consideration is propos...