The classical H. Poincaré Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem can be reduced to a center problem for some ordinary differential equation whose coefficients are trigonometric polynomials depending polynomially on the coeffic...

The [Formula: see text] and [Formula: see text]-approximation schemes are introduced to approximate the popular generalized Cornu spirals with the help of the parametric rational cubic trigonometric Bézier curves. The [Formula: see text]-approximation scheme has two free parameters whereas [Formula: see text]-approximation scheme has four free parameters. To approximate the generalized Cornu sp...

In this paper, we consider a family of closed planar algebraic curves C which are given in parametrization form via a trigonometric polynomial p. When C is the boundary of a compact convex set, the polynomial p represents the support function of this set. Our aim is to examine properties of the degree of the defining polynomial of this family of curves in terms of the degree of p. Thanks to the...

1 We define b-compatibility for planar curves and propose three 2 ball morphing techniques (b-morphs) between pairs of b3 compatible curves. B-morphs use the automatic ball-map 4 correspondence, proposed by Chazel et al. [11], from which 5 they derive vertex trajectories (Linear, Circular, Parabolic). 6 All are symmetric, meeting both curves with the same an7 gle, which is a right angle for the...

By using the trigonometric uniform splines of order 3 with a real tension factor, a numericalmethod is developed for solving a linear second order boundary value problems (2VBP) withDirichlet, Neumann and Cauchy types boundary conditions. The moment at the knots isapproximated by central finite-difference method. The order of convergence of the methodand the theory is illustrated by solving tes...

In this paper, we give several simple methods for drawing a whole rational surface (without base points) as several B ezier patches. The rst two methods apply to surfaces speciied by triangular control nets and partition the real projective plane RP 2 into four and six triangles respectively. The third method applies to surfaces speciied by rectangular control nets and partitions the torus RP 1...

This paper presents a probabilistic planner capable of nding paths for a exible surface patch. The planner is based on the Probabilistic Roadmap approach to path planning while the surface patch is modeled as a low degree B ezier surface. We assume that we are dealing with an elastic part and deene an approximate energy model for the part. The energy function penalizes excessive shear and bendi...

In this paper we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vall\'{e}e Poussin type wavelets able to detect singularities along boundary curves periodic characteristic functions. Motivated by recent results for discrete in two dimensions, provide lower and upper estimates the magnitude corresponding inner products. proof use localization pr...

The stochastic Fokas system (SFS), driven by multiplicative noise in the Itô sense, was investigated this study. Novel trigonometric, rational, hyperbolic, and elliptic solutions are found using a modified mapping method. Because is used to explain nonlinear pulse propagation monomode optical fibers, provided may be utilized analyze broad range of critical physical phenomena. In order impacts n...