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We study embedded rational curves in projective toric varieties. Generalizing results of the first author and Zotine for the case of lines, we show that any degree $d$ rational curve in a toric variety $X$ can be constructed from a special…
The purpose of the present paper is to present the main applications of a new method for the determination of the fractal structure of plane curves. It is focused on the inverse problem, that is, given a curve in the plane, find its fractal…
In this paper we develop the formalism of rational complex Bezier curves. This framework is a simple extension of the CAD paradigm, since it describes arc of curves in terms of control polygons and weights, which are extended to complex…
The existence of translated curves for quasiperiodically forced maps is established, under very mild regularity hypotheses, for rotation numbers of constant type. Among the translated curves, the invariant curves are characterized as the…
An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in (Linhart,…
Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…
The goal of inversion is to estimate the model which generates the data of observations with a specific modeling equation. One general approach to inversion is to use optimization methods which are algebraic in nature to define an objective…
We use twisted stable maps to compute the number of rational degree d plane curves having prescribed contacts to a smooth plane cubic.
Finding a computationally efficient algorithm for the inverse continuous wavelet transform is a fundamental topic in applications. In this paper, we show the convergence of the inverse wavelet transform.
Matrix weighted rational B\'{e}zier curves can represent complex curve shapes using small numbers of control points and clear geometric definitions of matrix weights. Explicit formulae are derived to convert matrix weighted rational…
We introduce and begin the topological study of real rational plane curves, all of whose inflection points are real. The existence of such curves is a corollary of results in the real Schubert calculus, and their study has consequences for…
A lima\c{c}on-like curve, allowing 2{\pi}-transition with monotone curvature between concentric curvature elements, is presented. The curve is 4th degree algebraic, 4th degree rational, and shares other common features with Pascal's…
Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any…
As in our previous work [1] we address the problem to determine the splitting of the normal bundle of rational curves. With apolarity theory we are able to characterize some particular subvarieties in some Hilbert scheme of rational curves,…
We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan…
It is believed that some numerical technique must be employed for the determination of the system parameters of a visual binary or a star with a planet because the relevant equations are not only highly nonlinear but also transcendental…
We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational…
In this paper, we give an elementary new method for determining the rational points on algebraic curves using torsion packets. We also provide examples of curves for which all rational points can be completely determined by our method.
The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the curvature error. Parametric polynomial curves of low degree are used and a geometric continuity is…