A rational parametrization of an algebraic curve establishes a rational correspondence of this curve with the affine or projective line. This correspondence is a birational equivalence if the parametrization is proper. So, intuitively speaking, a rational parametrization determines a linear tracing of the curve, when the parameter takes values in the algebraic closure of the ground field. Such ...

In this paper we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutio...

Rational cubic and bicubic trigonometric schemes are developed to conserve monotonicity of curve and surface data, respectively. The rational cubic function has four parameters in each subinterval, while the rational bicubic partially blended function has eight parameters in each rectangular patch. The monotonicity of curve and surface data is retained by developing constraints on some of these...

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the KušnirenkoBernštein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic r...

In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta’s Main Conj...

We consider the algebraic curve defined by $y^m = f(x)$ where $m \geq 2$ and $f(x)$ is a rational function over $\mathbb{F}_q$. We extend the concept of pure gap to {\bf c}-gap and obtain a criterion to decide when an $s$-tuple is a {\bf c}-gap at $s$ rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.

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...

A new heuristic method of geometric Hermite interpolation is presented to construct a planar cubic rational Bézier curve with two points and two unit tangent directions. The integral, which shows the change rate of the curvature, is taken as the energy function to measure the fairness of the parameter curve. Hence the curvature of the new curve is more stable. Since that the energy function of ...