Abstract— A smooth curve interpolation is very significant in computer graphics or in data visualization. In the present paper -piecewise rational cubic spline function with tension parameter is considered which produces a monotonic interplant to a given monotonic data set. The parameters in the description of the spline curve can be used to modify the shape of the curve, locally and globally. ...

The μ-invariant μ = (μ1, μ2, μ3) of a rational space curve gives important information about the curve. In this paper, we describe the structure of all parameterizations that have the same μ-type, what we call a μ-stratum, and as well the closure of strata. Many of our results are based on papers by the second author that appeared in the commutative algebra literature. We also present new resul...

Any smooth projective Fano variety in characteristic zero is rationally connected and hence contains a very free rational curve. In positive characteristic a smooth projective Fano variety is rationally chain connected. However, it is not known whether such varieties are separably rationally connected, or equivalently, whether they have a very free rational curve. This is an open question even ...

Akiyama and Goto have proposed a cryptosystem based on rational points on curves over function fields (stated in the equivalent form of sections of fibrations on surfaces). It is easy to construct a curve passing through a few given points, but finding the points, given only the curve, is hard. We show how to break their original cryptosystem by using algebraic points instead of rational points...

Two equations are presented which express the derivative of a rational Bézier curve in terms of its control points and weights. These equations are natural generalisations of the non-rational case and various properties are found from them. Bounds on the magnitude of the derivative and the direction of the derivative (the hodograph) are obtained. Expressions for the second derivative, curvature...

Schoof's algorithm is used to nd a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve de ned over a nite eld. By realizing e cient combination of several improvements, such as Atkin-Elkies's method, the isogeny cycles method, and trial search by match-and-sort techniques, we can count the number of rational points on an...

It is well known that irreducible algebraic plane curves having a singularity of maximum multiplicity are rational and can be parametrized by lines. In this paper, given a tolerance ¿ 0 and an -irreducible algebraic plane curve C of degree d having an -singularity of multiplicity d−1, we provide an algorithm that computes a proper parametrization of a rational curve that is exactly parametrizab...

We use a combination of both symbolic and numerical techniques to construct a degree bounded C-continuous, rational B-spline -approximations of real algebraic surface-surface intersection curves. The algebraic surfaces could be either in implicit or rational parametric form. At singular points we use the classical Newton power series factorizations to determine the distinct branches of the spac...

In a previous paper ([10]), the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and Q( √ −3). Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points o...

We construct a family of infinitely many elliptic curves over Q with a nontrivial rational 2-torsion point and with rank ≥ 6, which is parametrized by the rational points of an elliptic curve of rank ≥ 1.