1. I N T R O D U C T I O N Rational parametrization of curves and surfaces is a very interesting problem in mathematics. For curves, it is shown that a curve of genus zero is a rational by Noether. As for surfaces, there is Noether 's theorem that an algebraic surface S is rational if and only if it contains an irreducible rational curve C with dim ICI >1. Although there exist some techniques t...

In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the Computer Aided Geometric Design Community. The analysis of the validity of these techniques has been a fruitful ground for Commutative Algebraists and Algebraic Geometers, and several results have been obtained so far. Yet, a lot of research is still be...

By using some elementary inequalities, authors in this paper makes further improvement for estimating the heights of Bézier curve and rational Bézier curve. And the termination criterion for subdivision of the rational Bézier curve is also improved. The conclusion of the extreme value problem is thus further confirmed.

We give a deterministic polynomial-time algorithm that computes a nontrivial rational point on an elliptic curve over a finite field, given a Weierstrass equation for the curve. For this, we reduce the problem to the task of finding a rational point on a curve of genus zero.

By the Mordell-Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎There is no known algorithm for finding the rank of this group‎. ‎This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

An algorithm is presented to approximate planar ooset curves within an arbitrary tolerance > 0. Given a planar parametric curve C(t) and an ooset radius r, the circle of radius r is rst approximated by piecewise quadratic B ezier curve segments within the tolerance. The exact ooset curve C r (t) is then approximated by the convolution of C(t) with the quadratic B ezier curve segments. For a pol...

This paper focuses on the blending of a plane with surfaces of revolution relying on Dupin cyclides, which are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century. A general algorithm is presented for the construction of two kinds of blends: pillar and recipient. This algorithm uses Rational Quadric Bézier Curves (RQBCs) to model ...

Minkowski Pythagorean hodograph curves are polynomial curves with polynomial speed, measured with respect to Minkowski norm. Curves of this special class are particularly well suited for representing medial axis transforms of planar domains. In the present paper we generalize this polynomial class to a rational class of curves in Minkowski 3-space. We show that any rational Minkowski Pythagorea...

Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle. MSC classes: 37F10, 30D05. A simpl...