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In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…

We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…

In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…

We present a method for computing all the symmetries of a rational ruled surface defined by a rational parametrization which works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the…

We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization…

A method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit…

In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in math.AG/0210096, where implicit equations are obtained as…

An efficient computer algorithm is described for the perspective drawing of a wide class of surfaces. The class includes surfaces corresponding lo single-valued, continuous functions which are defined over rectangular domains. The algorithm…

A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit…

We present a matrix-based algorithm for deciding if the parametrization of a curve or a surface is invertible or not, and for computing the inverse of the parametrization if it exists.

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from…

The ruled surface is a typical modeling surface in computer aided geometric design. It is usually given in the standard parametric form. However, it can also be in the forms than the standard one. For these forms, it is necessary to…

One distinguishing feature of rational curves is that they have algebraic parameterizations. Arc spaces are a way of describing approximations to parameterizations of all curves in some fixed space. Playing on these descriptions, this paper…

In this article we show how to compute a matrix representation and the implicit equation by means of the method developed in [Botbol: arXiv:1007.3437], using the computer algebra system Macaulay2 \cite{M2}. As it is probably the most…

This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space…

Given a family of rational curves depending on a real parameter, defined by its parametric equations, we provide an algorithm to compute a finite partition of the parameter space (${\Bbb R}$, in general) so that the shape of the family…

Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points,…

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…

In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and…