J. Rafael Sendra

Symbolic parametrization of curves

If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essential to be able to represent these geometric objects in an appropriate way. For some applications an implicit representation by algebraic equations is desirable, whereas for others an explicit or parametric representation is more suitable. Therefore, transformation algorithms from one representation to the other are of utmost importance. We investigate the transformation of an implicit representation of a plane algebraic curve into a parametric representation. Various methods for computing a rational parametrization, if one exists, are described. As a new idea we introduce the concept of working with classes of conjugate (singular or simple) points on curves. All the necessary operations, like determining the multiplicity and the character of the singular points or passing a linear system of curves through these points, can be applied to such classes of conjugate points. Using this idea one can parametrize a curve if one knows only one simple point on it. We do not propose any new method for finding such a simple point. By classical methods a rational point on a rational curve can be computed, if such a point exists. Otherwise, one can express the coordinates of such a point in an algebraic extension of degree 2 over the ground field.

Parametric generalized offsets to hypersurfaces

In this paper we extend the classical notion of offset to the concept of generalized offset to hypersurfaces. In addition, we present a complete theoretical analysis of the rationality and unirationality of generalized offsets. Characterizations for deciding whether the generalized offset to a hypersurface is parametric or it has two parametric components are given. As an application, an algorithm to analyse the rationality of the components of the generalized offset to a plane curve or to a surface, and to compute rational parametrizations of its rational components, is outlined.

Parametrization of algebraic curves over optimal field extensions

Abstract In this paper we investigate the problem of determining rational parametrizations of plane algebraic curves over an algebraic extension of least degree over the field of definition. This problem reduces to the problem of finding simple points with coordinates in the field of definition on algebraic curves of genus 0. Consequently we are also able to decide parametrizability over the reals. We generalize a classical theorem of Hilbert and Hurwitz about birational transformations. An efficient algorithm for computing such optimal parametrizations is presented.

An Algebraic Approach to Lens Distortion by Line Rectification

A very important property of the usual pinhole model for camera projection is that 3D lines in the scene are projected in 2D lines. Unfortunately, wide-angle lenses (specially low-cost lenses) may introduce a strong barrel distortion which makes the usual pinhole model fail. Lens distortion models try to correct such distortion. In this paper, we propose an algebraic approach to the estimation of the lens distortion parameters based on the rectification of lines in the image. Using the proposed method, the lens distortion parameters are obtained by minimizing a 4 total-degree polynomial in several variables. We perform numerical experiments using calibration patterns and real scenes to show the performance of the proposed method.

Tracing index of rational curve parametrizations

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 a parametrization might trace the curve once or several times. We formally introduce the concept of the tracing index of a parametrization, we show how to compute it, and we relate it to the degree of rational reparametrizations as well as to the degree of the curve. In addition, we show how to apply these results to the case of real curves, where we introduce the notion of real tracing index.

Genus formula for generalized offset curves

In this paper, we present a formula for computing the genus of irreducible generalized offset curves to projective irreducible plane curves with only affine ordinary singularities over an algebraically closed field. The formula expresses the genus of the offset by means of the degree and the genus of the original curve.

Computation of the topology of real algebraic space curves

An algorithm for computing the topology of a real algebraic space curve C, implicitly defined as the intersection of two surfaces, is presented. Given C, the algorithm generates a space graph which is topologically equivalent to the real variety on the Euclidean space. The algorithm is based on the computation of the graphs of at most two projections of C. For this purpose, we introduce the notion of space general position for space curves, we show that any curve under the above conditions can always be linearly transformed to be in general position, and we present effective methods for checking whether space general position has been reached.

Parametrization of approximate algebraic curves by lines

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 e > 0 and an e-irreducible algebraic plane curve L of degree d having an e-singularity of multiplicity d - 1, we provide an algorithm that computes a proper parametrization of a rational curve that is exactly parametrizable by lines. Furthermore, the error analysis shows that under certain initial conditions that ensures that points are projectively well defined, the output curve lies within the offset region of L at distance at most 2√2e1(2d)exp(2).

Properness and Inversion of Rational Parametrizations of Surfaces

Abstract. In this paper we characterize the properness of rational parametrizations of hypersurfaces by means of the existence of intersection points of some additional algebraic hypersurfaces directly generated from the parametrization over a field of rational functions. More precisely, if V is a hypersurface over an algebraically closed field ? of characteristic zero and is a rational parametrization of V, then the characterization is given in terms of the intersection points of the hypersurfaces defined by xiqi(t¯)−pi(t¯), i=1,...,n over the algebraic closure of ?(V). In addition, for the case of surfaces we show how these results can be stated algorithmically. As a consequence we present an algorithmic criteria to decide whether a given rational parametrization is proper. Furthermore, if the parametrization is proper, the algorithm also computes the inverse of the parametrization. Moreover, for surfaces the auxiliary hypersurfaces turn to be plane curves over ?(V), and hence the algorithm is essentially based on resultants. We have implemented these ideas, and we have empirically compared our method with the method based on Gröbner basis.

Algebraic Lens Distortion Model Estimation

A very important property of the usual pinhole model for camera projection is that 3D lines in the scene are projected to 2D lines. Unfortunately, wide-angle lenses (specially low-cost lenses) may introduce a strong barrel distortion, which makes the usual pinhole model fail. Lens distortion models try to correct such distortion. We propose an algebraic approach to the estimation of the lens distortion parameters based on the rectification of lines in the image. Using the proposed method, the lens distortion parameters are obtained by minimizing a 4 total-degree polynomial in several variables. We perform numerical experiments using calibration patterns and real scenes to show the performance of the proposed method. Source Code The source code, the code documentation, and the online demo are accessible at the IPOL web page of this article1. In this page, an implementation is available for download. Compilation and usage instructions are included in the README.txt file of the archive.