A Direct Approach to Computing theμ-basis of Planar Rational Curves (original) (raw)

A Grobner Bases Approach to the Detection of Improperly Parameterized Rational Curve

American Journal of Computational Mathematics, 2013

This paper propose an algorithm for the detection of improper parameterization of rational curves using the concept of Gröbner bases. The advantage of the proposed algorithm lies in the fact that the Gröbner bases can operate in both univariate and multivariate fields with specified ordering.

Exact symbolic–numeric computation of planar algebraic curves

Theoretical Computer Science, 2013

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project Cgal. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, Lgp and Maple's Isolate for polynomial system solving, and Cgal's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.

The moving line ideal basis of planar rational curves

Computer Aided Geometric Design, 1998

This paper shows that the ideal of any degree ~z planar rational curve can be generated by two polynomials that are each linear in x, .l/and degree nl and ne (~> hi) in t, n.i + 7z2 = ~. The value of m is fixed for a given rational curve, and serves to split all degree ,~ curves into Ln/2~ + I equivalence classes. These classes bear on the determinantal form of the implicit equation of the rational curve. © 1998 Elsevier Science B.V.

Computing parameterizations of rational algebraic curves

Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '94, 1994

In this paper I want to present a new method for computing parametrizations of algebraic curves. Basically this method is a direct application of integral basis computation. Examples show that this method is faster than older methods.

Implicitizing Rational Curves by the Method of Moving Algebraic Curves

Journal of Symbolic Computation, 1997

A function F (x, y, t) that assigns to each parameter t an algebraic curve F (x, y, t) = 0 is called a moving curve. A moving curve F (x, y, t) is said to follow a rational curve x = x(t)/w(t), y = y(t)/w(t) if F (x(t)/w(t), y(t)/w(t), t) is identically zero. A new technique for finding the implicit equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2n with no base points the method of moving conics generates the implicit equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y, whereas standard resultant methods generate the implicit equation as the determinant of a 2n × 2n matrix where each entry is a linear polynomial in x and y. Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.

Automatic parameterization of rational curves and surfaces IV: algebraic space curves

ACM Transactions on Graphics, 1989

For an irreducible algebraic space curve C that is implicitly defined as the intersection of two algebraic surfaces, f (x, y, z) = 0 and g(r, y, z) = 0, there always exists a birational correspondence between the points of C and the points of an irreducible plane curve P, whose genus is the same as that of C. Thus C is rational iff the genus of P is zero. Given an irreducible space curve C = ( f n g), with f and g not tangent along C, we present a method of obtaining a projected irreducible plane curve P together with birational maps between the points of P and C. Together with [4], this method yields an algorithm to compute the genus of C, and if the genus is zero, the rational parametric equations for C. As a biproduct, this method also yields the implicit and parametric equations of a rational surface S containing the space curve C.

Computations with algebraic curves

Lecture Notes in Computer Science, 1989

We present a variety of computational techniques dealing with algebraic curves both in the plane and in space. Our main results are polynomial time algorithms (1) to compute the genus of plane algebraic curves, (2) to compute the rational parametric equations for implicitly defined rational plane algebraic curves of arbitrary degree, (3) to compute birational mappings between points on irreducible space curves and points on projected plane curves and thereby to compute the genus and rational parametric equations for implicitly defined rational space curves of arbitrary degree, (4) to check for the faithfulness (one to one) of parameterizations.

Approximate parameterization by planar rational curves

2004

We describe a method for approximate parameterization of a planar algebraic curve by a rational Bézier (spline) curve. After briefly discussing exact methods for parameterization and methods for rational interpolation, we describe a new technique for rational parameterization. Our approach is based on the minimization of a suitable--nonlinear objective function, which takes both the distance from the curve and the positivity of the weight function (i.e., the numerator of the rational parametric representation) into account. The solution is computed by using an SQP-type optimization technique. In addition, we use a region--growing--type approach in order to obtain a good initial solution, which is crucial for the convergence of the nonlinear optimization procedure.

Fast computation of determinants of Bézout matrices and application to curve implicitization

Linear Algebra and its Applications, 2009

When using bivariate polynomial interpolation for computing the implicit equation of a rational plane algebraic curve given by its parametric equations, the generation of the interpolation data is the most costly of the two stages of the process. In this work a new way of generating those interpolation data with less computational cost is presented. The method is based on an efficient computation of the determinants of certain constant Bézout matrices.