Rational Hausdorff divisors: A new approach to the approximate parametrization of curves
Journal of Computational and Applied Mathematics
In this paper, we introduce the notion of rational Hausdorff divisor, we analyze the dimension and irreducibility of its associated linear system of curves, and we prove that all irreducible real curves belonging to the linear system are rational and are at finite Hausdorff distance among them. As a consequence, we provide a projective linear subspace where all (irreducible) elements are solutions of the approximate parametrization problem for a given algebraic plane curve. Furthermore, we
... ify the linear system with a plane curve that is shown to be rational and we develop algorithms to parametrize it analyzing its fields of parametrization. Therefore, we present a generic answer to the approximate parametrization problem. In addition, we introduce the notion of Hausdorff curve, and we prove that every irreducible Hausdorff curve can always be parametrized with a generic rational parametrization having coefficients depending on as many parameters as the degree of the input curve. 8' of 8 instead of 8, where the property P does not hold anymore neither the associated objects 8\ exist; for instance, a perturbation of a Q-reducible polynomial will be, in general, Q-irreducible and, therefore, the application of the existing polynomial factorization algorithms will just not solve our problem. One may try to recover the original unperturbed entity 8. Since, this is essentially impossible, a more realistic version of the problem is to determine a new object 8" near 8', satisfying P, as well as computing the associated objects 8" to 8". An algorithm solving a problem of the above type is called an approximate algorithm; a solution of the illustrating example on polynomial factorization is given e.g. in  . One may distinguish two main phases when dealing with this type of problems. On one hand, the development of a theoretical reasoning that yields an algorithm and, on the other, providing an analysis of the distance between input and output in terms of a given tolerance. The distance used depends on the particular treated problem. For instance, in the factorization problem, if the input is/ e C[xi,... ,x n ], and the output isg e C[xi,... ,x n ], one requires that ||/ -g||oo is smaller than the tolerance. Algebraic varieties (that is, sets of points whose coordinates are zeros of some polynomials), as for instance algebraic curves and algebraic surfaces, are the main objects in algebraic geometry. Therefore, in approximate algebraic geometry (which is our case), in general, since one usually deals with sets, the Hausdorff distance is used. Indeed, the Hausdorff distance has proven to be an appropriate tool for measuring the resemblance between two geometric objects, becoming in consequence a widely used tool in fields as computer aided design, pattern matching and pattern recognition (see e.g. [2, 3] ); at the end of Section 1.2 we recall the definition of Hausdorff distance. One can find in the literature papers treating this type of problems (say related to algebraic geometry and commutative algebra) with the same, or similar, strategies; see  for a general overview on approximate commutative algebra. Approximate algorithms for computing polynomial gcds can be found in    , the polynomial factorization problem is addressed in [8, 9, 1, 10] . For algebraic varieties there also exist approximate solutions: see    for the implicitization problem, see    for the study and analysis of singularities, and for the parametrization problem see ; approximate parametrization is the central topic of this paper and it will be commented in detail in the next subsections of this introduction. Approximate parametrization problem The approximate parametrization problem can be stated as follows (we state it for plane curves, but it can be similarly stated for space curves, see , and for surfaces, see ). Approximate parametrization problem for plane curves: Given the implicit equation of a non-rational real plane curve C and a tolerance e > 0, decide whether there exists a rational real plane curve C at finite small distance (i.e. small related to the tolerance e) to the input curve C and, in the affirmative case, compute a real rational parametrization ofC.