Using implicit equations of parametric curves and surfaces without computing them: Polynomial algebra by values

Abstract

The availability of the implicit equation of a plane curve or of a 3D surface can be very useful in order to solve many geometric problems involving the considered curve or surface: for example, when dealing with the point position problem or answering intersection questions. On the other hand, it is well known that in most cases, even for moderate degrees, the implicit equation is either difficult to compute or, if computed, the high degree and the big size of the coefficients makes extremely difficult its use in practice.We will show that, for several problems involving plane curves, 3D surfaces and some of their constructions (for example, offsets), it is possible to use the implicit equation (or, more precisely, its properties) without needing to explicitly determine it. We replace the computation of the implicit equation by the evaluation of the considered parameterizations in a set of points and its use, in order to deal with the considered geometric problems, is translated into one or several generalized eigenvalue problems on matrix pencils (depending again on several evaluations of the considered parameterizations). This is the so called "Polynomial Algebra by Values" approach where the huge polynomial equations coming from Elimination Theory (e.g., using resultants) are replaced by big structured and sparse numerical matrices. For these matrices there are well known numerical techniques allowing to provide the results we need to answer the geometric questions on the considered curves and surfaces.

Computing the topology of a real algebraic plane curve whose defining equations are available only “by values”

“…Observe first that as a consequence of the special form of p(t, s) and q(t, s), the degree of B(s) is d − 2 (for details, see Diaz-Toca et al (2012)). Then, let τ = (τ 1 < .…”

“…In many practical situations the polynomial f (x, y) or the polynomials in (x(t), y(t)) appear presented in a non-expanded form which if expanded in the standard monomial basis produce polynomials of high degree, with big coefficients whose manipulation is a difficult task. For example, when computing the intersection of two surfaces, or when computing offset curves (for more examples see Diaz-Toca et al (2012)). Otherwise, if the polynomials describing the considered curve are presented by their values then they can be easily evaluated at any desired point with a reasonable computational cost.…”

Efficient computation of dual space and directional multiplicity of an isolated point

“…In implicit representations, one has access to evaluation of the level set function, rather than parameterized coordinates, therefore evaluation at arbitrary points in space is the basic tool at hand to be used for computations on this representation. For instance the topology or even the change of representation to implicit form is possible using solely the evaluation at arbitrary points [2,6,8]. Horner's scheme provides a stable and fast method to evaluate a polynomial in monomial representation.…”

Computing the common zeros of two bivariate functions via Bézout resultants

“…Computational complexity: The complexity of resultant methods based on Sylvester and Bézout matrices grows like the degree of p and q to the power of 6. Therefore, computations quickly become unfeasible when the degrees are larger than, say, 30, and this is one reason why the literature concentrates on examples with degree 20, or smaller [1,9,17,25,32]. Typically, we reduce the complexity to quartic scaling, and sometimes even further, by using domain subdivision, which is beneficial for both accuracy and efficiency (see section 4).…”