328 research outputs found
Resultants and Moving Surfaces
We prove a conjectured relationship among resultants and the determinants
arising in the formulation of the method of moving surfaces for computing the
implicit equation of rational surfaces formulated by Sederberg. In addition, we
extend the validity of this method to the case of not properly parametrized
surfaces without base points.Comment: 21 pages, LaTex, uses academic.cls. To appear: Journal of Symbolic
Computatio
Implicitization of curves and (hyper)surfaces using predicted support
We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation.
For predicting the implicit support, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory.
Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial.
We implement our methods on Maple, and some on Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces.
We apply our approach to approximate implicitization,
and quantify the accuracy of the approximate output,
which turns out to be satisfactory on all tested examples; we also relate our measures to Hausdorff distance.
In building a square or rectangular matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complexes offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice.
We compare our prototype to existing software and find that it is rather competitive
A partial solution to the problem of proper reparametrization for rational surfaces
Given an algebraically closed field K, and a rational parametrization P of an algebraic surface V ⊂ K3 , we consider the problem of computing a proper rational parametrization Q from P (reparametrization problem). More precisely, we present an algorithm that computes a rational parametrization Q of V such that the degree of the rational map induced by Q is less than the degree induced by P. The properness of the output parametrization Q is analyzed. In particular, if the degree of the map induced by Q is one, then Q is proper and the reparametrization problem is solved. The algorithm works if at least one of two auxiliary parametrizations defined from P is not proper
A univariate resultant based implicitation algorithm for surfaces
In this paper, we present a new algorithm for computing the implicit equation of a rational surface V
from a rational parametrization P(t). The algorithm is valid independent of the existence of base points,
and is based on the computation of polynomial gcds and univariate resultants. Moreover, we prove that the
resultant-based formula provides a power of the implicit equation. In addition, performing a suitable linear
change of parameters, we prove that this power is indeed the degree of the rational map induced by the
parametrization. We also present formulas for computing the partial degrees of the implicit equation.Ministerio de EducaciĂłn y cienci
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