8,012 research outputs found
Computation of the singularities of parametric plane curves
Given an algebraic plane curve C defined by a rational parametrization P(t), we present formulae
for the computation of the degree of C, and the multiplicity of a point. Using the results presented
in [Sendra, J.R., Winkler, F., 2001. Tracing index of rational curve parametrizations. Computer Aided
Geometric Design 18 (8), 771–795], the formulae simply involve the computation of the degree of a
rational function directly determined from P(t). Furthermore, we provide a method for computing the
singularities of C and analyzing the non-ordinary ones without knowing its defining polynomial. This
approach generalizes the results in [Abhyankar, S., 1990. Algebraic geometry for scientists and engineers.
In: Mathematical Surveys and Monographs, vol. 35. American Mathematical Society; van den Essen,
A., Yu, J.-T., 1997. The D-resultants, singularities and the degree of unfaithfulness. Proceedings of the
American Mathematical Society 25, 689–695; Gutierrez, J., Rubio, R., Yu, J.-T., 2002. D-Resultant for
rational functions. Proceedings of the American Mathematical Society 130 (8), 2237–2246] and [Park, H.,
2002. Effective computation of singularities of parametric affine curves. Journal of Pure and Applied
Algebra 173, 49–58].Ministerio de Educación y CienciaComunidad de MadridUniversidad de Alcal
Effective computation of singularities of parametric affine curves
AbstractLet k be a field of characteristic zero and f(t),g(t) be polynomials in k[t]. For a plane curve parameterized by x=f(t),y=g(t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x1=f1(t),…,xn=fn(t) over an arbitrary ground field k, where f1,…,fn∈k[t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included
Topology of 2D and 3D Rational Curves
In this paper we present algorithms for computing the topology of planar and
space rational curves defined by a parametrization. The algorithms given here
work directly with the parametrization of the curve, and do not require to
compute or use the implicit equation of the curve (in the case of planar
curves) or of any projection (in the case of space curves). Moreover, these
algorithms have been implemented in Maple; the examples considered and the
timings obtained show good performance skills.Comment: 26 pages, 19 figure
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
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