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823
Implicitization of rational Parametric Equations
 Journal of Symbolic Computation
, 1992
"... Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational parametric equations. (1) Find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) Decide whether the param ..."
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Cited by 21 (6 self)
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Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational parametric equations. (1) Find a basis of the implicit prime ideal determined by a set of rational parametric equations. (2) Decide whether
Implicitization of Rational Parametric Surfaces
, 1996
"... A generalized projective implicitization theorem is presented that can be used to solve the implicitization of rational parametric curves and surfaces in an affine space. The Groebner bases technique is used to implement the algorithm. The algorithm has the advantages that it can handle base points ..."
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Cited by 2 (0 self)
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A generalized projective implicitization theorem is presented that can be used to solve the implicitization of rational parametric curves and surfaces in an affine space. The Groebner bases technique is used to implement the algorithm. The algorithm has the advantages that it can handle base points
On rationally parametrized modular equations
"... Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic transformati ..."
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Cited by 15 (0 self)
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Abstract. The classical theory of elliptic modular equations is reformulated and extended, and many new rationally parametrized modular equations are discovered. Each arises in the context of a family of elliptic curves attached to a genuszero congruence subgroup Γ0(N), as an algebraic
Implicit Representation of Rational Parametric Surfaces
 J. SYMBOLIC COMPUTATION
, 1992
"... this paper we present algorithms to implicitize rational parametric surfaces with and without base points. One of the strength of the algorithms lies in the fact that we do not use multivariate factorization. The base points blow up to rational curves on the surface and we present techniques to comp ..."
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Cited by 27 (6 self)
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this paper we present algorithms to implicitize rational parametric surfaces with and without base points. One of the strength of the algorithms lies in the fact that we do not use multivariate factorization. The base points blow up to rational curves on the surface and we present techniques
Universal rational parametrizations and toric varieties
 IN ALGEBRAIC GEOMETRY AND GEOMETRIC MODELING, CONTEMPORARY MATHEMATICS
, 2003
"... This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We giv ..."
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Cited by 4 (3 self)
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This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We
Computing rational parametrizations of canal surfaces
 Journal of Symbolic Computation
, 1997
"... A canal surface is the envelope of a oneparameter set of spheres with radii r(t) and centers m(t). It is shown that any canal surface to a rational spine curve m(t) and a rational radius function r(t) possesses rational parametrizations. We derive algorithms for the computation of these parametriza ..."
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Cited by 32 (7 self)
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A canal surface is the envelope of a oneparameter set of spheres with radii r(t) and centers m(t). It is shown that any canal surface to a rational spine curve m(t) and a rational radius function r(t) possesses rational parametrizations. We derive algorithms for the computation
Rational Parametrizations of Neural Networks
"... A connection is drawn between rational functions, the realization theory of dynamical systems, and feedforward neural networks. This allows us to parametrize single hidden layer scalar neural networks with (almost) arbitrary analytic activation functions in terms of strictly proper rational function ..."
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A connection is drawn between rational functions, the realization theory of dynamical systems, and feedforward neural networks. This allows us to parametrize single hidden layer scalar neural networks with (almost) arbitrary analytic activation functions in terms of strictly proper rational
Implicitization of rational parametric curves and surfaces
 Proc. AAECC8
, 1990
"... In this paper we use Gröbner bases for the implicitization of rational parametric curves and surfaces in 3Dspace. We prove that the implicit form of a curve or surface given by the rational parametrization x1:= p1 q1 x2:= p2 q2 x3:= p3 q3 where the p’s and q’s are univariate polynomials in y1 or b ..."
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Cited by 5 (0 self)
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In this paper we use Gröbner bases for the implicitization of rational parametric curves and surfaces in 3Dspace. We prove that the implicit form of a curve or surface given by the rational parametrization x1:= p1 q1 x2:= p2 q2 x3:= p3 q3 where the p’s and q’s are univariate polynomials in y1
Rational Parametrizations of Real Cubic Surfaces
, 1998
"... Real cubic algebraic surfaces may be described by either implicit or parametric equations. Each of these representations has strengths and weaknesses and have been used extensively in computer graphics. Applications involving both representations include the efficient computation of surface intersec ..."
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intersections, and triangulation of curved surfaces. One particularly useful representation is the rational parametrization, where the three spatial coordinates are given by rational functions of two parameters. Rational parametrizations speed up many computations, and their relatively simple structure allows
Results 1  10
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823