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Polynomial parametrization of curves without affine singularities

2002
*
Computer Aided Geometric Design
*

This class is characterized by the existence

doi:10.1016/s0167-8396(02)00090-0
fatcat:xicpfzytbzaubg34fts4fub3km
*of*a*polynomial**parametrization*and by the absence*of**affine**singularities*. ... This paper gives an algorithm for computing proper*polynomial**parametrizations*for a particular class*of**curves*. ... Acknowledgements We want to thank Vladimir Shprilrain who gave us this linelike point*of*view. ...##
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Corrigendum to: "Polynomial parametrization of curves without affine singularities"

2002
*
Computer Aided Geometric Design
*

It is well-known that any

doi:10.1016/s0167-8396(02)00160-7
fatcat:ga3h5dfebzehtkqvglfmy3652q
*polynomial**curve*(see Lausch and Nöbauer, 1973) has a proper*polynomial**parametrization*. We need to recall some facts from the literature. ... A*curve*is called*polynomial*if and only if there is a*parametrization*Such, a*parametrization*is called proper if and only if t may be expressed as a rational function in x, y. ... A plane algebraic*curve*is the zero set*of*a*polynomial*F (x, y) ∈ K [x, y] . ...##
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Page 1670 of Mathematical Reviews Vol. , Issue 2003C
[page]

2003
*
Mathematical Reviews
*

(A-LINZ-SC; Linz)

*Polynomial**parametrization**of**curves**without**affine**singularities*. (English summary) Comput. Aided Geom. Design 19 (2002), no. 3, 223-234. ... This class is characterized by the existence*of*a*polynomial*parametri- zation and by the absence*of**affine**singularities*. ...##
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Detecting real singularities of a space curve from a real rational parametrization

2009
*
Journal of symbolic computation
*

In this paper we give an algorithm that detects real

doi:10.1016/j.jsc.2007.09.002
fatcat:br54vbbgtjgebj3kd2zytybxii
*singularities*, including*singularities*at infinity, and counts local branches and multiplicities*of*real rational*curves*in the*affine*n-space*without*... This allows us to find all real parameters corresponding to the real*singularities*between the solutions*of*a system*of**polynomials*in one variable. ... For*curves*in the*affine*n-space, Park (see Park (2002) ) gives a method which computes the*singularities**of**polynomially**parametrized**curves*over fields*of*characteristic zero by means*of*Gröbner basis ...##
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Computation of the singularities of parametric plane curves

2007
*
Journal of symbolic computation
*

Furthermore, we provide a method for computing the

doi:10.1016/j.jsc.2007.06.001
fatcat:criogsvw2je35bcxynsxx5smum
*singularities**of*C and analyzing the non-ordinary ones*without*knowing its defining*polynomial*. ... Tracing index*of*rational*curve**parametrizations*. ... Moreover, we focus on the problem*of*computing and analyzing all the*singularities**of*a rational plane*curve**without*knowing its defining*polynomial*. ...##
###
Parametrizing Algebraic Curves
[article]

2011
*
arXiv
*
pre-print

We present the technique

arXiv:1108.6219v1
fatcat:viz5kjt6vjbl3jue3vfawx2n3e
*of**parametrization**of*plane algebraic*curves*from a number theorist's point*of*view and present Kapferer's simple and beautiful (but little known) proof that nonsingular*curves*...*of*degree > 2 cannot be*parametrized*by rational functions. ...*Curves**without**Parametrization*After having given lots*of*examples for*parametrized*families*of*rational solutions*of*certain diophantine equations we now turn to the problem*of*showing that certain*curves*...##
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A note on the rational parameterization of algebraic curves inMathematica ®

2009
*
Journal of Interdisciplinary Mathematics
*

The main aim

doi:10.1080/09720502.2009.10700627
fatcat:niei7enxlzdf5lxwq6wxj6buzq
*of*the paper is to describe an efficient algorithm which can be used for finding rational parameterizations*of*special classes*of*algebraic*curves*. ... The choice*of*an appropriate representation*of*geometric objects (explicit,*parametric*, or implicit one) is a fundamental issue for the development*of*efficient algorithms. ... Webcomputing portal, where the implementation is accessible, is supported by the grant 1N04078*of*the Ministry*of*Education, Youth and Sports. ...##
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Computations with algebraic curves
[chapter]

1989
*
Lecture Notes in Computer Science
*

Our main results are

doi:10.1007/3-540-51084-2_26
fatcat:c4vw5bojjradpd4vxvviaschuu
*polynomial*time algorithms (1) to compute the genus*of*plane algebraic*curves*, (2) to compute the rational*parametric*equations for implicitly defined rational plane algebraic*curves*...*of*arbitrary degree, (3) to compute birational mappings between points on irreducible space*curves*and points on projected plane*curves*and thereby to compute the genus and rational*parametric*equations ... Further, if C(t) is a nonzero*polynomial*, its roots give the*singular*points with multiplicities*of*the*affine**curve*. ...##
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Symbolic parametrization of curves

1991
*
Journal of symbolic computation
*

As a new idea we introduce the concept

doi:10.1016/s0747-7171(08)80144-7
fatcat:lknq45rd4rbkjilgtb3ghabmc4
*of*working with classes*of*conjugate (*singular*or simple) points on*curves*. ... We investigate the transformation*of*an implicit representation*of*a plane algebraic*curve*into a*parametric*representation. ... Schicho for interesting discussions on the subject*of*pazametrization. We are indebted to the anonymous referees for some valuable suggestions. ...##
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Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves
[article]

2009
*
arXiv
*
pre-print

*affine*ϵ-rational plane

*curves*,

*without*exact

*singularities*at infinity, by means

*of*linear systems

*of*(d-2)-degree

*curves*. ... In this paper, given a tolerance ϵ>0 and an ϵ-irreducible algebraic

*affine*plane

*curve*C

*of*proper degree d, we introduce the notion

*of*ϵ-rationality, and we provide an algorithm to

*parametrize*approximately ... In this paper, we generalize the ideas in [17] to the case

*of*

*affine*plane

*curves*

*without*

*singularities*at infinity. ...

##
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Radical parametrizations of algebraic curves by adjoint curves

2011
*
Journal of symbolic computation
*

plane

doi:10.1016/j.jsc.2011.05.005
fatcat:gwrgtnnaqzfatj6f6xkhqvzn2q
*curve**of*degree d ≤ 5 and every irreducible*singular*plane*curve**of*degree 6. ... In addition, we also present an algorithm for*parametrizing*by radicals any irreducible plane*curve**of*degree d having at least a point*of*multiplicity d-r, with 1≤ r ≤ 4 and, as a consequence, every irreducible ... Acknowledgements The authors thank Gian Pietro Pirola for his useful remarks, especially for pointing out the work*of*Oscar Zariski on this topic. ...##
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Expressive curves
[article]

2020
*
arXiv
*
pre-print

We prove that a plane

arXiv:2006.14066v2
fatcat:hrdemhz3yvaalns4tcg4tyiywi
*curve*C is expressive if (a) each irreducible component*of*C can be*parametrized*by real*polynomials*(either ordinary or trigonometric), (b) all*singular*points*of*C in the*affine*... These are the*curves*whose defining*polynomial*has the smallest number*of*critical points allowed by the topology*of*the set*of*real points*of*a*curve*. ... We used Sage to compute resultants, and Desmos to draw*curves*. While cataloguing expressive*curves**of*degrees ≤ 4, we made use*of*the classifications produced by A. Korchagin and D. Weinberg [28] . ...##
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Manifold splines

2006
*
Graphical Models
*

As a result, our new spline surface defined over any manifold is a piecewise

doi:10.1016/j.gmod.2006.03.004
fatcat:vlkc4cify5ffjf7dflyxy36cwa
*polynomial*surface with high*parametric*continuity*without*the need for any patching and/or trimming operations. ... We study the*affine*structure*of*domain manifolds in depth and prove that the existence*of*manifold splines is equivalent to the existence*of*a manifold's*affine*atlas. ... Fig. 11 . 11 Manifold spline examples: (A) Holomorphic 1-form x which induces the*affine*atlas A; (B)*parametric*domain manifold M with*singular*points Z marked; (C)*Polynomial*spline F defined on the ...##
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Manifold splines

2005
*
Proceedings of the 2005 ACM symposium on Solid and physical modeling - SPM '05
*

As a result, our new spline surface defined over any manifold is a piecewise

doi:10.1145/1060244.1060249
dblp:conf/sma/GuHQ05
fatcat:inyekbp2gvf3tlerj6qoiv236i
*polynomial*surface with high*parametric*continuity*without*the need for any patching and/or trimming operations. ... We study the*affine*structure*of*domain manifolds in depth and prove that the existence*of*manifold splines is equivalent to the existence*of*a manifold's*affine*atlas. ... Fig. 11 . 11 Manifold spline examples: (A) Holomorphic 1-form x which induces the*affine*atlas A; (B)*parametric*domain manifold M with*singular*points Z marked; (C)*Polynomial*spline F defined on the ...##
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Automatic parameterization of rational curves and surfaces III: Algebraic plane curves

1988
*
Computer Aided Geometric Design
*

The genus is compuled by a complete analysis

doi:10.1016/0167-8396(88)90011-8
fatcat:uwa2xbhypzgv3botwwosb5ugri
*of*the*singularities**of*plane algebraic*curves*, using*affine*quadratic transformations. ... We consider algorithms to compute the genus and rational*parametric*equations, for implicitly defined irreducible rational plane algebraic*curves**of*arbitrary degree. ... Rational parameterization techniques for irreducible algebraic space*curves*which are specified by two*polynomial*equations in space,*without*conditions on the rationality*of*the defining surfaces, are ...
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