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Polynomial parametrization of curves without affine singularities
2002
Computer Aided Geometric Design
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This class is characterized by the existence 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. ...
doi:10.1016/s0167-8396(02)00090-0
fatcat:xicpfzytbzaubg34fts4fub3km
Corrigendum to: "Polynomial parametrization of curves without affine singularities"
2002
Computer Aided Geometric Design
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It is well-known that any 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] . ...
doi:10.1016/s0167-8396(02)00160-7
fatcat:ga3h5dfebzehtkqvglfmy3652q
Page 1670 of Mathematical Reviews Vol. , Issue 2003C
[page]
2003
Mathematical Reviews
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(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. ...
Detecting real singularities of a space curve from a real rational parametrization
2009
Journal of symbolic computation
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In this paper we give an algorithm that detects real 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 ...
doi:10.1016/j.jsc.2007.09.002
fatcat:br54vbbgtjgebj3kd2zytybxii
Computation of the singularities of parametric plane curves
2007
Journal of symbolic computation
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Furthermore, we provide a method for computing the 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. ...
doi:10.1016/j.jsc.2007.06.001
fatcat:criogsvw2je35bcxynsxx5smum
Parametrizing Algebraic Curves
[article]
2011
arXiv
pre-print
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We present the technique 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 ...
arXiv:1108.6219v1
fatcat:viz5kjt6vjbl3jue3vfawx2n3e
A note on the rational parameterization of algebraic curves inMathematica ®
2009
Journal of Interdisciplinary Mathematics
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The main aim 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. ...
doi:10.1080/09720502.2009.10700627
fatcat:niei7enxlzdf5lxwq6wxj6buzq
Computations with algebraic curves
[chapter]
1989
Lecture Notes in Computer Science
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Our main results are 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. ...
doi:10.1007/3-540-51084-2_26
fatcat:c4vw5bojjradpd4vxvviaschuu
Symbolic parametrization of curves
1991
Journal of symbolic computation
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As a new idea we introduce the concept 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. ...
doi:10.1016/s0747-7171(08)80144-7
fatcat:lknq45rd4rbkjilgtb3ghabmc4
Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves
[article]
2009
arXiv
pre-print
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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. ...
arXiv:0901.0320v1
fatcat:xtissooypzfo5furlcd2yn6puy
Radical parametrizations of algebraic curves by adjoint curves
2011
Journal of symbolic computation
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plane 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. ...
doi:10.1016/j.jsc.2011.05.005
fatcat:gwrgtnnaqzfatj6f6xkhqvzn2q
Expressive curves
[article]
2020
arXiv
pre-print
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We prove that a plane 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] . ...
arXiv:2006.14066v2
fatcat:hrdemhz3yvaalns4tcg4tyiywi
Manifold splines
2006
Graphical Models
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As a result, our new spline surface defined over any manifold is a piecewise 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 ...
doi:10.1016/j.gmod.2006.03.004
fatcat:vlkc4cify5ffjf7dflyxy36cwa
Manifold splines
2005
Proceedings of the 2005 ACM symposium on Solid and physical modeling - SPM '05
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As a result, our new spline surface defined over any manifold is a piecewise 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 ...
doi:10.1145/1060244.1060249
dblp:conf/sma/GuHQ05
fatcat:inyekbp2gvf3tlerj6qoiv236i
Automatic parameterization of rational curves and surfaces III: Algebraic plane curves
1988
Computer Aided Geometric Design
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The genus is compuled by a complete analysis 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 ...
doi:10.1016/0167-8396(88)90011-8
fatcat:uwa2xbhypzgv3botwwosb5ugri
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