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### Polynomial parametrization of curves without affine singularities

J Gutierrez
2002 Computer Aided Geometric Design
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.  ...

### Corrigendum to: "Polynomial parametrization of curves without affine singularities"

J. Gutierrez, R. Rubio, J. Schicho
2002 Computer Aided Geometric Design
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] .  ...

### 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.  ...

### Detecting real singularities of a space curve from a real rational parametrization

R. Rubio, J.M. Serradilla, M.P. Vélez
2009 Journal of symbolic computation
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  ...

### Computation of the singularities of parametric plane curves

Sonia Pérez-Díaz
2007 Journal of symbolic computation
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.  ...

### Parametrizing Algebraic Curves [article]

Franz Lemmermeyer
2011 arXiv   pre-print
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  ...

### A note on the rational parameterization of algebraic curves inMathematica ®

Michal Bizzarri, Miroslav Lávička
2009 Journal of Interdisciplinary Mathematics
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.  ...

### Computations with algebraic curves [chapter]

Shreeram S. Abhyankar, Chanderjit L. Bajaj
1989 Lecture Notes in Computer Science
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.  ...

### Symbolic parametrization of curves

J. Rafael Sendra, Franz Winkler
1991 Journal of symbolic computation
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.  ...

### Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves [article]

Sonia Perez-Diaz, Sonia L. Rueda, Juana Sendra, J. Rafael Sendra
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  to the case of affine plane curves without singularities at infinity.  ...

J. Rafael Sendra, David Sevilla
2011 Journal of symbolic computation
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.  ...

### Expressive curves [article]

Sergey Fomin, Eugenii Shustin
2020 arXiv   pre-print
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  .  ...

### Manifold splines

Xianfeng Gu, Ying He, Hong Qin
2006 Graphical Models
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  ...

### Manifold splines

Xianfeng Gu, Ying He, Hong Qin
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 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  ...

### Automatic parameterization of rational curves and surfaces III: Algebraic plane curves

Shreeram S. Abhyankar, Chanderjit L. Bajaj
1988 Computer Aided Geometric Design
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  ...
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