Implicitization of Parametric Hypersurfaces via Points

Ferruccio Orecchia; Isabella Ramella

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche (2018)

  • Volume: 85, Issue: 1, page 201-204
  • ISSN: 0370-3568

Abstract

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Given a parametric polynomial representation of an algebraic hypersurface \mathbf{S} in the projective space we give a new algorithm for finding the implicit cartesian equation of \mathbf{S}.The algorithm is based on finding a suitable finite number of points on \mathbf{S} and computing, by linear algebra, the equation of the hypersurface of least degree that passes through the points. In particular the algorithm works for plane curves and surfaces in the ordinary three-dimensional space. Using C++ the algorithm has been implemented on an intel Pentium running Linux. Since our algorithm is based only on computations of linear algebra it reveals very efficient if compared with others that do not use linear algebra for the computations.

How to cite

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Orecchia, Ferruccio, and Ramella, Isabella. "Implicitization of Parametric Hypersurfaces via Points." Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche 85.1 (2018): 201-204. <http://eudml.org/doc/296734>.

@article{Orecchia2018,
abstract = {Given a parametric polynomial representation of an algebraic hypersurface $\mathbf\{S\}$ in the projective space we give a new algorithm for finding the implicit cartesian equation of $\mathbf\{S\}$.The algorithm is based on finding a suitable finite number of points on $\mathbf\{S\}$ and computing, by linear algebra, the equation of the hypersurface of least degree that passes through the points. In particular the algorithm works for plane curves and surfaces in the ordinary three-dimensional space. Using C++ the algorithm has been implemented on an intel Pentium running Linux. Since our algorithm is based only on computations of linear algebra it reveals very efficient if compared with others that do not use linear algebra for the computations.},
author = {Orecchia, Ferruccio, Ramella, Isabella},
journal = {Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche},
keywords = {Implicitization; Hypersurfaces},
language = {eng},
month = {12},
number = {1},
pages = {201-204},
publisher = {Società Nazione di Scienze, Lettere e Arti in Napoli; Giannini},
title = {Implicitization of Parametric Hypersurfaces via Points},
url = {http://eudml.org/doc/296734},
volume = {85},
year = {2018},
}

TY - JOUR
AU - Orecchia, Ferruccio
AU - Ramella, Isabella
TI - Implicitization of Parametric Hypersurfaces via Points
JO - Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche
DA - 2018/12//
PB - Società Nazione di Scienze, Lettere e Arti in Napoli; Giannini
VL - 85
IS - 1
SP - 201
EP - 204
AB - Given a parametric polynomial representation of an algebraic hypersurface $\mathbf{S}$ in the projective space we give a new algorithm for finding the implicit cartesian equation of $\mathbf{S}$.The algorithm is based on finding a suitable finite number of points on $\mathbf{S}$ and computing, by linear algebra, the equation of the hypersurface of least degree that passes through the points. In particular the algorithm works for plane curves and surfaces in the ordinary three-dimensional space. Using C++ the algorithm has been implemented on an intel Pentium running Linux. Since our algorithm is based only on computations of linear algebra it reveals very efficient if compared with others that do not use linear algebra for the computations.
LA - eng
KW - Implicitization; Hypersurfaces
UR - http://eudml.org/doc/296734
ER -

References

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  1. Abbott, J., Bigatti, A.M., Robbiano, L. (2016) Implicitization of Hypersurfaces. J. Symb. Comput.30, 137-149. Zbl1390.13080MR3594322DOI10.1016/j.jsc.2016.11.002
  2. Albano, G., Cioffi, F., Orecchia, F., Ramella, F. (2000) Minimally generating ideals of rational parametric curves in polynomial time. J. Symb. Comput.30, 137-149. Zbl0980.14033MR1777168DOI10.1006/jsco.1999.0354
  3. Atiyah, M. F., Macdonald, I. G. (1969) Introduction to Commutative Algebra. Reading, Mass. London-Don Mills, Ont.Addison-Wesley Publishing Company, Inc. Zbl0175.03601MR242802
  4. Geramita, A. V., Orecchia, F. (1981) On the Cohen-Macaulay type of s-lines. J. Algebra70, 89-133. Zbl0464.14007MR618382DOI10.1016/0021-8693(81)90247-7
  5. Hartshorne, R. (1977) Algebraic Geometry, Graduate Texts in Mathematics52New York, Springer-Verlag. Zbl0367.14001MR463157
  6. Orecchia, F. (1981) Points in generic position and conductors of curves with ordinary singularities, J. London Math. Soc.24, 85–96. Zbl0492.14017MR623673DOI10.1112/jlms/s2-24.1.85

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