Page 138 - Geometric Modeling and Algebraic Geometry

7 Curve Parametrization Exploiting the Newton Polygon 139
7.7.6 Parametrization of the conic

The conic C can be parametrized easily (e.g. by stereographic projection) if one

point on it is known. In our case the origin is contained in C and we compute the
following parametrization over Q using a pencil of lines through the origin:

−9
t 2 +12t
ψ 2 : A C : t → −9t
1
Q
t 2 +12t
Composing both maps finally yields a parametrization of the input curve:
" #
−(12+12t+t )t
2
ψ −1 ◦ ψ 2 : A C : t → 3(2+3t)(6+t)
1
Q 2
1 −(6+9t+t )(6+t)
12t
Acknowledgments
The authors were supported by the FWF (Austrian Science Fund) in the frame of the
research projects SFB 1303.

References

1. Tobias Beck and Josef Schicho. Parametrization of Algebraic Curves Defined by Sparse
Equations. AAECC, 2006. Accepted for publication. Preprint available as RICAM Report
2005–08 at http://www.ricam.oeaw.ac.at/publications/reports.
2. Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The
user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and
number theory (London, 1993).
3. David Cox. What is a toric variety? In Topics in Algebraic Geometry and Geometric
Modeling, volume 334 of Contemporary Mathematics, pages 203–223. American Mathe-
matical Society, Providence, Rhode Island, 2003. Workshop on Algebraic Geometry and
Geometric Modeling (Vilnius, 2002).
4. David A. Cox. Toric varieties and toric resolutions. In Resolution of singularities (Ober-
gurgl, 1997), volume 181 of Progr. Math., pages 259–284. Birkh¨ auser, Basel, 2000.
5. William Fulton. Introduction to toric varieties, volume 131 of Annals of Mathematics
Studies. Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures
in Geometry.
6. G.-M. Greuel, C. Lossen, and M. Schulze. Three algorithms in algebraic geometry, coding
theory and singularity theory. In Applications of algebraic geometry to coding theory,
physics and computation (Eilat, 2001), volume 36 of NATO Sci. Ser. II Math. Phys. Chem.,
pages 161–194. Kluwer Acad. Publ., Dordrecht, 2001.
7. Gert-Martin Greuel and Gerhard Pfister. A Singular introduction to commutative algebra.
Springer-Verlag, Berlin, 2002. With contributions by Olaf Bachmann, Christoph Lossen
and Hans Sch¨ onemann, With 1 CD-ROM (Windows, Macintosh, and UNIX).
8. Ga´ etan Hach´ e and Dominique Le Brigand. Effective construction of algebraic geometry
codes. IEEE Trans. Inform. Theory, 41(6, part 1):1615–1628, 1995. Special issue on
algebraic geometry codes.