Abstract
We classify the homogeneous polynomials in three variables whose toric polar linear system defines a Cremona transformation. This classification includes, as a proper subset, the classification of toric surface patches from geometric modeling which have linear precision. Besides the well-known tensor product patches and Bézier triangles, we identify a family of toric patches with trapezoidal shape, each of which has linear precision. Furthermore, Bézier triangles and tensor product patches are special cases of trapezoidal patches.
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References
E. Casas-Alvero, Singularities of Plane Curves. London Mathematical Society Lecture Note Series, vol. 276 (Cambridge University Press, Cambridge, 2000).
D. Cox, What is a toric variety? in Topics in Algebraic Geometry and Geometric Modeling. Contemporary Mathematics, vol. 334 (Amer. Math. Soc., Providence, 2003), pp. 203–223.
D. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, 3rd edn. Undergraduate Texts in Mathematics (Springer, New York, 2007).
I.V. Dolgachev, Polar Cremona transformations, Mich. Math. J. 48, 191–202 (2000). Dedicated to William Fulton on the occasion of his 60th birthday.
L. García-Puente, F. Sottile, Linear precision for parametric patches, Adv. Comput. Math. (2009, to appear).
R. Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling (Morgan Kaufmann/Academic Press, San Diego, 2002).
R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977).
K. Karčiauskas, R. Krasauskas, Comparison of different multisided patches using algebraic geometry, in Curve and Surface Design, ed. by P.-J. Laurent, P. Sablonniere, L.L. Schumaker. Saint-Malo, 1999 (Vanderbilt University Press, Nashville, 2000), pp. 163–172.
R. Krasauskas, Toric surface patches, Adv. Comput. Math. 17(1–2), 89–133 (2002). Advances in Geometrical Algorithms and Representations.
R. Krasauskas, Bézier patches on almost toric surfaces, in Algebraic Geometry and Geometric Modeling. Mathematics and Visualization (Springer, Berlin, 2006), pp. 135–150.
R. Krasauskas, Minimal rational parametrizations of canal surfaces, Computing 79(2–4), 281–290 (2007).
H. Pottmann, J. Wallner, Computational Line Geometry. Mathematics and Visualization (Springer, Berlin, 2001).
F. Sottile, Toric ideals, real toric varieties, and the moment map, in Topics in Algebraic Geometry and Geometric Modeling. Contemporary Mathematics, vol. 334 (Amer. Math. Soc., Providence, 2003), pp. 225–240.
B. Sturmfels, Solving Systems of Polynomial Equations. CBMS Regional Conference Series in Mathematics, vol. 97 (Amer. Math. Soc., Providence, 2002).
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Communicated by Wolfgang Dahmen and Herbert Edelsbrunner.
Work of Sottile supported by NSF grants CAREER DMS-0538734 and DMS-0701050, the Institute for Mathematics and its Applications, and Texas Advanced Research Program under Grant No. 010366-0054-2007.
Work of Graf von Bothmer supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
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Graf von Bothmer, HC., Ranestad, K. & Sottile, F. Linear Precision for Toric Surface Patches. Found Comput Math 10, 37–66 (2010). https://doi.org/10.1007/s10208-009-9052-6
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DOI: https://doi.org/10.1007/s10208-009-9052-6