Linear Precision for Toric Surface Patches

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. 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. H.-C. Graf von Bothmer Mathematisches Institut, Georg-August-Universitiät Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany e-mail: [email protected] K. Ranestad Matematisk Institutt, Universitetet i Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway e-mail: [email protected] F. Sottile ( ) Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA e-mail: [email protected] 38 Found Comput Math (2010) 10: 37–66