19 June 2009 ArXiv.org:0806.3230LINEAR PRECISION FOR TORIC SURFACE PATCHESHANS-CHRISTIAN GRAF VON BOTHMER, KRISTIAN RANESTAD, AND FRANK SOTTILEAbstract. We classify the homogeneous polynomials in three variables whose toricpolar linear system defines a Cremona transformation. This classification includes, asa proper subset, the classification of toric surface patches from geometric modelingwhich have linear precision. Besides the well-known tensor product patches andB´ezier triangles, we identify a family of toric patches with trapezoidal shape, each ofwhich has linear precision. Furthermore, B´ezier triangles and tensor product patchesare special cases of trapezoidal patches.Communicated by Wolfgang Dahmen and Herbert EdelsbrunnerIntroductionWhile the basic units in the geometric modeling of surfaces are B´ezier trianglesand rectangular tensor product patches, some applications call for multi-sided C∞patches (see [8] for a discussion). Krasauskas’s toric B´ezier patches [10] are a flexibleand mathematically appealing system of such patches. These are based on real toricvarieties from algebraic geometry, may have shape any polytope ∆ with integer vertices,and they include the classical B´ezier patches as special cases. For descriptions ofmultisided patches and toric patches, see [6].More precisely, given a set of lattice points in Znwith convex hull ∆, Krasauskasdefined toric B´ezier functions, which are polynomial blending functions associated toeach lattice point. This collection of lattice points and toric B´ezier functions, togetherwith a positive weight associated to each lattice point is atoric patch. Choosing also acontrol point in Rdfor each lattice point leads to a map Φ: ∆ → Rdwhose image maybe used in modeling. If we choose the lattice points themselves as control points weobtain the tautological map τ : ∆ → ∆, which is a bijection. If the tautological maphas a rational inverse, then the toric patch has linear precision.The lattice points and weights of a toric patch are encoded in a homogeneous multi-variate polynomial F (x0, . . . , xn) with positive coefficients, with every such polynomialcorresponding to a toric patch. In [5] it was shown that the toric patch given by F haslinear precision if and only if the associated toric polar linear system,T (F ) =Dx0∂F∂x0, x1∂F∂x1, . . . , xn∂F∂xnE,2000 Mathematics Subject Classification. 14M25, 65D17.Key words and phrases. B´ezier patches, geometric modeling, linear precision, Cremona transfor-mation, toric patch.Work of Sottile supported by NSF grants CAREER DMS-0538734 and DMS-0701050, the Institutefor Mathematics and its Applications, and Texas Advanced Research Program under Grant No.010366-0054-2007.12 H-C. GRAF V. BOTHMER, K. RANESTAD, AND F. SOTTILEdefines a birational mapΦF: Pn−→ Pn. This follows from the existence of a rationalreparameterization transforming the tautological map into ΦF. The polar linear systemis toric because the derivations xi∂∂xiare vector fields on the torus (C×)n⊂ Pn.When T (F ) defines a birational map, we say that F defines atoric polar Cremonatransformation. We seek to classify all such homogeneous polynomials F withoutthe restriction that the coefficients are positive or even real. This is a variant ofthe classification of homogeneous polynomials F whose polar linear system (whichis generated by the partial derivatives∂F∂xi) defines a birational map. Dolgachev [4]classified all such square free polynomials in 3 variables and those in 4 variables thatare products of linear forms.Definition. Two polynomials F and G are calledequivalent if they can be transformedinto each other by successive invertible monomial substitutions, multiplications withLaurent monomials, or scalings of the variables.The property of defining a toric polar Cremona transformation is preserved underthis equivalence. Our main result is the classification (up to equivalence) of homoge-neous polynomials in three variables that define toric polar Cremona transformations.Theorem 1. A homogeneous polynomial F in three variables that defines a toric polarCremona transformation is equivalent to one of the following(1) (x + z)a(y + z)bfor a, b ≥ 1,(2) (x + z)a¡(x + z)d+ yzd−1¢bfor a ≥ 0 and b, d ≥ 1, or(3)¡x2+ y2+ z2− 2(xy + xz + yz)¢d, for d ≥ 1.When a = 0 and d = 1 in (2), we obtain the polynomial (x+y+z)b, which correspondsto a B´ezier triangular patch of degree b used in geometric modeling. Similarly, thepolynomials F in (1) correspond to tensor product patches, which are also commonin geometric modeling. These are also recovered from the polynomials in (2) whend = 0, after multiplying by zb. Less-known in geometric modeling aretrapezoidalpatches, which correspond to the polynomials of (2) for general parameters a, b, d.Their blending functions and weights are given in Example 1.13.Corollary 2. The only toric surface patches possessing linear precision are tensorproduct patches, B´ezier triangles, and the trapezoidal patches of Example 1.13.The polynomials of Theorem 1(3) cannot arise in geometric modeling, for they arenot equivalent to a polynomial with positive coefficients.We remark that the notion of linear precision used here and in [5] is more restrictivethan typically used in geometric modeling. There, linear precision often means thatthere are control points in ∆ so that the resulting map ∆ → ∆ is the identity. Weinclude these control points in our definition of a patch to give a precise definitionthat enables the mathematical study of linear precision. Nevertheless, this restrictiveclassification will form the basis for a more thourough study of the general notion oflinear precision for toric patches.In Section 1, we review definitions and results from [5] about linear precision fortoric patches, including Proposition 1.4 which asserts that a toric patch has linearprecision if and only if a polynomial associated to the patch defines a toric polarCremona transformation, showing that Corollary 2 follows from Theorem 1. We alsoshow directly that polynomials associated to B´ezier triangles, tensor product patches,LINEAR PRECISION FOR TORIC SURFACE PATCHES 3and trapezoidal patches define toric polar Cremona transformations. In particular,this implies that trapezoidal patches have linear precision. In Section 2, we provethat the above equivalence preserves the property of defining a toric polar Cremonatransformation. Then we give our