Projective Geometry of Wachspress Coordinates
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Projective Geometry of Wachspress Coordinates Kathlén Kohn1 · Kristian Ranestad2 Received: 9 September 2019 / Revised: 9 September 2019 / Accepted: 8 October 2019 © The Author(s) 2019
Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic. Keywords Polytopes · Hyperplane arrangements · Barycentric coordinates · Wachspress varieties · Adjoint hypersurfaces · Segre classes of monomial schemes · Uniform probability distributions
Communicated by Teresa Krick.
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Kathlén Kohn [email protected] Kristian Ranestad [email protected]
1
Institutionen för Matematik, KTH, Lindstedtsvägen 25, 10044 Stockholm, Sweden
2
Matematisk institutt, Universitetet i Oslo, PB 1053, Blindern, 0316 Oslo, Norway
123
Foundations of Computational Mathematics
Mathematics Subject Classification 14J70 · 52B10 · 52B11 · 52C35 · 14N20 · 14N30
1 Introduction Barycentric coordinates on convex polytopes have many applications, such as mesh parameterizations in geometric modeling, deformations in computer graphics, or polyhedral finite element methods. Whereas barycentric coordinates are uniquely defined on simplices, there are different versions of barycentric coordinates on more general convex polytopes. For instance, mean value coordinates and Wachspress coordinates are both commonly used in practice. A nice overview on different versions of barycentric coordinates, their history, and their applications is [4]. This article provides an algebro-geometric study of Wachspress coordinates, which were first introduced for polygons by Wachspress
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