User:MWinter4/Wachspress coordinates

In mathematics and geometric modeling Wachspress coordinates are a type of generalized barycentric coordinates on convex polytopes. That is, for a point in a convex polytope with vertices , they produce a canonical way to express as a convex combination . They were initially defined by Wachspress in dimendion two, and subsequently generalized by Warren to convex polytopes of general dimension and combinatorics.

Wachspress coordinates are rational coordinates, that is, each coordinate is given as a rational function over the polytope:

where the and are polynomials and is required for normalization. Wachspress showed that generalized barycentric coordinates can in general not be polynomials, and so Wachspress coordinates are in a sense as simple as possible. In fact, Warren showed that they are the unique rational generalized barycentric coordinates of lowest possible degree. The degree of is exactly , where is the number of facets of the polytope, and is its dimension. The degree of is .

Wachspress coordinates are affine invariant, which is best seen from their definition via relative cone volumes.

Applications

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  • Positive geometry
  • Algebraic statistics
  • Finite element basis
  • ...

Properties

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Construction via cone volumes

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Assume that   contains the origin in its interior. To compute the Wachspress coordinates of the origin in the polytope let   be its polar dual. For a vertex   in  , let   be the facet of   dual to  , and   the cone over   with apex at  . The Wachspress coordinate   of the origin is the volume of this cone relative to the volume of the polar dual:

 

The cone volumes clearly add up to the volume of   and so  . To compute the Wachspress coordinates for any other interior point   of the polytope, perform the above computation for the translate  . Since relative volumes are affinely invariant, the Wachspress coordinates too are affinely invariant (i.e. they do not change if the polytope and the point are transformed by the same affine transformation).

Relation to Colin de Verdière matrices

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Suppose that   contains the origin in its interior. For a vector   the generalized polar dual is

 

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Wachspres variety

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The Wachspress coordinates describe a map from   to the standard simplex  . The image of this map is the graph of a rational function in   and hence an affine variety, the Wachspress variety. Its ideal is called the Wachspress ideal. The Wachspress variety is smooth (in  ) and of codimension  . It is cut out by   polynomials of degree  :

 

Wachspress map

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References

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