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 $$x$$ in a convex polytope $$P\subset\Bbb R^d$$ with $$n$$ vertices $$v_1,...,v_n$$, they produce a canonical way to express $$x$$ as a convex combination $$\alpha_1(x) v_1+\cdots + \alpha_n(x) v_n$$. 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:


 * $$\alpha_i(x)=\frac{p_i(x)}{q(x)},$$

where the $$p_i$$ and $$q$$ are polynomials and $$\textstyle q(x)=\sum_i p_i(x)$$ 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 $$p_i$$ is exactly $$m-d$$, where $$m$$ is the number of facets of the polytope, and $$d$$ is its dimension. The degree of $$q$$ is $$m-d-1$$.

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

Applications

 * Positive geometry
 * Algebraic statistics
 * Finite element basis

Construction via cone volumes
Assume that $$P$$ contains the origin in its interior. To compute the Wachspress coordinates of the origin in the polytope let $$P^\circ$$ be its polar dual. For a vertex $$v_i$$ in $$P$$, let $$F_i$$ be the facet of $$P^\circ$$ dual to $$v_i$$, and $$C_i$$ the cone over $$F_i$$ with apex at $$x$$. The Wachspress coordinate $$\alpha_i(0)$$ of the origin is the volume of this cone relative to the volume of the polar dual:


 * $$\alpha_i(0)=\frac{\mathrm{vol}(C_i)}{\mathrm{vol}(P^\circ)}=\frac{\mathrm{vol}_{d-1}(F_i)}{||v_i||\cdot \mathrm{vol}(P^\circ)}.$$

The cone volumes clearly add up to the volume of $$P^\circ$$ and so $$\alpha_1(0)+\cdots+\alpha_n(0)=1$$. To compute the Wachspress coordinates for any other interior point $$x$$ of the polytope, perform the above computation for the translate $$P-x$$. 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
Suppose that $$P$$ contains the origin in its interior. For a vector $$\mathbf c=(c_1,...,c_n)\in\Bbb R^n$$ the generalized polar dual is


 * $$P^\circ(\mathbf c)=\{x\in\mathbb R^d\mid \langle x,v_i\rangle\le c_i \text{ for all } i\in\{1,...,n\}\}$$

...


 * $$\mathrm{vol}(P^\circ(\mathbf c))\,=\,\mathrm{vol}(P) \,+\, (\mathbf c-\mathbf 1)^\top \tilde\alpha \,+\, (\mathbf c-\mathbf 1)^\top \tilde M(\mathbf c-\mathbf 1)\,+\,o(\|\mathbf c-\mathbf 1\|^2).$$

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