polarzonoid-package: The Polar Zonoid Z_n in \mathbb{R}^{2n+1}
In polarzonoid: Compute Maps and Properties of Polar Zonoids

polarzonoid-package

R Documentation

The Polar Zonoid `Z_n` in `\mathbb{R}^{2n+1}`

Description

In each odd dimension is a special convex body - the polar zonoid - which is generated by trigonometric polynomials. The package has some applications of the polar zonoid, including the properties of spaces of arcs on the circle and 3x3 rotation matrices.

Introduction

A zonoid is a special type of convex body, see Bolker. Among its many properties, a zonoid is centrally symmetric. A zonoid has many equivalent definitions, but in this package a zonoid Z in \mathbb{R}^m is defined by m real-valued functions f_1,f_2,...,f_m on the circle \mathbb{S}^1. These functions are called the generators of Z. When the generators are piecewise constant, one obtains a zonotope. For the precise definition of a zonoid, see the User Guide vignette.

For an integer n \ge 0, we define the polar zonoid Z_n by taking the generators to be the 2n{+}1 functions:

\cos(\theta),\sin(\theta), \cos(2\theta),\sin(2\theta), ... ,\cos(n\theta),\sin(n\theta), 1 ~~~~~~~~ \theta \in [0,2\pi]

These functions are the standard basis of the trignometric polynomials. Note that it is convenient for us to put the constant function 1 last, instead of the usual convention of putting it first. The polar zonoid is a straighforward generalization of the polar zonohedron, see Chilton and Coxeter. In this paper, it is shown that as the number of sides of the polar zonohedron goes to \infty, the zonohedron converges to Z_1 \subseteq \mathbb{R}^3.

Let A_n be the space of n or fewer disjoint arcs in the circle. From properties of trigonometric polynomials, it can be shown that there is a natural homeomorphism A_n ~~ \rightleftarrows ~~ \partial Z_n. For a proof of this, including the definition of the topology of A_n, see the User Guide vignette. Among those properties is the fact that a trigonometric polynomial of degree n has at most 2n roots. It is clear that 2n roots define a set of n disjoint arcs, in two different ways.

In the special case n=0, we define A_0 to be the 2 improper arcs: the empty arc and the full circle. We have the inclusions:

A_0 \subseteq A_1 \subseteq A_2 \subseteq ... \subseteq A_n

Now the boundary \partial Z_n is trivially homemorphic to the sphere \mathbb{S}^{2n}. This is true for any convex body in \mathbb{R}^{2n+1}. Thus we have homeomorphisms:

A_n ~~ \rightleftarrows ~~ \partial Z_n ~~ \rightleftarrows ~~ \mathbb{S}^{2n}

where the symbol \rightleftarrows denotes a homeomorphism. The bulk of the API for this package is the numerical calculation of these maps. The maps that go from left to right are straightforward and implemented for all n. The inverse maps are much more complicated and, in this version of the package, are only implemented for n = 0,1,2,3. Since Z_n is determined by the single parameter n, there is no need to have an object for Z_n in the package API. The parameter n can be inferred from the dimension of vector and matrix function arguments, or in some cases can be given explicitly.

As a sanity check, note that n arcs have 2n endpoints, so we expect A_n to be a space of dimension 2n. The fact that it is a simple manifold like \mathbb{S}^{2n} is somewhat surprising. However, in the simple cases n = 0 and 1, it is easy to visualize; see the User Guide for details.