---
title: "Polarzonoid User Guide"
author: "Glenn Davis"
date: "`r Sys.Date()`"


output: 
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 2
    number_sections: true

header-includes:
  - \newcommand{\abs}[1]{\lvert #1 \rvert}
  - \newcommand{\Vw}{\large\mathscr{V} \normalsize_w}  
    
bibliography: bibliography.bib

# csl: iso690-numeric-brackets-cs.csl
csl: personal.csl
# csl: institute-of-mathematical-statistics.csl
# csl: transactions-on-mathematical-software.csl
vignette: >
  %\VignetteIndexEntry{Polarzonoid User Guide}
  %\VignetteEngine{knitr::rmarkdown}
---

<script type="text/x-mathjax-config">
MathJax.Hub.Config({ TeX: { equationNumbers: {autoNumber: "all"} } });
</script>


```{css, echo=FALSE}
body {
  max-width: 725px;     /* make wider, default is 700px */
}
  h1{
  font-size: 20pt;    /* make the level 1 headers smaller */
}
```


```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
old_opt = options( width=144 )

require("rgl",quietly=TRUE)
rgl::setupKnitr(autoprint = TRUE)
```

<br><br>



# Introduction

```{r, echo=FALSE, message=FALSE}
library(polarzonoid)
library(zonohedra,quietly=TRUE)
```

The mail goal of this vignette is to show that there are homeomorphisms
\begin{equation*}
A_n  ~~ \longleftrightarrow  ~~ \partial Z_n ~~ \longleftrightarrow  ~~ \mathbb{S}^{2n}
\end{equation*}
Where $A_n$ is a space of arcs and $\partial Z_n$ is the boundary of a _polar zonoid_.
Both of these spaces are defined below.

In this User Guide we start with very general zonoids,
and after two sections specialize to the polar zonoid.



<br><br>

#  Zonoids with Generating Functions

For this package we only consider zonoids defined by generating functions in $L^1( \mathbb{S}^1 )$,
where the unit circle $\mathbb{S}^1$ has the usual Lesbesgue measure.
I do not know whether more exotic zonoids exist.
For a very good survey of the theory of zonoids, see @Bolker1969.

Let $w_1, w_2, ..., w_m$ be absolutely integrable functions from $\mathbb{S}^1$ to $\mathbb{R}$,
i.e. each $w_i \in L^1( \mathbb{S}^1 )$.
For $\theta \in \mathbb{S}^1$, let $w(\theta) := (w_1(\theta), ... , w_m(\theta)) \in \mathbb{R}^n$.
The letter $w$ denotes "weight".

Define the zonoid $Z_w$ by
\begin{equation}\label{eqn:defZ1}
Z_w := \left\{ \int_V  w(\theta) \, d\theta ~|~ V \subseteq \mathbb{S}^1  \right\}
~~~=~~~
\left\{ \int_{\mathbb{S}^1} \mathbf{1}_{V}(\theta) w(\theta) \, d\theta ~|~ V \subseteq \mathbb{S}^1  \right\}
\end{equation}


where $V$ varies over all measurable subsets of $\mathbb{S}^1$.
The function $\mathbf{1}_{V}$ is the indicator function of $V$.
The functions $w_1(\theta), ... , w_m(\theta)$ are the _generating functions_ of $Z_w$,
and $w(\theta)$ is the _generating function_.

The function $w(\theta)$ induces a vector measure on the circle:
$\mu_w(V) :=  \int_V w(\theta) \, d\theta$.
This measure is _atom free_.
The measure viewpoint gives a simpler alternative definition:
\begin{equation}\label{eqn:defZ2}
Z_w  := \left\{ \mu_w(V) ~|~ V \subseteq \mathbb{S}^1  \right\}
\end{equation}

This is the classical definition of a general zonoid, except with $\mu_w$ replaced by
an arbitrary atom free vector measure $\mu$.
Given a zonoid $Z_\mu$ which is the range of $\mu$, the measure $\mu$ is not unique.
It was shown in @Rickert1967 that two zonoids are equal if and only if
the total measures of the corresponding spaces are equal,
and certain induced measures on the projective space $\mathbb{R}\text{P}^{n-1}$ are equal.
This was reproved in @Bolker1969.
By a classical theorem of Lyapunov, a zonoid is convex and closed, see Theorem 5.5 of @RudinFA.

The generating function $w$ is far from unique;
for most $w$, with very few exceptions, it is easy to find many $w' \neq w$
for which $Z_{w'} = Z_w$.

Now let $Q^\infty$ be the set of measurable functions on $\mathbb{S}^1$ whose essential range
is in $[0,1]$.
In fancier terms, $Q^\infty$ is the non-negative part of the unit ball in  $L^\infty( \mathbb{S}^1 )$.
The indicator functions $\mathbf{1}_{V}$ are in $Q^\infty$;
in fact $Q^\infty$ is like a cube and the indicator functions $\mathbf{1}_{V}$
are the vertices of that cube.
It is an amazing fact that, in equation (\ref{eqn:defZ1}),
if the vertices $\mathbf{1}_{V}$ are replaced by arbitrary $f \in Q^\infty$
the resulting set does not get any larger !
Therefore we also have:
\begin{equation}
Z_w  = \left\{ \int_{\mathbb{S}^1} f(\theta) w(\theta) \, d\theta ~|~ f \in Q^\infty  \right\}
\end{equation}

This is another classic theorem of Lyapunov;
for the proof, see Theorem 5.5 in @RudinFA.
In control theory this amazing fact is related to the "bang-bang principle".


**SIDE NOTE:**
The previous concepts actually have a physical application in colorimetry.
The variable $\theta$ is wavelength.
The function $f$ is reflectance, which must be between 0 and 1.
The functions $w_1(\theta), w_2(\theta), w_3(\theta)$ are responsivities for
short, medium, and long wavelength bands.
$Z_w$ is the _object color solid_.
In reference @Koenderink, which is a scientific book on colorimetry,
the theorem of Lyapunov and the "bang-bang principle"
are mentioned in Chapter 7 Notes 4 and 5, page 321.
The first bang is 0, and the second bang is 1.


<br><br>

#  The Boundary of Zonoids with Generating Functions

We want to derive an expression similar to (\ref{eqn:defZ2}),
but for $\partial Z_w$ instead of $Z_w$.

Let $c = (c_1,...,c_m)$ be a fixed non-zero vector and consider maximizing the inner product
of $\langle c , z \rangle$ over all $z \in Z_w$.
By (\ref{eqn:defZ2}) any such $z$ is $\mu_w(V)$ for some $V \subset \mathbb{S}^1$ and we have:
\begin{equation}
\left\langle  c, \mu_w(V) \right\rangle
~=~ \left\langle  c, \int_V  w(\theta) \, d\theta \right\rangle
~=~  \int_V \left\langle  c,  w(\theta) \right\rangle  \, d\theta
~=~  \int_V c_1 w_1(\theta) + ... +  c_m w_m(\theta)  \, d\theta
\end{equation}
This expression is clearly maximized when $V$ is the set of $\theta$
where the combination $c_1 w_1(\theta) + ... +  c_m w_m(\theta)$ is non-negative.
Denote this set by:
\begin{equation}\label{eqn:Vc}
V_c  :=  \left\{  \theta ~|~ c_1 w_1(\theta) + ... +  c_m w_m(\theta) \ge 0  \right\}
\end{equation}
Since the maximum of the inner product must be taken on $\partial Z_w$,
choosing this $V_c$ means that $\mu_w(V_c) \in \partial Z_w$.
Note that $V_c$ might be empty, and it might be the entire circle.

To proceed further, we require from now on  that $w$ has this simplifying property:

**Property S**.
Every non-trivial linear combination $c_1 w_1(\theta) + ... +  c_m w_m(\theta)$
is non-zero for almost all $\theta$.

Think of this property as a very strong form of linear independence.
Not only is the linear combination non-zero on a set of positive measure,
it is non-zero at almost every point.

As a consequence of Property S, the inner product with $c$
is _uniquely_ maximized by $\mu_w(V_c)$.
In other words, if $V \neq V_c$ then
$\langle c , \mu_w(V) \rangle < \langle c , \mu_w(V_c) \rangle$.
In the terminology of convex bodies, every _face_ of $Z_w$ is single point.

Now define
\begin{equation}
\Vw ~:=~ \left\{ V_c ~|~ c \ne 0 \} ~=~ \{ V_c ~|~ c \in \mathbb{S}^{2n} \right\}
\end{equation}
$\Vw$ is a family of subsets of $\mathbb{S}^1$ that depends only on $w$.
The last equality is true because multiplying $c$ by a positive scalar does not change $V_c$.
We make $\Vw$ into a topological space by identify $V_c$ with its
indicator function $\mathbf{1}_{V_c} \subset L^1( \mathbb{S}^1 )$.
So we can think of $\Vw$ as a subset of $L^1( \mathbb{S}^1 )$ and let $\Vw$ inherit the topology
from the bigger space.


**Theorem A:**
If $w$ has Property S, then the map
\begin{equation}\label 
\Vw \to \partial Z_w
\end{equation}
given by  $V_c \mapsto \mu_w(V_c)$, is a homeomorphism.


**Proof:**
The map $V_c \mapsto \mu_w(V_c) = \int_{V_c} w \, d\theta$ is clearly continuous.
Recall that by Property S, if $V \neq V_c$,
then $\langle c , \mu_w(V) \rangle < \langle c , \mu_w(V_c) \rangle$.

First the surjective part.
Let $b \in \partial Z_w$.
Since $b \in Z_w$, $b = \mu_w(V)$ for some $V \subset \mathbb{S}^1$.
Since $Z_w$ is convex, there is a supporting hyperplane of $Z_w$ at $b$
with non-zero outward-pointing normal $c$.
if $V \neq V_c$, then
$\langle c , b \rangle   ~=~  \langle c , \mu_w(V) \rangle < \langle c , \mu_w(V_c) \rangle$.
But point $b$ maximizes the inner product with $c$.
This is a contradiction and so $V = V_c$ and $b = \mu_w(V_c)$.

Now the injective part.
Let $V_c,V_{c'} \in \Vw$.
If $V_{c'} \neq V_c$, then
$\langle c , \mu(V_{c'}) \rangle  <  \langle c , \mu(V_c) \rangle$
and so $\mu(V_{c'}) \neq  \mu(V_c)$.
$\square$

Without Property S, the conclusion of the theorem is definitely false.
For example, if the range of $w$ is finite, i.e. if $w$ is a _simple function_,
then $\Vw$ is finite too.
But $\partial Z_w$ is obviously not.
In this case the zonoid is a _zonotope_ - a zonoid which is also a polytope.
Many equivalent conditions for zonotopes are discussed by Bolker in
Theorem 3.3 of @Bolker1969.
The image of the map in the theorem is simply the vertices of $\partial Z_w$.
We hope to discuss _polar zonotopes_ in a future version of this package.

We summarize with an expression for $\partial Z_w$.
If $w$ has Property S, then
\begin{equation}\label{eqn:bZw}
\partial Z_w  = \left\{ \int_{V_c} w(\theta)  \, d \theta  ~|~ c \neq 0  \right\} ~=~ \left\{ \mu_w(V_c) ~|~ V_c \in \Vw \right\}
\end{equation}
Compare this expression with (\ref{eqn:defZ2}).
In that expression for $Z_w$, $V$ ranges over _all_ $V \subset \mathbb{S}^1$,
but for $\partial Z_w$ only the sets $V_c$ are used.

Given a particular generating function $w$,
the first step in describing $\partial Z_w$ is to describe the family $\Vw$
of subsets of $\mathbb{S}^1$.
In the next section we do this for a particularly simple and special $w$.


<br><br>

#  Trigonometric Polynomials and The Polar Zonoid

We are finally ready to define the polar zonoid.
Fix $n \ge 1$ and let $w(\theta)$ be given by the $m = 2n+1$ functions:
\begin{equation}\label{eqn:w}
w(\theta) := ( \cos \theta, \sin \theta, \cos 2\theta, \sin 2\theta, ..., \cos n\theta, \sin n\theta, 1)
\end{equation}
These $m$ functions are a basis for the _trigonometric polynomials_ of order $n$.
From now on in this vignette, $w$ means this particular function.

We call the generated zonoid the _polar zonoid_ of order $n$ in $\mathbb{R}^{2n+1}$
and denote it by $Z_n$.
To describe $\partial Z_n$, by (\ref{eqn:Vc}) and (\ref{eqn:bZw}) we must describe the sets
\begin{equation}
V_{a,b} := 
\{ \theta ~|~ a_1\cos \theta + b_1 \sin \theta  ~+  ... ~+ a_n \cos n\theta + b_n \sin n\theta + a_0 \ge 0 \}
\end{equation}
where not all $a_i$ and $b_i$ are 0.
Note we have switched from $c_i$ to $a_i$ and $b_i$ because this is the classical
notation for trigonometric polynomials.
We also put the constant term last instead of first,
because we think of the last coordinate as "vertical" and the others as "horizontal".

For later use, we state without proof that the product of two trigonometric polynomials
of order $n_1$ and $n_2$ is a trigonometric polynomial of order $n_1 + n_2$.

**NOTE:**
The term _polar zonoid_ was chosen because it is a generalization
of the _polar zonohedron_, see @Chilton1963.
_Warning:_ In other contexts, i.e. the theory of convex bodies,
the term "polar zonoid" is short for "polar of a zonoid",
which is an entirely different concept.

It is well-known that the trignometric polynomials have most $2n$ roots,
see Chapter X, Theorem 1.7 in @Zygmund2.
Thus Property S is trivially true, and the previous Theorem applies.
Since the trigonometric polynomials are continuous,
we can conclude that each set $V_{a,b}$ is the union of $n$ or fewer arcs
(possibly no arcs at all) in the circle, _or_ the entire circle.

Conversely, given any $n$ disjoint closed arcs,
there is a trigonometric polynomial of order $n$
whose roots are the $2n$ endpoints of those arcs,
and is non-negative on those arcs.

We conclude that for the above $w$ of trigonometric functions:
\begin{equation}
\Vw = \text{ the space of } n \text{ or fewer pairwise disjoint closed arcs in the circle } \mathbb{S}^1
\end{equation}

We investigate families of arcs in the next section.



<br><br>

# Spaces of Arcs $A_n$

*Definition:* $A_n$ is the space of $n$ _or fewer_ pairwise disjoint closed arcs in the circle $\mathbb{S}^1$.

$A_n$ is the same as $\Vw$ in the previous section,
and from now on we use this new notation.

For example, here are 2 plots of arcs on the circle, the left with 1 arc and the right with 3 arcs.

```{r, echo=TRUE,  message=TRUE,  warning=TRUE, fig.width=8, fig.height=4, fig.cap='Figure 3.0 examples of collections of pairwise disjoint arcs', out.width="95%", cache=FALSE }

oldpar = par( mfrow=c(1,2)  , omi=c(0.1,0.1,0.1,0.1), mai=c(0.45,0.5,0.1,0) )
arcmat = matrix( c(1.5,2.9), nrow=1, ncol=2, byrow=TRUE )
plotarcs( arcmat, main=NA )

arcmat = matrix( c(1.5,2, pi,0.5, 5,pi/4 ), nrow=3, ncol=2, byrow=TRUE )
plotarcs( arcmat, main=NA ) ;  par( oldpar )
```


Both of these represent *points* in $A_3$.
The left plot also represents a *point* in $A_1$.
We define $A_0$ to be the set of 2 _improper_ arcs: the empty arc, and the full circle.

We have proper inclusions:
\begin{equation}
A_0 \subset A_1  \subset A_2 \subset A_3  \subset A_4  ~~~ ...
\end{equation}

We make $A_n$ into a topological space by identify a set of disjoint arcs $V$ with its
indicator function $\mathbf{1}_{V} \subset L^1( \mathbb{S}^1 )$.
We think of $A_n$ as a subset of this $L^1$
and let $A_n$ inherit the topology from the bigger space.

In more detail, $A_n$ is a metric space with metric defined as follows.
Let $V$ and $W$ be sets of arcs in $A_n$, so $V , W \subset \mathbb{S}^1$.
Define the metric
$d(V,W) := \int_{\mathbb{S}^1} \lvert \mathbf{1}_{V} - \mathbf{1}_{W} \rvert d\theta$.

The metric can be viewed in another way; the symmetric difference of $V$ and $W$ is
also a collection of arcs, and $d(V,W)$ is the total length of the arcs in this
symmetric difference.
The corresponding functions in the package are `arcsdistance()` and `arcssymmdiff()`.
With this choice of metric, an arc of length 0 is considered to be the same as the empty arc,
but it _is_ allowed in the software.

There is a useful relationship between the uniqueness of supporting hyperplanes and
the number of arcs.

**Theorem:**
For $V \in A_n$, $V$ has exactly $n$ arcs if and only if the supporting
hyperplane of $Z_n$ at $\mu_w(V)$ is unique.

**Proof:**
Suppose $V \in A_n$ has exactly $n$ arcs, i.e. that $V \in A_n - A_{n-1}$.
Let $T(\theta)$ be the trigonometric polynomial of order $n$ with roots at the $2n$ endpoints,
and orientation that matches $V$.
Then all the roots of $T(\theta)$ of simple which means that $T(\theta)$ is unique
up to a positive multiple.
The coefficients of $T(\theta)$ are the normal to supporting hyperplane and this
implies that the hyperplane is unique.

Suppose that $V \in A_n$ has exactly $n'$ arcs with $n' < n$.
The corresponding polynomial $T(\theta)$ has order $n'$ with $2n'$ roots.
Take a trignometric polynomial $T_\delta(\theta)$ of order $n - n'$ that is everywhere positive
and not constant. 
This is easy - just take all $a_i$ and $b_i$ small and non-zero for $i> 0$,
and $a_0$ sufficiently large.
Then the product of $T(\theta)$ and $T_\delta(\theta)$ is order $n$
and has exactly the same arcs.
The coefficients of the product are a normal vector in a different direction
and thus there is another supporting hyperplane.
$\square$.

The theorem says that if a point in $\partial Z_n$ is the image of a point
in $A_n - A_{n-1}$, then $\partial Z_n$ is smooth at that point.
But if it is the image of a point in $A_{n-1}$, then  $\partial Z_n$ has a "crease" there.

For example, when $n=1$, $\partial Z_1$ is homeomorphic to $\mathbb{S}^2$
and it has cusps at 2 "poles" corresponding to the 2 points in $A_0$.
An approximation of it is illustrated in this figure:
```{r,  echo=TRUE,  message=TRUE,  warning=TRUE, fig.align='center', fig.width=7.5, fig.height=5, fig.cap='polar zonohedron with 25 generators &emsp;&emsp; [this is an interactive WebGL widget, try it !]',   fig.keep='none', fig.show='hide', out.width="100%", cache=FALSE }
plot( zonohedra::polarzonohedron( 25, height=2*pi ), type='f', falpha=1  )
rgl::arc3d( 1.005*c(2,0,pi), 1.005*c(0,2,pi), center=c(0,0,pi), base=c(0,1), color='red', lwd=3 )
rgl::rglwidget( webgl=TRUE )
```
From Theorem A there is a homeomorphism $A_n  \to  \partial Z_n$.
For $n{=}1$, the description of it is fairly easy.
For $n{=}1$ there is a single arc in the circle.
The center of the arc maps to the "longitude" of the boundary $\partial Z_1$,
and the length of the arc maps to the "latitude".
Length=0 (the empty arc) goes to the black cusp $(0,0,0)$ at the "south pole"
and Length=$2\pi$ (the full circle) goes to the white cusp $(0,0,2\pi)$ at the "north pole".
Semicircles, where Length=$\pi$, go to the red "equator".
This is a plot of a polar zonohedron, with $M=25$ generators,
that approximates the true polar zonoid.
As $M \to \infty$, the polar zonohedron converges to the polar zonoid,
in the Hausdorff metric.
This is shown at the end of @Chilton1963.
The surface is the surface of revolution of the curve $2\sin(z/2)$.

**SIDE NOTE:**
These investigations were largely inspired by a
colorimetry paper by P. Centore @Centore2018,
which started me thinking about the space $A_2$.



<br><br>

# The Homeomorphism $A_n \to \partial Z_n$

From **Theorem A** we have a homemomorphism:
\begin{equation}
A_n  \to  \partial Z_n
\end{equation}
given by $V \mapsto \int_V w(\theta) \, d\theta$ and $w(\theta)$ is given by (\ref{eqn:w}).
The goal of this section is to compute simple expressions for this mapping.

Let all arcs be oriented in the counter-clockwise direction.
Take a single arc starting at $\theta_0$ and with length $L$,
and denote it by $[\theta_0,\theta_0+L]$.
For simplicity, compute the map for only 3 coordinate functions of $w(\theta)$:
$\left( \cos k\theta, \sin k\theta, 1 \right)$.
\begin{align*}
[\theta_0,\theta_0{+}L] &\mapsto \left( \int_{\theta_0}^{\theta_0+L} \cos k\theta \, d\theta , \int_{\theta_0}^{\theta_0{+}L} \sin k\theta \, d\theta  , \int_{\theta_0}^{\theta_0+L} 1 \, d\theta  \right) \\
&\mapsto  \left( (\sin( k(\theta_0{+}L) ) - \sin(k\theta_0) )/k ) ,  (-\cos( k(\theta_0{+}L) ) + \cos(k\theta_0) )/k )  , L \right)
\end{align*}
Expressions like these are used in the function `boundaryfromarcs()`.

It is far simpler to change the first 2 coordinates to complex numbers:
\begin{align*}
[\theta_0,\theta_0{+}L] &\mapsto \left( \int_{\theta_0}^{\theta_0+L} e^{ik\theta} \, d\theta ,  \int_{\theta_0}^{\theta_0+L} 1 \, d\theta  \right)  \\
&\mapsto  \left( (e^{ik(\theta_0{+}L)} - e^{ik\theta_0} ) / (ik)  , L \right)
\end{align*}
We leave it to the reader to check that the real and complex expressions are equivalent.

Now take the complex number representation a little further.
Let the same arc start at $u$ and go counterclockwise to $v$ with length $L$,
where $u$ and $v$ are complex points on the unit circle.
So that $u = e^{i\theta_0}$ and $v = e^{i(\theta_0{+}L)} = e^{i\theta_0} e^{iL}$.
Now denote the same arc by $[u,v]$.
Making these substitutions gives:
\begin{equation}
[u,v] \mapsto \left( \frac{v^k - u^k}{ik} , L \right)
\end{equation}
If we have a disjoint union of 2 arcs $V = [u_1,v_1] \sqcup [u_2,v_2]$
with lengths $l_1$ and $l_2$, then we take the sum.
\begin{equation}
[u_1,v_1] \sqcup [u_2,v_2] \mapsto \left( \frac{(v_1^k - u_1^k) + (v_2^k - u_2^k)}{ik} , l_1 + l_2 \right)
\end{equation}

If we denote the other coordinates on the right side by $z_k$ as in:
\begin{equation}
[u_1,v_1] \sqcup [u_2,v_2] \mapsto \left( z_1, ..., z_n , l_1 + l_2 \right)
\end{equation}
then we identify points on the right side with points in $\mathbb{R}^{2n+1}$,
and we have:
\begin{equation}
z_k = \frac{(v_1^k - u_1^k) + (v_2^k - u_2^k)}{ik} \hspace{20pt} k = 1,...,n
\end{equation}
The generalization to more than 2 disjoint arcs is straightforward.
These equations are used extensively in the functions `boundaryfromsphere()`
and `arcsfromboundary()`,
see the [Implicitization](implicitization.html) vignette.



<br><br>

# The Homeomorphism $\partial Z_n \to  \mathbb{S}^{2n}$

$Z_n$ is a convex body so it is homeomorphic to the ball $B^{2n+1}$,
and its boundary is homeomorphic to $\mathbb{S}^{2n}$.
An explict map is trivial to construct.
The center of $Z_n$ is $c = (0,...0,\pi)$ in the interior of $Z_n$.
For $b \in \partial Z_n$ use the projection onto the unit sphere
centered at $c$: $b \mapsto (b - c)/||b -c|| \in\mathbb{S}^{2n}$.
This is computed in the function `spherefromboundary()`.

The inverse $\mathbb{S}^{2n} \to \partial Z_n$ is much harder since now we
are given a ray based at $c$ and must compute where that ray intersects $\partial Z_n$.
This is computed iteratively in `boundaryfromsphere()`
and depends on an implicit function for $\partial Z_n$,
for details see the [Implicitization](implicitization.html) vignette.
This function is only implemented for $n$ = 1, 2, and 3.

The space of arcs $A_n$ has a natural involution $\gamma$,
which is the complementation operation, given by
$\gamma(V) = \operatorname{Cl}(\mathbb{S}^1 - V)$.
The closure $\operatorname{Cl}$ is only necessary to create closed arcs.
Note that if $V$ has $k$ disjoint closed arcs, then so does $\gamma(V)$.
This involution is conjugate to the antipodal map on $\mathbb{S}^{2n}$
under the composition $A_n \to \partial Z_n \to \mathbb{S}^{2n}$.
These properties are discussed further in the
[Real Projective Spaces and 3x3 Rotation Matrices](rotations.html) vignette.


<br><br>

# References

<div id="refs"></div>



<br><br>

# Appendix - Logging

Logging is performed using the package **logger**, see @logger.
This is a powerful package that allows a separate configuration
for logging from within **polarzonoid**, and that is what I have done.
During package loading, the logging threshold is changed from `INFO` to `WARN`.
To change it back again, one can execute:  
`log_threshold( INFO, namespace="polarzonoid" )`

The layout callback functions is customized;
it adds the name of the calling function to the message.
To install your own layout function, you can execute:  
`log_layout( <your function>, namespace="polarzonoid" )`

The appender callback functions is also customized;
it comes to an immediate stop if the message level is `ERROR` or `FATAL`.
To return to the default behavior, you can execute:  
`log_appender( appender_console, namespace="polarzonoid" )`

The formatter callback function is forced to be `formatter_sprintf()`;
this should not be changed.



<br><br>

# Session Information

This document was prepared
`r format(Sys.Date(), "%a %b %d, %Y")`
with the following configuration:
<pre>
```{r, echo=FALSE, results='asis'}
options(old_opt)
sessionInfo()
```
</pre>
