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<p><br><br></p>
<h1 id="introduction">Introduction</h1>
<p>The focus of this vignette is the <code>zonohedron()</code> constructor
and specifically its tolerance argument <code>e2</code>,
whose default value is <code>1.e-10</code>.</p>
<p>One goal of the <strong>zonohedra</strong> package is to handle all possible
zonogon facets, not just the parallelograms in the generic case.
The input to the constructor is matrix whose columns are the generators
of the zonohedron.
The generators of a specific facet span a plane, and adding another
generator increases the span to all of \(\mathbb{R}^3\).
Stated another way, the set of generators of a specific facet has rank 2,
and is maximal with respect to this property.
So a naive way of determining the facets is to examine <em>all</em> subsets
of the generators and determine whether each one [... truncated]
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<p><br><br></p>
<h1 id="introduction">Introduction</h1>
<p>The focus of this vignette are the two
functions <code>raytrace()</code> and <code>raytrace2trans()</code>.
The former is for the boundary of the zonohedron
and the latter is for the associated 2-transition surface.
We revisit the example at the end of section 6
in Scott Burns’ paper @Burns2021,
which is also illustrated in the 1nm plot from Figure 8.
His example is from colorimetry, where the boundary of the zonohedron
is the set of optimal colors
and the 2-transition surface is the set of Schrödinger colors
(both for Illuminant E).
The correspondence for the optimal colors was discovered by
Paul Centore, see @Centore2013.</p>
<p>Other featured functions are <code>invertboundary()</code>, <code>inside()</code> and <code>inside2trans()</code>.</p>
<pre><code class="language-r">l [... truncated]
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<p>\newcommand{\argmax}{\mathop{\mathrm{argmax}}\limits}
\newcommand{\max}{\mathop{\mathrm{max}}\limits}</p>
<p>\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{assumption}{Assumption}</p>
<p>\begin{theorem}
A test theorem
\end{theorem}</p>
<p>This vignette is a long-winded mathematical exposition on zonotopes
that concentrates on inversion.
Discussion of software is delayed until the end.
Featured functions are:
<code>zonoseg()</code>, <code>zonogon()</code>, <code>zonohedron()</code>, <code>invert()</code>, and <code>invertboundary()</code>.</p>
<h1 id="basic-concepts">Basic Concepts</h1>
<p>The emphasis in this vignette are the concepts needed to understand
the inversion functions in the <strong>zonohedra</strong> package.
Much of this is based on @Ziegler2012.</p>
<h2 id="supporting-hyperplanes">supporting hyperplanes</h2>
<p>A <em>supporting hyperplane</em> of a  [... truncated]
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