Entities
========

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Throughout this text, unless otherwise specified,
any polygon :math:`P \subseteq \mathbb{R}^{2}` is assumed to be simple and closed.
We also assume all polygons have nonzero area.

We define :math:`\mathcal{L}` to be a tuple
:math:`\mathcal{L} := (L, \mathcal{E}, \mathcal{F})` where
:math:`L \subseteq \mathbb{R}^{2}` is a polygon and :math:`\mathcal{E}` and
:math:`\mathcal{F}` have the following properties:

* :math:`\mathcal{E} := \{E_{1}, \cdots, E_{\sigma(\mathcal{E})}\}` where

  * :math:`\forall i, E_{i}` is closed,
  * :math:`\forall i, E_{i} \subseteq \partial{L}`,
  * :math:`i \neq j \implies E_{i} \cap E_{j} = \varnothing`.

* :math:`\mathcal{F} := \{F_{1}, \cdots, F_{\sigma(\mathcal{F})}\}` where

  * :math:`\forall i, F_{i}` is closed,
  * :math:`\forall i, F_{i} \subseteq \partial{L}`,
  * :math:`i \neq j \implies F_{i} \cap F_{j} = \varnothing`.

* :math:`\tilde{E} \cup \tilde{F} = \partial{L}` where
  :math:`\tilde{E} := \bigcup_{i} E_{i}`
  and :math:`\tilde{F} := \bigcup_{j} F_{j}`.

* :math:`\forall i, j, |E_{i} \cap F_{j}| \in \mathbb{N}_{0}`.

That is, each :math:`E_{i}` and :math:`F_{i}` is a closed connected curve that
is a part of the boundary of :math:`L`.

.. note::

   Although the definitions of :math:`\mathcal{E}` and :math:`\mathcal{F}` are
   completely symmetric, and thus notation-wise interchangable, they are
   derived from some external information not relavent to this document.

.. note::

   Also, definition-wise, only one of :math:`\mathcal{E}` and
   :math:`\mathcal{F}` is necessary, but is included anyway for convenience
   of the following discourse.

We also define an object associated with :math:`\mathcal{L}` to be
any polygon :math:`M \subseteq L`.

Finally, let :math:`w_{r}, w_{e} \in \mathbb{R}` be such that
:math:`w_{r} \leq w_{e}`.

.. warning::

   The constraint that :math:`w_{r} \leq w_{e}` is an additional constraint for
   easing formalization, not a necessary condition enforced by any constraint.

Now, define :math:`\mathcal{T}` as a tuple, with some abuse of notation,

.. math::

   \begin{align}
   \mathcal{T} := &~ (\mathcal{L}, \mathcal{M}, w_{r}, w_{e}) \\
   = &~ (L, \mathcal{E}, \mathcal{F}, \mathcal{M}, w_{r}, w_{e}) \\
   = &~ (L, E_{1}, \cdots, E_{\sigma(\mathcal{E})}, F_{1}, \cdots, F_{\sigma(\mathcal{F})}, M_{1}, \cdots, M_{\sigma(\mathcal{M})}, w_{r}, w_{e}),
   \end{align}

where each :math:`M_{i}` is an object associated with :math:`\mathcal{L}`.
As with :math:`\tilde{E}` and :math:`\tilde{F}`, we define
:math:`\tilde{M} := \bigcup_{i} M_{i}`.

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Let :math:`\mathcal{T}, w_{r},` and :math:`w_{e}` be given. Let
:math:`\rho_{r} := \frac{1}{2} w_{r}` and :math:`\rho_{e} := \frac{1}{2} w_{e}`.
We define

.. math::

   \begin{align}
     \mathcal{R}(\mathcal{T}) := &~ L \backslash (N_{\rho_{r}}(\partial{L}) \cup N_{\rho_{r}}(\tilde{M})) \\
     = &~ \{R_{1}, \cdots, R_{\sigma(\mathcal{R})}\}
   \end{align}

where each :math:`R_{i}` is connected and
:math:`i \neq j \implies R_{i} \cap R_{j} = \varnothing`.

.. note::

   It is possible for :math:`R_{i}` *not* to be simply connected.

We also define

.. math::

   \mathcal{R}^{*}(\mathcal{T}) := \{R_{1}^{*}, \cdots, R_{\sigma(\mathcal{R})}^{*}\}

where :math:`R_{i}^{*} := N_{\rho_{r}}(R_{i})` for each :math:`i`.

.. note::

   It is possible to have :math:`i \neq j` where :math:`R_{i}^{*} \cap R_{j}^{*} \neq \varnothing`.

Similarly, we define

.. math::

   \begin{align}
     \mathcal{Q}(\mathcal{T}) := &~ \mathbb{R}^{2} \backslash (N_{\rho_{r}}(\tilde{F}) \cup N_{\rho_{r}}(\tilde{M})) \\
     = &~ \{Q_{1}, \cdots, Q_{\sigma(\mathcal{Q})}\}
   \end{align}

where each :math:`Q_{i}` is connected and :math:`i \neq j \implies Q_{i} \cap Q_{j} = \varnothing`.

Finally, we also similarly define

.. math::

   \mathcal{Q}^{*}(\mathcal{T}) := \{Q_{1}^{*}, \cdots, Q_{\sigma(\mathcal{Q})}^{*}\}

where :math:`Q_{i}^{*} := N_{\rho_{r}}(Q_{i})` for each :math:`i`.

.. note::

   By definition, we have

   .. math::

      \forall i, \exists ! j ~\text{such that}~ R_{i} \subseteq Q_{j}.

   Since :math:`\mathcal{Q}` is pairwise disjoint, it follows

   .. math::

      \forall i, \exists ! j ~\text{such that}~ R_{i} \cap Q_{j} \neq \varnothing.

Since :math:`L` is bounded, :math:`\exists !\hat{Q} \in \{Q_{1}, \cdots, Q_{\sigma(\mathcal{Q})}\}`
such that :math:`A(\hat{Q}) = \infty`.

As long as the context is clear, we may simply denote these as
:math:`\mathcal{R}, \mathcal{R}^{*}, \mathcal{Q},` and :math:`\mathcal{Q}^{*}`.

.. attention::

   **Note redacted.**

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We define :math:`\mathcal{P}` as a tuple of polygons
:math:`\mathcal{P} := (S, C)`, where
:math:`S \subseteq \mathbb{R}^{2}` and
:math:`C \subseteq \mathbb{R}^{2}` of :math:`\mathcal{P}`.

We say :math:`\mathcal{P}` is in :math:`\mathcal{T}` if
:math:`S \in \mathcal{M}`.