Entities

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

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

  • \(\mathcal{E} := \{E_{1}, \cdots, E_{\sigma(\mathcal{E})}\}\) where

    • \(\forall i, E_{i}\) is closed,

    • \(\forall i, E_{i} \subseteq \partial{L}\),

    • \(i \neq j \implies E_{i} \cap E_{j} = \varnothing\).

  • \(\mathcal{F} := \{F_{1}, \cdots, F_{\sigma(\mathcal{F})}\}\) where

    • \(\forall i, F_{i}\) is closed,

    • \(\forall i, F_{i} \subseteq \partial{L}\),

    • \(i \neq j \implies F_{i} \cap F_{j} = \varnothing\).

  • \(\tilde{E} \cup \tilde{F} = \partial{L}\) where \(\tilde{E} := \bigcup_{i} E_{i}\) and \(\tilde{F} := \bigcup_{j} F_{j}\).

  • \(\forall i, j, |E_{i} \cap F_{j}| \in \mathbb{N}_{0}\).

That is, each \(E_{i}\) and \(F_{i}\) is a closed connected curve that is a part of the boundary of \(L\).

Note

Although the definitions of \(\mathcal{E}\) and \(\mathcal{F}\) 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 \(\mathcal{E}\) and \(\mathcal{F}\) is necessary, but is included anyway for convenience of the following discourse.

We also define an object associated with \(\mathcal{L}\) to be any polygon \(M \subseteq L\).

Finally, let \(w_{r}, w_{e} \in \mathbb{R}\) be such that \(w_{r} \leq w_{e}\).

Warning

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

Now, define \(\mathcal{T}\) as a tuple, with some abuse of notation,

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

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

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

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

where each \(R_{i}\) is connected and \(i \neq j \implies R_{i} \cap R_{j} = \varnothing\).

Note

It is possible for \(R_{i}\) not to be simply connected.

We also define

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

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

Note

It is possible to have \(i \neq j\) where \(R_{i}^{*} \cap R_{j}^{*} \neq \varnothing\).

Similarly, we define

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

where each \(Q_{i}\) is connected and \(i \neq j \implies Q_{i} \cap Q_{j} = \varnothing\).

Finally, we also similarly define

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

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

Note

By definition, we have

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

Since \(\mathcal{Q}\) is pairwise disjoint, it follows

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

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

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

Attention

Note redacted.

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

We say \(\mathcal{P}\) is in \(\mathcal{T}\) if \(S \in \mathcal{M}\).