Entities¶
Redacted Header¶
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,
where each \(M_{i}\) is an object associated with \(\mathcal{L}\). As with \(\tilde{E}\) and \(\tilde{F}\), we define \(\tilde{M} := \bigcup_{i} M_{i}\).
Redacted Header¶
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
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
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
where each \(Q_{i}\) is connected and \(i \neq j \implies Q_{i} \cap Q_{j} = \varnothing\).
Finally, we also similarly define
where \(Q_{i}^{*} := N_{\rho_{r}}(Q_{i})\) for each \(i\).
Note
By definition, we have
Since \(\mathcal{Q}\) is pairwise disjoint, it follows
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.
Redacted Header¶
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 \(S \in \mathcal{M}\).