Techinical
==========

There are two main technical concerns in implementing the functions
defined previously.

- An unbounded region is not directly representable.
- Rounding errors need to be handled and/or avoided.

Unbounded region
----------------

Given polygon :math:`L`, let :math:`B` be its bounding rectangle,
defined by its two opposite points :math:`(x_{0}, y_{0})`
and :math:`(x_{1}, y_{1})` where :math:`x_{0} < x_{1}` and
:math:`y_{0} < y_{1}`.

Additionally, given :math:`e`, in turn, we define :math:`\hat{B}` as a rectangle
defined by points :math:`(x_{0} - 4 \rho_{e}, y_{0} - 4 \rho_{e})` and
:math:`(x_{1} + 4 \rho_{e}, y_{1} + 4 \rho_{e})`.

If we change the definition of :math:`\mathcal{Q}(\mathcal{T})` as

.. math::

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

Then we can define :math:`\hat{Q}` of :math:`\mathcal{T}` as
the one that includes the point :math:`u := (x_{0} - 2 \rho_{e}, y_{0} - 2 \rho_{e})`.

.. note::

   The use of constants :math:`2` and :math:`4` is to simplify expressions
   while avoiding possible rounding errors and/or edge cases.

Rounding errors
---------------

As for set relations between regions :math:`A, B \subseteq \mathbb{R}^{2}`,
it would be preferrable to have :math:`A \subseteq B^{\circ}` instead of
:math:`A \subseteq B` whenever possible.

Hence, in determining :math:`R_{k}`'s, we actually want as a condition

.. math::

   R_{k} \subseteq Q^{\circ}.

To avoid rounding errors, we further tweak the definitions as the following.

.. math::

   \begin{align}
     \mathcal{R}(\mathcal{T}) := &~ L \backslash (N_{\rho_{r} - \varepsilon}(\partial{L}) \cup N_{\rho_{r} - \varepsilon}(\tilde{M})) \\
     = &~ \{R_{1}, \cdots, R_{\sigma(\mathcal{R})}\} \\
     \mathcal{Q}(\mathcal{T}) := &~ \hat{B} \backslash (N_{\rho_{r} - 2 \varepsilon}(\tilde{F}) \cup N_{\rho_{r} - 2 \varepsilon}(\tilde{M})) \\
     = &~ \{Q_{1}, \cdots, Q_{\sigma(\mathcal{Q})}\}
   \end{align}

where :math:`\varepsilon` is some fixed threshold.

Although not strictly necessary, for each :math:`i` and :math:`j`,
we also alter the definitions of :math:`R_{i}^{*}`'s
and :math:`Q_{j}^{*}`'s to

.. math::

   \begin{align}
     R_{i}^{*} := &~ N_{\rho_{r} - 3 \varepsilon}(R_{i}) \\
     Q_{j}^{*} := &~ N_{\rho_{r} - 3 \varepsilon}(Q_{j}).
   \end{align}