====== Trilateration ======
<WRAP alert>
This method of interpreting hydrophone data has been deprecated in favor of [[cs::hydrophones:pinger_bearing:start| this method]].
</WRAP>

<WRAP info>
Before reading this page, make sure to check out the **Problem Setup** section of [[cs:hydrophones:trilateration_setup:start|this page]].
</WRAP>

This page is a summary of how we use the hydrophones to figure out our position.

Note that $\delta$, $\epsilon$, and $\zeta$ are defined as:


$h_0$ is at location $(0,0,0)$ \\
$h_x$ is at location $(\delta,0,0)$ \\
$h_y$ is at location $(0,\epsilon,0)$ \\
$h_z$ is at location $(0,0,\zeta)$ \\

The primary results from [[cs:hydrophones:trilateration_setup:start|this derivation]] are equations $\ref{eq:xyz}$ and $\ref{eq:p0_initial}$.
$$
\begin{equation} \label{eq:xyz} 
x = \frac{\Delta x (2p_0 - \Delta x) + \delta^2}{2 \delta} \\
y = \frac{\Delta y (2p_0 - \Delta y) + \epsilon^2}{2 \epsilon} \\
z = \frac{\Delta z (2p_0 - \Delta z) + \zeta^2}{2 \zeta} 
\end{equation}$$

$$ \begin{equation}\label{eq:p0_initial}
0 = p_0^2(a_x + a_y + a_z - 1) + p_0(b_x + b_y + b_z) + (c_x+c_y+c_z)
\end{equation}
$$

With variable definitions given by $\ref{eq:variable_definitions}$.

$$ \begin{equation} \label{eq:variable_definitions}
a_x = \left(\frac{\Delta x}{\delta}\right)^2 \qquad
b_x = \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) \qquad
c_x = \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\


a_y = \left(\frac{\Delta y}{\epsilon}\right)^2 \qquad 
b_y = \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) \qquad
c_y = \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\

a_z = \left(\frac{\Delta z}{\zeta}\right)^2 \qquad 
b_z = \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) \qquad
c_z = \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2
\end{equation}
$$

Let us simplify eq. $\ref{eq:p0_initial}$ using the following substitution:
$$
a = (a_x + a_y + a_z - 1) \\
b = (b_x + b_y + b_z) \\
c = (c_x+c_y+c_z)
$$

This gives us eq. $\ref{eq:p0_initial_simple}$, which is an ordinary quadratic equation.
$$ \begin{equation}\label{eq:p0_initial_simple}
0 = p_0^2 a + p_0 b + c
\end{equation}
$$

Applying the quadratic formula to eq. $\ref{eq:p0_initial_simple}$, we can solve for $p_0$.

$$ \begin{equation}\label{eq:p0_solved}
p_0 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\end{equation}
$$

This will give us two possible solutions for $p_0$. We can combine this result with eq. $\ref{eq:xyz}$ to solve for $x$, $y$, and $z$.

====== Reversing the Problem ======
Here we describe how the simulator takes the position of the sub and calculates fake hydrophone timing data. <WRAP todo>
Need figure this part out!
</WRAP>