Trilateration

This method of interpreting hydrophone data has been deprecated in favor of this method.

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 this derivation are equations $\ref{eq:xyz}$ and $\ref{eq:p0_initial}$. $$$$\label{eq:xyz} x = \frac{\Delta x (2p_0 - \Delta x) + \delta^2}{2 \delta} \\ y = \frac{\Delta y (2p_0 - \Delta y) + \epsilon^2}{2 \epsilon} \\ z = \frac{\Delta z (2p_0 - \Delta z) + \zeta^2}{2 \zeta}$$$$ $$$$\label{eq:p0_initial} 0 = p_0^2(a_x + a_y + a_z - 1) + p_0(b_x + b_y + b_z) + (c_x+c_y+c_z)$$$$ With variable definitions given by $\ref{eq:variable_definitions}$.

$$$$\label{eq:variable_definitions} a_x = \left(\frac{\Delta x}{\delta}\right)^2 \qquad b_x = \frac{\Delta x}{\delta^2}(\delta^2 -\Delta x^2) \qquad c_x = \left(\frac{\Delta x^2 - \delta^2}{2 \delta}\right)^2 \\ a_y = \left(\frac{\Delta y}{\epsilon}\right)^2 \qquad b_y = \frac{\Delta y}{\epsilon^2}(\epsilon^2 -\Delta y^2) \qquad c_y = \left(\frac{\Delta y^2 - \epsilon^2}{2 \epsilon}\right)^2 \\ a_z = \left(\frac{\Delta z}{\zeta}\right)^2 \qquad b_z = \frac{\Delta z}{\zeta^2}(\zeta^2 -\Delta z^2) \qquad c_z = \left(\frac{\Delta z^2 - \zeta^2}{2 \zeta}\right)^2$$$$ Let us simplify eq. $\ref{eq:p0_initial}$ using the following substitution: $$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. $$$$\label{eq:p0_initial_simple} 0 = p_0^2 a + p_0 b + c$$$$ Applying the quadratic formula to eq. $\ref{eq:p0_initial_simple}$, we can solve for $p_0$.

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

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.

Need figure this part out!