[math]\underline{\textbf{Navigation}}[/math]
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This same process can be applied to the side walls for the detector. For the sidewalls, we have approximated them as lines following the equation
[math]x=cot\ 29.5^{\circ}\ y + 0.09156[/math]
Parameterizing this
[math]r \mapsto {y\ cot\ 29.5^{\circ} + 0.09156, y, 0}[/math]
[math]t \mapsto {t\ cos\ 29.5^{\circ} + 0.09156, t\ sin\ 29.5^{\circ} , 0}[/math]
[math]\begin{bmatrix}
x'' \\
y'' \\
z''
\end{bmatrix}=
\begin{bmatrix}
cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\
sin\ 6^{\circ} & cos\ 6^{\circ}& 0 \\
0 & 0 & 1
\end{bmatrix}\cdot
\begin{bmatrix}
x' \\
y' \\
z'
\end{bmatrix}[/math]
[math]\begin{bmatrix}
x'' \\
y'' \\
z''
\end{bmatrix}=
\begin{bmatrix}
cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\
sin\ 6^{\circ} & cos\ 6^{\circ}& 0 \\
0 & 0 & 1
\end{bmatrix}\cdot
\begin{bmatrix}
t\ cos\ 29.5^{\circ}+0.09156 \\
t sin 29.5^{\circ}\\
0
\end{bmatrix}[/math]
[math]\begin{bmatrix}
x'' \\
y'' \\
z''
\end{bmatrix}=
\begin{bmatrix}
0.09156\ cos\ 6^{\circ}+t\ cos\ 6 ^{\circ}cos\ 29.5^{\circ}-t\ sin\ 6 ^{\circ}sin\ 29.5^{\circ} \\
t\ cos\ 6 ^{\circ}sin\ 29.5^{\circ}+0.09156\ sin\ 6^{\circ}+t\ cos\ 29.5^{\circ}sin\ 6^{\circ} \\
0
\end{bmatrix}[/math]
[math]\begin{bmatrix}
x'' \\
y'' \\
z''
\end{bmatrix}=
\begin{bmatrix}
0.09156\ cos\ 6^{\circ}+t\ (cos\ 6^{\circ}cos\ 29.5^{\circ}- sin\ 6 ^{\circ}sin\ 29.5^{\circ}) \\
0.09156\ sin\ 6 ^{\circ}+t\ (sin\ 6^{\circ} cos\ 29.5^{\circ}+cos\ 6 ^{\circ}sin\ 29.5^{\circ}) \\
0
\end{bmatrix}[/math]
Using the equation for y we can solve for t
[math]y''=0.09156\ sin\ 6^{\circ}+t (sin\ 6^{\circ} cos\ 29.5^{\circ}+cos\ 6 ^{\circ}sin\ 29.5^{\circ}) \Rightarrow t=\frac{y''-0.09156\ sin\ 6 ^{\circ}}{sin\ 6^{\circ} cos\ 29.5^{\circ}+cos\ 6^{\circ}sin\ 29.5^{\circ}}[/math]
Substituting this into the expression for x
[math]x''=0.09156\ cos\ 6^{\circ}+t\ (cos\ 6^{\circ}cos\ 29.5^{\circ}- sin\ 6^{\circ} sin\ 29.5^{\circ})[/math]
[math]x''=0.09156\ cos\ 6 ^{\circ}+\frac{y''-0.09156\ sin\ 6^{\circ}}{sin\ 6^{\circ} cos\ 29.5^{\circ}+cos\ 6^{\circ}sin\ 29.5^{\circ}} (cos\ 6^{\circ}cos\ 29.5^{\circ}- sin\ 6^{\circ} sin\ 29.5^{\circ})[/math]
[math]x''=0.09156\ cos\ 6^{\circ}+\frac{y''-0.09156\ sin\ 6^{\circ}}{sin\ 6^{\circ} cos\ 29.5^{\circ}+cos\ 6 ^{\circ}sin\ 29.5^{\circ}} (cos\ 6 ^{\circ}cos\ 29.5^{\circ}- sin\ 6^{\circ}sin\ 29.5^{\circ})[/math]
[math]x''=(0.994522)0.09156+\frac{y''-0.09156 (0.104528) }{0.0909769+.489726} (0.865588- 0.051472)[/math]
[math]x''=(0.091058)+\frac{y''-.0095706 }{0.580703} (.814116)[/math]
[math]x''=(0.091058)+(y''-.0095706 ) (1.401949)[/math]
[math]x''=1.401949\ y''-.013417+.091058[/math]
[math]x''=1.401949\ y''+.077641[/math]
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