Transformation Matrix
The Euler angles can be applied using a transformation matrix
[math]\left(
\begin{array}{ccc}
\cos (\theta ) & 0 & -\sin (\theta ) \\
0 & 1 & 0 \\
\sin (\theta ) & 0 & \cos (\theta ) \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
z \\
\end{array}
\right)[/math][math]=\left(
\begin{array}{c}
x \cos (\theta )-z \sin (\theta ) \\
y \\
z \cos (\theta )+x \sin (\theta ) \\
\end{array}
\right)[/math]
For event #29, in sector 3, the location of the first interaction is given by
Converting -25 degrees to radians,
[math]\theta =-0.436332[/math]
which is the rotation the detectors are rotated from the y axis.
[math]\left(
\begin{array}{ccc}
\cos (\theta ) & 0 & -\sin (\theta ) \\
0 & 1 & 0 \\
\sin (\theta ) & 0 & \cos (\theta ) \\
\end{array}
\right).\left(
\begin{array}{c}
-15.76 \\
0 \\
237.43 \\
\end{array}
\right)[/math][math]=\left(
\begin{array}{c}
86.0588 \\
0. \\
221.845 \\
\end{array}
\right)[/math]
Finding [math]\phi =\frac{120\ 2 \pi }{360};[/math] since "sector -1" =3-1=2*60=120 degrees
[math]\left(
\begin{array}{ccc}
\cos (\phi ) & -\sin (\phi ) & 0 \\
\sin (\phi ) & \cos (\phi ) & 0 \\
0 & 0 & 1 \\
\end{array}
\right).\left(
\begin{array}{c}
86.0588 \\
0. \\
221.845 \\
\end{array}
\right)[/math][math]=\left(
\begin{array}{c}
-43.0294 \\
74.5291 \\
221.845 \\
\end{array}
\right)[/math]
This shows how the coordinates are transformed and explains the validity of using the TBTracking information to obtain a phi angle in the lab frame.
