Right Hand Wall

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 {cot 29.5^{\circ}\ y + 0.09156, y, 0}[/math]

[math]t \mapsto {cos 29.5^{\circ}\ t + 0.09156, t sin\ 29.5^{\circ} , 0}[/math]

(x y z

)=(cos 6\[Degree] -sin 6\[Degree] 0 sin 6\[Degree] cos 6\[Degree] 0 0 0 1

) . (x' y' z'

)

(x y z

)=(cos 6\[Degree] -sin 6\[Degree] 0 sin 6\[Degree] cos 6\[Degree] 0 0 0 1

) . (t cos 29.5\[Degree]+0.09156 t sin 29.5\[Degree] 0

)

(x y z

)= (0.09156cos 6 \[Degree]+t cos 6 \[Degree]cos 29.5\[Degree]-t sin 6 \[Degree]sin 29.5\[Degree] t cos 6 \[Degree]sin 29.5\[Degree]+0.09156 sin 6 \[Degree]+t cos 29.5\[Degree]sin 6 \[Degree] 0

)

(x y z

)= (0.09156cos 6 \[Degree]+t (cos 6 \[Degree]cos 29.5\[Degree]- sin 6 \[Degree]sin 29.5\[Degree]) 0.09156 sin 6 \[Degree]+t (sin 6 \[Degree] cos 29.5\[Degree]+cos 6 \[Degree]sin 29.5\[Degree]) 0

)

Using the equation for y we can solve for t

[math]y''=0.09156\ sin\ 6^{\circ}+t (sin 6 \[Degree] cos 29.5^{\circ}+cos 6 ^{\circ}sin 29.5^{\circ}) -\gt t=(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''=0.09156cos 6^{\circ}+t (cos 6^{\circ}cos 29.5^{\circ}- sin 6^{\circ} sin 29.5^{\circ})=0.09156cos 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]

rightRotated = 
  ContourPlot[x2 == 1.401949 y + 0.077641, {y, -1, 1}, {x2, 0, 1.8}, 
   Frame -> {True, True, False, False}, 
       PlotLabel -> 
    "Right side limit of DC as a function of X and Y", 
   FrameLabel -> {"y (meters)", "x (meters)"}, 
   ContourStyle -> Black, 
       PlotLegends -> Automatic];