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| − | We can define the constraints of the plane the DC is in
| + | <center><math>\underline{\textbf{Navigation}}</math> |
| | | | |
| − | <pre> | + | [[Wire_Number_Function|<math>\vartriangleleft </math>]] |
| − | right = ContourPlot[
| + | [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]] |
| − | x2 == Cot[29.5 \[Degree]] y + .09156, {y, -1, 1}, {x2, 0, 1.8},
| + | [[The_Wires|<math>\vartriangleright </math>]] |
| − | Frame -> {True, True, False, False},
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| − | PlotLabel -> "Right side limit of DC as a function of X and Y",
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| − | FrameLabel -> {"y (meters)", "x (meters)"}, ContourStyle -> Black,
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| − | PlotLegends -> Automatic];
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| | | | |
| − | left = ContourPlot[
| + | </center> |
| − | x2 == -Cot[29.5 \[Degree]] y + .09156, {y, -1, 1}, {x2, 0, 1.8},
| |
| − | Frame -> {True, True, False, False},
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| − | PlotLabel -> "Right side limit of DC as a function of X and Y",
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| − | FrameLabel -> {"y (meters)", "x (meters)"}, ContourStyle -> Black,
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| − | PlotLegends -> Automatic];
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| | | | |
| − | </pre> | + | <center>[[File:Part1d1.png]]</center> |
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| | | | |
| − | We can define the x coordinate of the wires as they cross the midpoint plane as shown earlier.
| + | ---- |
| | | | |
| − | <pre>
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| − | x0forWires[number_] := .23168 + .01337*(number);
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| − | </pre>
| |
| | | | |
| | + | <center><math>\underline{\textbf{Navigation}}</math> |
| | | | |
| − | We can define the point midway between two parallel lines as the point where one wire is recorded versus its next highest neighbor
| + | [[Wire_Number_Function|<math>\vartriangleleft </math>]] |
| | + | [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]] |
| | + | [[The_Wires|<math>\vartriangleright </math>]] |
| | | | |
| − | <pre>
| + | </center> |
| − | x0forWireMiddles[
| |
| − | number_] := ((.23168 + .01337*(number)) + (.23168 + .01337*(number \
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| − | + 1)))/2;
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| − | </pre>
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| − | | |
| − | All of the conditions dependent on <math> \theta </math> and <math>\phi</math>
| |
| − | | |
| − | <pre>
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| − | | |
| − | \[CapitalDelta]a :=
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| − | FullSimplify[(R Sin[\[Theta] \[Degree]])/
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| − | 2 (Csc[65 \[Degree] - \[Theta] \[Degree]] -
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| − | Csc[115 \[Degree] - \[Theta] \[Degree]]), \[Theta] > 0];
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| − | e := Sin[25 \[Degree]]/Cos[\[Theta] \[Degree]];
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| − | a := FullSimplify[(R Sin[\[Theta] \[Degree]])/
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| − | 2 (Csc[65 \[Degree] - \[Theta] \[Degree]] +
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| − | Csc[115 \[Degree] - \[Theta] \[Degree]]), \[Theta] > 0];
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| − | rD1 := Simplify[(a e - \[CapitalDelta]a) Tan[
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| − | 65 \[Degree]] Cos[\[Theta] \[Degree]], \[Theta] > 0];
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| − | rD2 := Simplify[(a e + \[CapitalDelta]a) Tan[
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| − | 65 \[Degree]] Cos[\[Theta] \[Degree]], \[Theta] > 0];
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| − | xD1 := Simplify[rD1 Cos[\[Phi] \[Degree]]];
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| − | yD1 := Simplify[rD1 Sin[\[Phi] \[Degree]]];
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| − | zD1 := Simplify[rD1 Cot[\[Theta] \[Degree]], \[Theta] > 0];
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| − | xD2 := Simplify[rD2 Cos[\[Phi] \[Degree]], \[Theta] > 0];
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| − | yD2 := Simplify[rD2 Sin[\[Phi] \[Degree]], \[Theta] > 0];
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| − | zD2 := Simplify[rD2 Cot[\[Theta] \[Degree]], \[Theta] > 0];
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| − | xP := Simplify[(R Cos[\[Phi] \[Degree]])/(Cot[\[Theta] \[Degree]] +
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| − | Cos[\[Phi] \[Degree]] Cot[65 \[Degree]]), \[Theta] > 0];
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| − | yP := Simplify[(R Sin[\[Phi] \[Degree]])/(Cot[\[Theta] \[Degree]] +
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| − | Cos[\[Phi] \[Degree]] Cot[65 \[Degree]]), \[Theta] > 0];
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| − | zP := Simplify[(R Cot[\[Theta] \[Degree]])/(Cot[\[Theta] \[Degree]] +
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| − | Cos[\[Phi] \[Degree]] Cot[65 \[Degree]]), \[Theta] > 0];
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| − | x1 := Simplify[(rD2^2 - rD1^2 +
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| − | Cot[\[Theta] \[Degree]]^2 (rD2^2 - rD1^2) - 2 xP (xD2 - xD1) -
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| − | 2 yP (yD2 - yD1) - 2 zP (zD2 - zD1))/(4 a e) - a e, \[Theta] >
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| − | 0];
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| − | x := Simplify[x1 - \[CapitalDelta]a + a e, \[Theta] > 0];
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| − | xCenter := x + \[CapitalDelta]a;
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| − | n := -957.412/(Tan[\[Theta] \[Degree]] + 2.14437) + 430.626;
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| − | D2P := Simplify[((xD2 - xP)^2 + (yD2 - yP)^2 + (zD2 -
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| − | zP )^2)^.5, \[Theta] > 0] // N
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| − | D1P := Simplify[((xP - xD1)^2 + (yP - yD1)^2 + (zP -
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| − | zD1)^2)^.5, \[Theta] > 0] // N;
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| − | y := Simplify[(D1P^2 - x1^2)^.5, \[Theta] > 0] // N;
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| − | b := Simplify[a Sqrt[1 - e^2], \[Theta] > 0] // N;
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| − | R = 2.52934271645;
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| − | </pre>
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| − | | |
| − | We can define the x-y position on the DC plane as a function of <math>\phi</math> for and limit the angles with the wall on the right and left hand sides. By symmetry these angles are equal, but opposite in sign.
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| − | | |
| − | <pre>
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| − | Limits = Table[\[Phi] /.
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| − | NSolve[Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a ==
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| − | Cot[29.5 \[Degree]] y + .09156 , \[Phi]], {\[Theta], 5, 40}];
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| − | Lim = Table[0, {rows, 1, 36}];
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| − | For[rows = 1, rows < 37, rows++,
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| − | For[i = 1, i < 5, i++,
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| − | If[Limits[[rows, i]] > 0 && Limits[[rows, i]] < 40,
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| − | Lim[[rows]] = Limits[[rows, i]];
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| − | ];
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| − | ];
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| − | ];
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| − | </pre>
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| − | | |
| − | The angle must have 4 subtracted to equal the line element numbering
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| − | | |
| − | <pre>
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| − | In[33]:= Lim[[7 - 4]]
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| − | | |
| − | Out[33]= 23.4101
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| − | </pre>
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| − | | |
| − | The limit of <math>\phi</math> increases in the plane of the DC sector changes as <math>\theta</math> increases.
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| − | | |
| − | <pre>
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| − | In[34]:= Lim[1]
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| − | | |
| − | Out[34]= {20.076, 22.024, 23.4101, 24.448, 25.255, 25.9008, 26.4297,
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| − | 26.8711, 27.2453, 27.5667, 27.846, 28.0911, 28.3079, 28.5013,
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| − | 28.675, 28.8319, 28.9744, 29.1044, 29.2237, 29.3336, 29.4352,
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| − | 29.5294, 29.6171, 29.699, 29.7757, 29.8477, 29.9155, 29.9795,
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| − | 30.0399, 30.0973, 30.1517, 30.2034, 30.2528, 30.2999, 30.3449,
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| − | 30.388}[1]
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| − | </pre>
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| − | | |
| − | This limiting condition will record the angle <math>\phi</math> in degrees within an array of wire number(1-112) horizontally and angle <math>\theta (5^{\circ}-40^{\circ} )</math> vertically
| |
| − | | |
| − | <pre>
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| − | RightSolutions =
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| − | Table[{\[Phi] /.
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| − | Solve[Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a ==
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| − | Tan[6 \[Degree]] y +
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| − | x0forWireMiddles[number] , \[Phi]]}, {\[Theta], 5,
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| − | 40}, {number, 1, 112}];
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| − | LineRight = Table[{\[Phi], columns}, {rows, 1, 36}, {columns, 1, 112}];
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| − | For[rows = 1, rows < 37, rows++,
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| − | For[columns = 1, columns < 113, columns++,
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| − | For[i = 1, i < 5, i++,
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| − | If[RightSolutions[[rows, columns, 1, i]] > 0 &&
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| − | RightSolutions[[rows, columns, 1, i]] < Lim[[rows]],
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| − | LineRight[[rows,
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| − | columns]] = {RightSolutions[[rows, columns, 1, i]],
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| − | columns + .5};
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| − | ];
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| − | ];
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| − | ];
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| − | ];
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| − | </pre>
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| − | | |
| − | At <math>\phi </math>=0, the condition of n=-959.637/(tan <math>\theta^{\circ}</math> +2.14437)+430.189 should be met
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| − | | |
| − | <pre>
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| − | In[38]:= f[\[Theta]for\[Phi]at0_] := -959.637/(
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| − | Tan[\[Theta]for\[Phi]at0 \[Degree]] + 2.14437) + 430.189
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| − | | |
| − | In[39]:= f[40]
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| − | | |
| − | Out[39]= 108.538
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| − | </pre>
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| − | | |
| − | | |
| − | This implies that for <math>\theta=40^{\circ}</math>, we know that the wire number to be 108.538. This corresponds to the position being in between 108.5 and 109.5, hence it falls into the 109 "bin".
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| − | | |
| − | Testing the geometry
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| − | <pre>
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| − | ClearAll[\[Theta]];
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| − | \[Theta] = 40;
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| − | ellipse40 =
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| − | ContourPlot[(x + \[CapitalDelta]a)^2/a^2 + y^2/b^2 == 1, {y, -1,
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| − | 1}, {x, 1.2, 1.7}, Frame -> {True, True, False, False},
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| − | PlotLabel ->
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| − | "XY position on DC as a function of \[Phi] for \[Theta]=40\
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| − | \[Degree]", FrameLabel -> {"y (meters)", "x (meters)"},
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| − | ContourStyle -> Red, PlotLegends -> Automatic];
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| − | Show[Table[
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| − | ContourPlot[
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| − | xWire == Tan[6 \[Degree]] yWire + x0forWires[number], {yWire, -1,
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| − | 1}, {xWire, 1.2, 1.7},
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| − | FrameLabel -> {"y(meters)", "x(meters)"}], {number, 70, 109}],
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| − | Table[ContourPlot[
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| − | xWire ==
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| − | Tan[6 \[Degree]] yWire + x0forWireMiddles[number2], {yWire, -1,
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| − | 1}, {xWire, 1.2, 1.7},
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| − | ContourStyle -> {Dashing[Large]}], {number2, 70,
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| − | 109}], bottom, right, left, ellipse40]
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| − | </pre>
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| − | | |
| − | [[File:40EllipseTest.png]]
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| − | | |
| − | | |
| − | Clearing wire numbers greater than the "wire number" at <math>\phi</math>=0
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| − | | |
| − | <pre>
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| − | For[rows = 1, rows < 37, rows++,
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| − | For[columns = 1, columns < 113, columns++,
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| − | If[LineRight[[rows, columns, 2]] > f[rows + 4],
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| − | LineRight[[rows, columns, 1]] = {\[Phi]}
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| − | ]
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| − | ]
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| − | ];
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| − | </pre>
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| − | | |
| − | The fact the equation for the y component in the plane of the detector uses a square root function, we know there must be two solutions. Due to the use of inverse trigonometric
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| − | functions within Mathematica, only the positive values of y will be used. We can find the angle \[Phi] where the elliptical xy path for \[Theta]=40\[Degree] crosses the midway point in between wires.
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| − | This corresponds to hits at a constant \[Theta] occurring on different wires. The elliptical path hits the right and left wall at \[Phi] = \[PlusMinus]24.9252 and will not cross additional wires lower than this
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| − | in the 1st quadrant. On the left hand side, the wire number limit is lower since the slope of this plane is positive.
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| − | | |
| − | | |
| − | [[File:SameWireDifferentPsi.png]]
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| − | | |
| − | By symmetry, we know that the positions within the 4th quadrant can be reflected into the 1st quadrant by taking the opposite slope of the wire function
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| − | | |
| − | <pre>
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| − | ClearAll[\[Theta]];
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| − | | |
| − | LeftSolutions =
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| − | Table[{\[Phi] /.
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| − | Solve[Sqrt[
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| − | a^2 (1 - y^2/b^2)] - \[CapitalDelta]a == -Tan[6 \[Degree]] y +
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| − | x0forWireMiddles[number] , \[Phi]], number}, {\[Theta], 5,
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| − | 40}, {number, 1, 112}];
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| − | LineLeft =
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| − | Table[{\[Phi], columns}, {rows, 1, 36}, {columns, 1, 112}];
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| − | For[rows = 1, rows < 37, rows++,
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| − | For[columns = 1, columns < 113, columns++,
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| − | For[i = 1, i < 7, i++,
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| − | If[LeftSolutions[[rows, columns, 1, i]] > 0 &&
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| − | LeftSolutions[[rows, columns, 1, i]] < Lim[[rows]],
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| − | LineLeft[[rows,
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| − | columns]] = {LeftSolutions[[rows, columns, 1, i]],
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| − | columns + .5};
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| − | ];
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| − | ];
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| − | For[i = 1, i < 6, i++,
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| − | If[LeftSolutions[[rows, columns, 1, i]] > 0 &&
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| − | LeftSolutions[[rows, columns, 1, i]] < Lim[[rows]],
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| − | LineLeft[[rows,
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| − | columns + 1]] = {LeftSolutions[[rows, columns, 1, i]],
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| − | columns + .5};
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| − | ];
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| − | ];
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| − | ];
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| − | ];
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| − | </pre>
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| − | | |
| − | We can combine the angles recorded in both quadrants
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| − | | |
| − | Collecting the \[Phi] angles as an x variable with the wire as the y variable for a constant value of \[Theta] first for the right hand side
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| − | | |
| − | <pre>
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| − | constant\[Theta]right = Table[0, {rows, 1, 36}, {columns, 1, 112}];
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| − | constant\[Theta]r = Table[0, {rows, 1, 36}, {columns, 1, 112}];
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| − | | |
| − | For[rows = 1, rows < 37, rows++,
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| − | For[columns = 1, columns < 113, columns++,
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| − | constant\[Theta]right[[rows, columns]] = LineRight[[rows, columns]]
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| − | ]
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| − | ]
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| − | </pre>
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| − | | |
| − | | |
| − | Flattening the list elements to numerical values
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| − | | |
| − | <pre>
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| − | For[rows = 1, rows < 37, rows++,
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| − | constant\[Theta]r[[rows]] =
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| − | Flatten[constant\[Theta]right[[rows]], 2]];
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| − | </pre>
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| − | | |
| − | Partitioning the data into {\[Phi],wire number} combinations that increase with respect to the <math>\phi</math> component
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| − | | |
| − | <pre>
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| − | For[rows = 1, rows < 37, rows++,
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| − | constant\[Theta]r[[rows]] = Partition[constant\[Theta]r[[rows]], 2]
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| − | ];
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| − | </pre> | |
| − | | |
| − | Sorting the x values from smallest to largest
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