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| − | We can define different arrays to collect the coordinates in the different frames using a passive transformation. Assuming that the intersection of the ellipse and sense wires is in the y-x plane, we will have a positive rotation, <math>R(\theta_{yx})</math>
| + | <center><math>\underline{\textbf{Navigation}}</math> |
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| − | <pre> | + | [[The_Ellipse|<math>\vartriangleleft </math>]] |
| | + | [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]] |
| | + | [[Parameterizing_the_Ellipse_Equation|<math>\vartriangleright </math>]] |
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| − | rFromYtoX = ( {
| + | </center> |
| − | {Cos[6 \[Degree]], -Sin[6 \[Degree]], 0},
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| − | {Sin[6 \[Degree]], Cos[6 \[Degree]], 0},
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| − | {0, 0, 1}
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| − | } );
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| − | rFromXtoY = ( {
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| − | {Cos[6 \[Degree]], Sin[6 \[Degree]], 0},
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| − | {-Sin[6 \[Degree]], Cos[6 \[Degree]], 0},
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| − | {0, 0, 1}
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| − | } );
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| − | yxPoints = constant\[Theta];
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| − | constant\[Theta]yx = constant\[Theta];
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| − | constant\[Theta]yxRotated = constant\[Theta];
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| − | constant\[Theta]xyz = constant\[Theta];
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| − | constant\[Theta]xyzRotated = constant\[Theta];
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| − | RowLengths = Table[{Nothing}, {i, 1, 36}];
| + | [[File:part1d2.png]] |
| − | For[rows = 1, rows < 37, rows++,
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| − | RowLengths[[rows]] = Length[constant\[Theta][[rows]]];
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| − | For[columns = 1, columns < RowLengths[[rows]] + 1, columns++,
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| − | \[Theta] = rows + 4;
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| − | \[Phi] = constant\[Theta][[rows, columns, 1]];
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| − | constant\[Theta]yx[[rows, columns]] = {y,
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| − | Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a};
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| − | constant\[Theta]xyz[[rows,
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| − | columns]] = {constant\[Theta]yx[[rows, columns, 2]],
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| − | constant\[Theta]yx[[rows, columns, 1]], 0};
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| − | If[constant\[Theta][[rows, columns, 1]] < 0,
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| − | constant\[Theta]yx[[rows, columns,
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| − | 1]] = -constant\[Theta]yx[[rows, columns, 1]];
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| − | constant\[Theta]xyz[[rows, columns,
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| − | 2]] = -constant\[Theta]xyz[[rows, columns, 2]];
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| − | ];
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| − | constant\[Theta]xyzRotated[[rows, columns]] =
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| − | rFromYtoX.{constant\[Theta]xyz[[rows, columns, 1]],
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| − | constant\[Theta]xyz[[rows, columns, 2]],
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| − | constant\[Theta]xyz[[rows, columns, 3]]};
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| − |
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| − | constant\[Theta]yxRotated[[rows,
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| − | columns]] = {constant\[Theta]xyzRotated[[rows, columns, 2]],
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| − | constant\[Theta]xyzRotated[[rows, columns, 1]]};
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| − | yxPoints[[rows, columns]] = {y,
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| − | Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a,
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| − | constant\[Theta][[rows, columns, 1]],
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| − | constant\[Theta][[rows, columns, 2]]};
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| − |
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| − | ]
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| − | ];
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| − | ClearAll[\[Theta], \[Phi]];
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| − | DesiredyxPoints = Desiredconstant\[Theta];
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| − | Desiredconstant\[Theta]yx = Desiredconstant\[Theta];
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| − | Desiredconstant\[Theta]yxRotated = Desiredconstant\[Theta];
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| − | Desiredconstant\[Theta]xyz = Desiredconstant\[Theta];
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| − | Desiredconstant\[Theta]xyzRotated = Desiredconstant\[Theta];
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| − | DesiredRowLengths = Table[{Nothing}, {i, 1, 36}];
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| − | For[rows = 1, rows < 37, rows++,
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| − | DesiredRowLengths[[rows]] =
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| − | Length[Desiredconstant\[Theta][[rows]]];
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| − | For[columns = 1, columns < DesiredRowLengths[[rows]] + 1,
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| − | columns++,
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| − | \[Theta] = rows + 4;
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| − | \[Phi] = Desiredconstant\[Theta][[rows, columns, 1]];
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| − | Desiredconstant\[Theta]yx[[rows, columns]] = {y,
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| − | Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a};
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| − | Desiredconstant\[Theta]xyz[[rows,
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| − | columns]] = {Desiredconstant\[Theta]yx[[rows, columns, 2]],
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| − | Desiredconstant\[Theta]yx[[rows, columns, 1]], 0};
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| − | If[Desiredconstant\[Theta][[rows, columns, 1]] < 0,
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| − | Desiredconstant\[Theta]yx[[rows, columns,
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| − | 1]] = -Desiredconstant\[Theta]yx[[rows, columns, 1]];
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| − | Desiredconstant\[Theta]xyz[[rows, columns,
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| − | 2]] = -Desiredconstant\[Theta]xyz[[rows, columns, 2]];
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| − | ];
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| − | Desiredconstant\[Theta]xyzRotated[[rows, columns]] =
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| − | rFromYtoX.{Desiredconstant\[Theta]xyz[[rows, columns, 1]],
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| − | Desiredconstant\[Theta]xyz[[rows, columns, 2]],
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| − | Desiredconstant\[Theta]xyz[[rows, columns, 3]]};
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| − |
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| − | Desiredconstant\[Theta]yxRotated[[rows,
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| − | columns]] = {Desiredconstant\[Theta]xyzRotated[[rows, columns,
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| − | 2]], Desiredconstant\[Theta]xyzRotated[[rows, columns, 1]]};
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| − | DesiredyxPoints[[rows, columns]] = {y,
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| − | Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a,
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| − | Desiredconstant\[Theta][[rows, columns, 1]],
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| − | Desiredconstant\[Theta][[rows, columns, 2]]};
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| − |
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| − | ]
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| − | ];
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| − | ClearAll[\[Theta], \[Phi]]
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| − | </pre>
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| | | | |
| | | | |
| − | The parameter's range changes from the DC frame with 0<t<2<math>\pi</math>, since
| + | ---- |
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| − | <center><math>x(t=0)=a\ cos\ t = a\ cos\ 0 =a = a\ cos\ 2 \pi =x(t=2 \pi)</math></center>
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| | | | |
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| − | <center><math>y(t=0)=b\ sin\ t = b\ sin\ 0 = 0 = b\ sin\ 2 \pi = y(t=2 \pi)</math></center>
| + | <center><math>\underline{\textbf{Navigation}}</math> |
| − |
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| − | In the frame of the wires, the x'' axis no longer is aligned with the semi-major axis, therefore for <math>\theta=40^{\circ}, \phi=0, t=0 </math> in the DC frame
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| | | | |
| | + | [[The_Ellipse|<math>\vartriangleleft </math>]] |
| | + | [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]] |
| | + | [[Parameterizing_the_Ellipse_Equation|<math>\vartriangleright </math>]] |
| | | | |
| − | <pre>
| + | </center> |
| − | In[153]:= ClearAll[X, \[Theta]];
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| − | \[Theta] = 40;
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| − | X = X /. Solve[(X + \[CapitalDelta]a)^2/a^2 == 1 && X > 0, X]
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| − | | |
| − | Out[155]= {1.68318}
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| − | </pre>
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| − | | |
| − | This gives the (x' , y')=(1.68318, 0) in the DC frame for<math> \phi=0^{\circ}, t=0</math>
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| − | | |
| − | <pre>
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| − | | |
| − | SemiMajorAxisRotatedReverse =
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| − | ContourPlot[X == -9.514 Y, {Y, -2, 2}, {X, 0, 1.8},
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| − | Frame -> {True, True, False, False},
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| − | PlotLabel ->
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| − | "Right side limit of DC as a function of X and Y",
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| − | FrameLabel -> {"y (meters)", "x (meters)"},
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| − | ContourStyle -> Green,
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| − | PlotLegends -> Automatic];
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| − | </pre>
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| − | | |
| − | Graphing the frame of the sector and the wires within the sector,
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| − | | |
| − | | |
| − | <pre>
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| − | DataXY = Table[
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| − | ListPlot[{constant\[Theta]yx[[i]]}, PlotStyle -> Black,
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| − | AxesLabel -> {"y", "x"},
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| − | PlotLabel ->
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| − | "DC Wire for Constant \[Theta] as a Function of \[Phi]"], {i, 1,
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| − | 36}];
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| − | DataXYRotated =
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| − | Table[ListPlot[{constant\[Theta]yxRotated[[i]]}, PlotStyle -> Black,
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| − | AxesLabel -> {"y'", "x'"},
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| − | PlotLabel ->
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| − | "DC Wire for Constant \[Theta] as a Function of \[Phi]"],
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| − | {i, 1, 36}];
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| − | DesiredDataXY =
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| − | Table[ListPlot[{Desiredconstant\[Theta]yx[[i]]}, PlotStyle -> Black,
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| − | AxesLabel -> {"y", "x"},
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| − | PlotLabel ->
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| − | "DC Wire for Constant \[Theta] as a Function of \[Phi]"], {i, 1,
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| − | 36}];
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| − | DesiredDataXYRotated =
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| − | Table[ListPlot[{Desiredconstant\[Theta]yxRotated[[i]]},
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| − | PlotStyle -> Black, AxesLabel -> {"y'", "x'"},
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| − | PlotLabel ->
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| − | "DC Wire for Constant \[Theta] as a Function of \[Phi]"],
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| − | {i, 1, 36}];
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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, 1.5, 1.7}, Frame -> {True, True, False, False},
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| − | PlotLabel ->
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| − | "X' & Y' position on DC sector plane as a function of \[Phi] for \
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| − | \[Theta]=40\[Degree]", FrameLabel -> {"y' (meters)", "x' (meters)"},
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| − | ContourStyle -> Red, PlotLegends -> Automatic];
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| − | ellipse40Rotated =
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| − | ContourPlot[(X Cos[6 \[Degree]] +
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| − | Y Sin[6 \[Degree]] + \[CapitalDelta]a)^2/
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| − | a^2 + (-X Sin[6 \[Degree]] + Y Cos[6 \[Degree]])^2/b^2 ==
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| − | 1, {Y, -1, 1}, {X, 1.4, 1.7}, Frame -> {True, True, False, False},
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| − | PlotLabel ->
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| − | "X'' & Y'' position on DC sector wires a function of \[Phi] for \
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| − | \[Theta]=40\[Degree]", FrameLabel -> {"y'' (meters)", "x'' (meters)"},
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| − | ContourStyle -> Red, PlotLegends -> Automatic];
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| − | Show[ellipse40,
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| − | Table[ContourPlot[
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| − | xWire == Tan[6 \[Degree]] yWire + x0forWires[number], {yWire, -1,
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| − | 1}, {xWire, 1.5, 1.7},
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| − | FrameLabel -> {"y(meters)", "x(meters)"}], {number, 89, 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, 1.5, 1.7},
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| − | ContourStyle -> {Dashing[Large]}], {number2, 89, 109}],
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| − | DataXY[[36]],
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| − | DesiredDataXY[[36]], left, right, SemiMajorAxis, \
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| − | SemiMajorAxisRotatedReverse]
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| − | Show[ellipse40Rotated,
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| − | Table[ContourPlot[
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| − | xWire == Cos[6 \[Degree]] x0forWires[number], {yWire, -1,
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| − | 1.2}, {xWire, 1.4, 1.7},
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| − | FrameLabel -> {"y(meters)", "x(meters)"}], {number, 89, 109}],
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| − | Table[ContourPlot[
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| − | xWire == Cos[6 \[Degree]] x0forWireMiddles[number2], {yWire, -1,
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| − | 1.2}, {xWire, 1.4, 1.7},
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| − | ContourStyle -> {Dashing[Large]}], {number2, 89, 109}],
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| − | DataXYRotated[[36]],
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| − | DesiredDataXYRotated[[36]], rightRotated, leftRotated, \
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| − | SemiMajorAxisRotated, SemiMajorAxis, SemiMajorAxisRotatedReverse]
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| − | ClearAll[\[Theta]];
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| − | </pre>
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| − | | |
| − | | |
| − | | |
| − | <center>[[File:x'y'plane.png]]</center>
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| − | | |
| − | | |
| − | <center>[[File:x2y2plane.png]]</center>
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