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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> |
| | | | |
| − | <pre> | + | [[The_Ellipse|<math>\vartriangleleft </math>]] |
| | + | [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]] |
| | + | [[Parameterizing_the_Ellipse_Equation|<math>\vartriangleright </math>]] |
| | | | |
| − | 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};
| |
| − | 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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| − | 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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| − | 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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| − | 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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| − | ClearAll[\[Theta], \[Phi]]
| |
| − | </pre>
| |
| | | | |
| | | | |
| − | The parameter's range changes from the DC frame with 0<t<2<math>\pi</math>, since
| + | ---- |
| | | | |
| − | <center><math>x(t=0)=a\ cos\ t = a\ cos\ 0 =a = a\ cos\ 2 \pi =x(t=2 \pi)</math></center>
| |
| | | | |
| | | | |
| − | <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> |
| − |
| |
| − | 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
| |
| | | | |
| | + | [[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]
| |
| − | | |
| − | Out[155]= {1.68318}
| |
| − | </pre>
| |
| − | | |
| − | This gives the (x' , y')=(1.68318, 0) in the DC frame for<math> \phi=0^{\circ}, t=0</math>
| |
| − | | |
| − | | |
| − | The slope m, with respect to x, is found as<math>\frac{ \Delta x}{\Delta y} = \frac{(1.67396 - 1.076)}{(0.175941 - 0.113092)}=\frac{0.59796}{0.0628481}</math>=9.514
| |
| − | | |
| − | | |
| − | Slope point form gives
| |
| − | | |
| − | <center><math>x-x_1=m(y-y_1) </math></center>
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| − | | |
| − | <center><math>\Rightarrow x - 1.67396 = 9.514(y - 0.17594) </math></center>
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| − | | |
| − | <center><math>\Rightarrow y = 9.514x-1.67396 + 1.67396</math></center>
| |
| − | | |
| − | <center><math>\Rightarrow y = 9.514x</math></center>
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| − | | |
| − | | |
| − | <pre>
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| − | SemiMajorAxis =
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| − | ContourPlot[Y == 0, {Y, -1, 1}, {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)"}, ContourStyle -> Red,
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| − | PlotLegends -> Automatic];
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| − | SemiMajorAxisRotated =
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| − | ContourPlot[X == 9.614 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)"}, ContourStyle -> Blue,
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| − | PlotLegends -> Automatic];
| |
| − | </pre>
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| − | | |
| − | The center
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| − | | |
| − | | |
| − | <pre>
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| − | In[161]:= rFromXtoY.{\[CapitalDelta]a, 0, 0} // MatrixForm
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| − | | |
| − | Out[161]//MatrixForm= \!\(
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| − | TagBox[
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| − | RowBox[{"(", "",
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| − | TagBox[GridBox[{
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| − | {"1.0760024073650314`"},
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| − | {
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| − | RowBox[{"-", "0.11309241017012851`"}]},
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| − | {"0.`"}
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| − | },
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| − | GridBoxAlignment->{
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| − | "Columns" -> {{Center}}, "ColumnsIndexed" -> {},
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| − | "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
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| − | GridBoxSpacings->{"Columns" -> {
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| − | Offset[0.27999999999999997`], {
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| − | Offset[0.5599999999999999]},
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| − | Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
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| − | Offset[0.2], {
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| − | Offset[0.4]},
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| − | Offset[0.2]}, "RowsIndexed" -> {}}],
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| − | Column], "", ")"}],
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| − | Function[BoxForm`e$,
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| − | MatrixForm[BoxForm`e$]]]\)
| |
| − | </pre>
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| − | | |
| − | The vertex
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| − | | |
| − | <pre>
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| − | In[162]:= rFromXtoY.{1.6831832367824053`, 0, 0} // MatrixForm
| |
| − | | |
| − | Out[162]//MatrixForm= \!\(
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| − | TagBox[
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| − | RowBox[{"(", "",
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| − | TagBox[GridBox[{
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| − | {"1.6739625828969429`"},
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| − | {
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| − | RowBox[{"-", "0.17594055713873974`"}]},
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| − | {"0.`"}
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| − | },
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| − | GridBoxAlignment->{
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| − | "Columns" -> {{Center}}, "ColumnsIndexed" -> {},
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| − | "Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
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| − | GridBoxSpacings->{"Columns" -> {
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| − | Offset[0.27999999999999997`], {
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| − | Offset[0.5599999999999999]},
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| − | Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
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| − | Offset[0.2], {
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| − | Offset[0.4]},
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| − | Offset[0.2]}, "RowsIndexed" -> {}}],
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| − | Column], "", ")"}],
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| − | Function[BoxForm`e$,
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| − | MatrixForm[BoxForm`e$]]]\)
| |
| − | </pre>
| |
| − | | |
| − | | |
| − | The slope m, with respect to x, is found as <math>\frac{\Delta\ x}{\Delta\ y} = \frac{1.67396 - 1.076}{-0.175941 + 0.113092}=\frac{0.59796}{-0.0628481}=-9.514</math>
| |
| − | | |
| − | | |
| − | Slope point form gives
| |
| − | | |
| − | <center><math>x' - x_1=m(y'-y_1) </math></center>
| |
| − | | |
| − | | |
| − | <center><math>\Rightarrow x' - 1.67396 = -9.514(y' + 0.17594) </math></center>
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| − | | |
| − | | |
| − | <center><math>\Rightarrow y' = 9.514x'-1.67396 + 1.67396</math></center>
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| − | | |
| − | | |
| − | <center><math>\Rightarrow y' = -9.514x'</math></center>
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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]];
| |
| − | </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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