Plotting Different Frames
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![X'y'plane.png](/./images/c/c0/X%27y%27plane.png)
![X2y2plane.png](/./images/a/a1/X2y2plane.png)
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,
rFromYtoX = ( { {Cos[6 \[Degree]], -Sin[6 \[Degree]], 0}, {Sin[6 \[Degree]], Cos[6 \[Degree]], 0}, {0, 0, 1} } ); rFromXtoY = ( { {Cos[6 \[Degree]], Sin[6 \[Degree]], 0}, {-Sin[6 \[Degree]], Cos[6 \[Degree]], 0}, {0, 0, 1} } ); yxPoints = constant\[Theta]; constant\[Theta]yx = constant\[Theta]; constant\[Theta]yxRotated = constant\[Theta]; constant\[Theta]xyz = constant\[Theta]; constant\[Theta]xyzRotated = constant\[Theta]; RowLengths = Table[{Nothing}, {i, 1, 36}]; For[rows = 1, rows < 37, rows++, RowLengths[[rows]] = Length[constant\[Theta][[rows]]]; For[columns = 1, columns < RowLengths[[rows]] + 1, columns++, \[Theta] = rows + 4; \[Phi] = constant\[Theta][[rows, columns, 1]]; constant\[Theta]yx[[rows, columns]] = {y, Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a}; constant\[Theta]xyz[[rows, columns]] = {constant\[Theta]yx[[rows, columns, 2]], constant\[Theta]yx[[rows, columns, 1]], 0}; If[constant\[Theta][[rows, columns, 1]] < 0, constant\[Theta]yx[[rows, columns, 1]] = -constant\[Theta]yx[[rows, columns, 1]]; constant\[Theta]xyz[[rows, columns, 2]] = -constant\[Theta]xyz[[rows, columns, 2]]; ]; constant\[Theta]xyzRotated[[rows, columns]] = rFromYtoX.{constant\[Theta]xyz[[rows, columns, 1]], constant\[Theta]xyz[[rows, columns, 2]], constant\[Theta]xyz[[rows, columns, 3]]}; constant\[Theta]yxRotated[[rows, columns]] = {constant\[Theta]xyzRotated[[rows, columns, 2]], constant\[Theta]xyzRotated[[rows, columns, 1]]}; yxPoints[[rows, columns]] = {y, Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a, constant\[Theta][[rows, columns, 1]], constant\[Theta][[rows, columns, 2]]}; ] ]; ClearAll[\[Theta], \[Phi]]; DesiredyxPoints = Desiredconstant\[Theta]; Desiredconstant\[Theta]yx = Desiredconstant\[Theta]; Desiredconstant\[Theta]yxRotated = Desiredconstant\[Theta]; Desiredconstant\[Theta]xyz = Desiredconstant\[Theta]; Desiredconstant\[Theta]xyzRotated = Desiredconstant\[Theta]; DesiredRowLengths = Table[{Nothing}, {i, 1, 36}]; For[rows = 1, rows < 37, rows++, DesiredRowLengths[[rows]] = Length[Desiredconstant\[Theta][[rows]]]; For[columns = 1, columns < DesiredRowLengths[[rows]] + 1, columns++, \[Theta] = rows + 4; \[Phi] = Desiredconstant\[Theta][[rows, columns, 1]]; Desiredconstant\[Theta]yx[[rows, columns]] = {y, Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a}; Desiredconstant\[Theta]xyz[[rows, columns]] = {Desiredconstant\[Theta]yx[[rows, columns, 2]], Desiredconstant\[Theta]yx[[rows, columns, 1]], 0}; If[Desiredconstant\[Theta][[rows, columns, 1]] < 0, Desiredconstant\[Theta]yx[[rows, columns, 1]] = -Desiredconstant\[Theta]yx[[rows, columns, 1]]; Desiredconstant\[Theta]xyz[[rows, columns, 2]] = -Desiredconstant\[Theta]xyz[[rows, columns, 2]]; ]; Desiredconstant\[Theta]xyzRotated[[rows, columns]] = rFromYtoX.{Desiredconstant\[Theta]xyz[[rows, columns, 1]], Desiredconstant\[Theta]xyz[[rows, columns, 2]], Desiredconstant\[Theta]xyz[[rows, columns, 3]]}; Desiredconstant\[Theta]yxRotated[[rows, columns]] = {Desiredconstant\[Theta]xyzRotated[[rows, columns, 2]], Desiredconstant\[Theta]xyzRotated[[rows, columns, 1]]}; DesiredyxPoints[[rows, columns]] = {y, Sqrt[a^2 (1 - y^2/b^2)] - \[CapitalDelta]a, Desiredconstant\[Theta][[rows, columns, 1]], Desiredconstant\[Theta][[rows, columns, 2]]}; ] ]; ClearAll[\[Theta], \[Phi]]
The parameter's range changes from the DC frame with 0<t<2 , since
In the frame of the wires, the x axis no longer is aligned with the semi-major axis, therefore for
in the DC frame
In[153]:= ClearAll[X, \[Theta]]; \[Theta] = 40; X = X /. Solve[(X + \[CapitalDelta]a)^2/a^2 == 1 && X > 0, X] Out[155]= {1.68318}
This gives the (x' , y')=(1.68318, 0) in the DC frame for
![X'y'plane.png](/./images/c/c0/X%27y%27plane.png)
![X2y2plane.png](/./images/a/a1/X2y2plane.png)