Difference between revisions of "Plotting Different Frames"

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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>
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<center><math>\underline{\textbf{Navigation}}</math>
  
<pre>
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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>]]
  
rFromYtoX = ( {
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</center>
    {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}];
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[[File:part1d2.png]]
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]]
 
</pre>
 
  
  
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>
 
  
  
<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]];
 
\[Theta] = 40;
 
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>
 

Latest revision as of 20:34, 15 May 2018