|
|
| (23 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| − | 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},
| |
| − | {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}];
| + | [[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\[Pi], 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>]] |
| | + | |
| | + | </center> |
