The Ellipse - Revision history

Vanwdani at 20:33, 15 May 2018

2018-05-15T20:33:57Z

Vanwdani at 18:47, 15 May 2017

2017-05-15T18:47:57Z

Vanwdani at 18:34, 15 May 2017

2017-05-15T18:34:00Z

Vanwdani at 18:05, 15 May 2017

2017-05-15T18:05:25Z

Vanwdani at 17:58, 15 May 2017

2017-05-15T17:58:43Z

Vanwdani at 17:24, 15 May 2017

2017-05-15T17:24:41Z

Vanwdani at 15:59, 1 May 2017

2017-05-01T15:59:57Z

Vanwdani at 18:34, 29 April 2017

2017-04-29T18:34:59Z

Vanwdani at 18:22, 29 April 2017

2017-04-29T18:22:00Z

Vanwdani at 18:09, 29 April 2017

2017-04-29T18:09:47Z

@@ Line 1: / Line 1: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Left_Hand_Wall|<math>\vartriangleleft </math>]]
@@ Line 139: / Line 139: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Left_Hand_Wall|<math>\vartriangleleft </math>]]

← Older revision		Revision as of 18:47, 15 May 2017
Line 134:		Line 134:

	This shows the preservation of the vector length of the offset from the origin on the x axis in the frame of the DC section. The equation shows the ellipse rotated so that the vertex and co-vertex are no longer parallel to the frame of reference x and y axis.		This shows the preservation of the vector length of the offset from the origin on the x axis in the frame of the DC section. The equation shows the ellipse rotated so that the vertex and co-vertex are no longer parallel to the frame of reference x and y axis.
		+
		+
		+	----
		+

	<center><math>\textbf{\underline{Navigation}}</math>		<center><math>\textbf{\underline{Navigation}}</math>

← Older revision		Revision as of 18:34, 15 May 2017
Line 134:		Line 134:

	This shows the preservation of the vector length of the offset from the origin on the x axis in the frame of the DC section. The equation shows the ellipse rotated so that the vertex and co-vertex are no longer parallel to the frame of reference x and y axis.		This shows the preservation of the vector length of the offset from the origin on the x axis in the frame of the DC section. The equation shows the ellipse rotated so that the vertex and co-vertex are no longer parallel to the frame of reference x and y axis.
		+
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Left_Hand_Wall\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\triangle </math>]]
		+	[[Plotting_Different_Frames\|<math>\vartriangleright </math>]]
		+
		+	</center>

@@ Line 1: / Line 1: @@
 <center><math>\textbf{\underline{Navigation}}</math>
-[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
+[[Left_Hand_Wall|<math>\vartriangleleft </math>]]
 [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]]
 [[Plotting_Different_Frames|<math>\vartriangleright </math>]]

@@ Line 3: / Line 3: @@
 [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
 [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]]
-[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance|<math>\vartriangleright </math>]]
+[[Plotting_Different_Frames|<math>\vartriangleright </math>]]
 </center>

← Older revision		Revision as of 17:24, 15 May 2017
Line 1:		Line 1:
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\triangle </math>]]
		+	[[CED_Verification_of_DC_Angle_Theta_and_Wire_Correspondance\|<math>\vartriangleright </math>]]
		+
		+	</center>
		+
		+
	Viewing the conic section <math>\phi</math> maps out on the DC sector plane, we know that it follows an elliptical path centered on it's x axis. Performing a passive rotation on points in the DC section plane does not physically change the position in space, i.e. passive rotations only give the components in a new coordinate system. Once such a rotation has been performed, the equation describing these points must be done within that plane.		Viewing the conic section <math>\phi</math> maps out on the DC sector plane, we know that it follows an elliptical path centered on it's x axis. Performing a passive rotation on points in the DC section plane does not physically change the position in space, i.e. passive rotations only give the components in a new coordinate system. Once such a rotation has been performed, the equation describing these points must be done within that plane.

@@ Line 108: / Line 108: @@
 \end{bmatrix}</math></center>
-Substituting this into the equation for an ellipse in the frame of the wires,
+Substituting this into the equation for an ellipse in the frame of the wires, that is parallel to the frame of reference x and y axis,
 <center><math>\frac{(x''+h'')^2}{a^2}+\frac{(y''+k'')^2}{b^2}=1</math></center>
-<center><math>\frac{((x''+\Delta a\ cos\ 6^{\circ})cos\ 6^{\circ}+(y''+\Delta a\ sin\ 6^{\circ})sin\ 6^{\circ})^2}{a^2}+\frac{(-(x''+\Delta a\ cos\ 6^{\circ})sin\ 6^{\circ}+(y''+\Delta a\ sin\ 6^{\circ})cos\ 6^{\circ})^2}{b^2}=1</math></center>
+<center><math>\frac{((x'+\Delta a\ cos\ 6^{\circ})cos\ 6^{\circ}+(y'+\Delta a\ sin\ 6^{\circ})sin\ 6^{\circ})^2}{a^2}+\frac{(-(x'+\Delta a\ cos\ 6^{\circ})sin\ 6^{\circ}+(y'+\Delta a\ sin\ 6^{\circ})cos\ 6^{\circ})^2}{b^2}=1</math></center>
-<center><math>\frac{(x''\ cos\ 6^{\circ}+\Delta a\ cos^2 6^{\circ}+y''\ sin\ 6^{\circ}+\Delta a\ sin^2 6^{\circ})^2}{a^2}+\frac{(-x''\ sin\ 6^{\circ}-\Delta a\ cos\ 6^{\circ}sin\ 6^{\circ}+y''\ cos\ 6^{\circ}+\Delta a\ sin\ 6^{\circ}cos\ 6^{\circ})^2}{b^2}=1</math></center>
+<center><math>\frac{(x'\ cos\ 6^{\circ}+\Delta a\ cos^2 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a\ sin^2 6^{\circ})^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}-\Delta a\ cos\ 6^{\circ}sin\ 6^{\circ}+y'\ cos\ 6^{\circ}+\Delta a\ sin\ 6^{\circ}cos\ 6^{\circ})^2}{b^2}=1</math></center>
-<center><math>\frac{(x''\ cos\ 6^{\circ}+y''\ sin\ 6^{\circ}+\Delta a (cos^2 6^{\circ}+ sin^2 6^{\circ}) )^2}{a^2}+\frac{(-x''\ sin\ 6^{\circ}+y''\ cos\ 6^{\circ}+\Delta a (cos\ 6^{\circ}sin\ 6^{\circ}-cos\ 6^{\circ}sin\ 6^{\circ}))^2}{b^2}=1</math></center>
+<center><math>\frac{(x'\ cos\ 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a (cos^2 6^{\circ}+ sin^2 6^{\circ}) )^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}+y'\ cos\ 6^{\circ}+\Delta a (cos\ 6^{\circ}sin\ 6^{\circ}-cos\ 6^{\circ}sin\ 6^{\circ}))^2}{b^2}=1</math></center>
-<center><math>\frac{(x''\ cos\ 6^{\circ}+y''\ sin\ 6^{\circ}+\Delta a )^2}{a^2}+\frac{(-x''\ sin\ 6^{\circ}+y''\ cos\ 6 ^{\circ})^2}{b^2}=1</math></center>
+<center><math>\frac{(x'\ cos\ 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a )^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}+y''\ cos\ 6 ^{\circ})^2}{b^2}=1</math></center>

← Older revision		Revision as of 18:22, 29 April 2017
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	<center><math>\frac{(x''+h'')^2}{a^2}+\frac{(y''+k'')^2}{b^2}=1</math></center>		<center><math>\frac{(x''+h'')^2}{a^2}+\frac{(y''+k'')^2}{b^2}=1</math></center>

		+
		+	<center><math>\frac{((x'+\Delta a\ cos\ 6^{\circ})cos\ 6^{\circ}+(y'+\Delta a\ sin\ 6^{\circ})sin\ 6^{\circ})^2}{a^2}+\frac{(-(x'+\Delta a\ cos\ 6^{\circ})sin\ 6^{\circ}+(y'+\Delta a\ sin\ 6^{\circ})cos\ 6^{\circ})^2}{b^2}=1</math></center>
		+

	<center><math>\frac{(x'\ cos\ 6^{\circ}+\Delta a\ cos^2 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a\ sin^2 6^{\circ})^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}-\Delta a\ cos\ 6^{\circ}sin\ 6^{\circ}+y'\ cos\ 6^{\circ}+\Delta a\ sin\ 6^{\circ}cos\ 6^{\circ})^2}{b^2}=1</math></center>		<center><math>\frac{(x'\ cos\ 6^{\circ}+\Delta a\ cos^2 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a\ sin^2 6^{\circ})^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}-\Delta a\ cos\ 6^{\circ}sin\ 6^{\circ}+y'\ cos\ 6^{\circ}+\Delta a\ sin\ 6^{\circ}cos\ 6^{\circ})^2}{b^2}=1</math></center>

← Older revision		Revision as of 18:09, 29 April 2017
Line 113:		Line 113:


−	<center><math>\frac{(x'\ cos\ 6^{\circ}+\Delta a\ cos^2 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a\ sin^2 6^{\circ})^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}~~+y'~~\ cos\ 6^{\circ}~~+\Delta a (cos~~\ 6^{\circ}~~sin~~\ 6^{\circ}~~-cos~~\ 6^{\circ}~~sin~~\ 6^{\circ}))^2}{b^2}=1</math></center>	+	<center><math>\frac{(x'\ cos\ 6^{\circ}+\Delta a\ cos^2 6^{\circ}+y'\ sin\ 6^{\circ}+\Delta a\ sin^2 6^{\circ})^2}{a^2}+\frac{(-x'\ sin\ 6^{\circ}-\Delta a\ cos\ 6^{\circ}sin\ 6^{\circ}+y'\ cos\ 6^{\circ}+\Delta a\ sin\ 6^{\circ}cos\ 6^{\circ})^2}{b^2}=1</math></center>