Parameterizing the Ellipse Equation - Revision history

Vanwdani at 20:38, 15 May 2018

2018-05-15T20:38:42Z

Vanwdani at 18:47, 15 May 2017

2017-05-15T18:47:24Z

Vanwdani at 18:33, 15 May 2017

2017-05-15T18:33:32Z

Vanwdani at 18:04, 15 May 2017

2017-05-15T18:04:26Z

Vanwdani at 17:59, 15 May 2017

2017-05-15T17:59:41Z

Vanwdani at 17:24, 15 May 2017

2017-05-15T17:24:06Z

Vanwdani at 22:02, 3 May 2017

2017-05-03T22:02:05Z

Vanwdani: Blanked the page

2017-05-03T21:12:57Z

Blanked the page

Vanwdani at 21:12, 3 May 2017

2017-05-03T21:12:19Z

Vanwdani: Created page with "File:part1d1.pdf The point that was the semi-major vertex, when rotated 6\[Degree] to the right becomes

 In[1603]:= rFromYtoX.{1.6831832367824053`, 0, 0} // MatrixFor…"

2017-05-03T21:10:41Z

Created page with "File:part1d1.pdf The point that was the semi-major vertex, when rotated 6\[Degree] to the right becomes <pre> In[1603]:= rFromYtoX.{1.6831832367824053`, 0, 0} // MatrixFor…"

New page

[[File:part1d1.pdf]]

The point that was the semi-major vertex, when rotated 6\[Degree] to the right becomes

<pre>
In[1603]:= rFromYtoX.{1.6831832367824053`, 0, 0} // MatrixForm

Out[1603]//MatrixForm= \!$
TagBox[
RowBox[{"(", "",
TagBox[GridBox[{
{"1.6739625828969429`"},
{"0.17594055713873974`"},
{"0.`"}
},
GridBoxAlignment->{
"Columns" -> {{Center}}, "ColumnsIndexed" -> {},
"Rows" -> {{Baseline}}, "RowsIndexed" -> {}},
GridBoxSpacings->{"Columns" -> {
Offset[0.27999999999999997`], {
Offset[0.5599999999999999]},
Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
Offset[0.2], {
Offset[0.4]},
Offset[0.2]}, "RowsIndexed" -> {}}],
Column], "", ")"}],
Function[BoxForm`e$,
MatrixForm[BoxForm`e$]]]$
</pre>

Solving for the ellipse parameter given the angle and the corresponding X'' and Y'' components

<pre>
X = 1.6739625828969429`;
Y = 0.17594055713873974`;
\[Theta] = 40;
t = t /. Solve[
X^2 (Cos[6 \[Degree]])^2 +
Y^2 (Sin[6 \[Degree]])^2 + \[CapitalDelta]a^2 +
2 X Y Cos[6 \[Degree]] Sin[6 \[Degree]] +
2 X \[CapitalDelta]a Cos[6 \[Degree]] +
2 Y \[CapitalDelta]a Sin[6 \[Degree]] +
X^2 (Sin[6 \[Degree]])^2 + Y^2 (Cos[6 \[Degree]])^2 -
2 X Y Sin[6 \[Degree]] Cos[6 \[Degree]] ==
a^2 (Cos[t])^2 (Cos[6 \[Degree]])^2 +
b^2 (Sin[t])^2 (Sin[6 \[Degree]])^2 + \[CapitalDelta]a^2 +
2 a b Cos[t] Cos[6 \[Degree]] Sin[t] Sin[6 \[Degree]] +
2 a \[CapitalDelta]a Cos[t] Cos[6 \[Degree]] +
2 b \[CapitalDelta]a Sin[t] Sin[6 \[Degree]] +
a^2 (Cos[t])^2 (Sin[6 \[Degree]])^2 +
b^2 (Sin[t])^2 (Cos[6 \[Degree]])^2 -
2 a b Cos[t] Sin[6 \[Degree]] Cos[6 \[Degree]] Sin[t], t]
ClearAll[X, Y, \[Theta]];
</pre>

<pre>{-1.31406, 1.4676, 3.06482 - 1.66742 I, 3.06482 + 1.66742 I}</pre>

We can see that at the vertex position, the parameter reaches it's minimum at 1.4676. This point reflects the right and left sides of the ellipse and the corresponding decreases in wire number as the parameter is increased.

@@ Line 1: / Line 1: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Plotting_Different_Frames|<math>\vartriangleleft </math>]]
@@ Line 12: / Line 12: @@
-<center><math>\textbf{\underline{Navigation}}</math>
+<center><math>\underline{\textbf{Navigation}}</math>
 [[Plotting_Different_Frames|<math>\vartriangleleft </math>]]

← Older revision		Revision as of 18:47, 15 May 2017
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	</center>		</center>
	[[File:part1d3.png]]		[[File:part1d3.png]]
		+
		+
		+	----

← Older revision		Revision as of 18:33, 15 May 2017
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	</center>		</center>
	[[File:part1d3.png]]		[[File:part1d3.png]]
		+
		+
		+	<center><math>\textbf{\underline{Navigation}}</math>
		+
		+	[[Plotting_Different_Frames\|<math>\vartriangleleft </math>]]
		+	[[VanWasshenova_Thesis#Determining_wire-theta_correspondence\|<math>\triangle </math>]]
		+	[[Change_in_Wire_Bin_Number\|<math>\vartriangleright </math>]]
		+
		+	</center>

@@ Line 1: / Line 1: @@
 <center><math>\textbf{\underline{Navigation}}</math>
-[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\vartriangleleft </math>]]
+[[Plotting_Different_Frames|<math>\vartriangleleft </math>]]
 [[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]]
 [[Change_in_Wire_Bin_Number|<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>]]
+[[Change_in_Wire_Bin_Number|<math>\vartriangleright </math>]]
 </center>
 [[File:part1d3.png]]

← 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>
	[[File:part1d3.png]]		[[File:part1d3.png]]

← Older revision	Revision as of 22:02, 3 May 2017
Line 1:	Line 1:
−	+	[[File:part1d3.png]]

← Older revision		Revision as of 21:12, 3 May 2017
Line 1:		Line 1:
−	~~[[File:part1d1.png]]~~

−
−	~~The point that was the semi-major vertex, when rotated 6\[Degree] to the right becomes~~
−
−	~~<pre>~~
−	~~In[1603]:= rFromYtoX.{1.6831832367824053`, 0, 0} // MatrixForm~~
−
−	~~Out[1603]//MatrixForm= \!\(~~
−	~~TagBox[~~
−	~~RowBox[{"(", "",~~
−	~~TagBox[GridBox[{~~
−	~~{"1.6739625828969429`"},~~
−	~~{"0.17594055713873974`"},~~
−	~~{"0.`"}~~
−	},
−	~~GridBoxAlignment->{~~
−	~~"Columns" -> {{Center}}, "ColumnsIndexed" -> {},~~
−	~~"Rows" -> {{Baseline}}, "RowsIndexed" -> {}},~~
−	~~GridBoxSpacings->{"Columns" -> {~~
−	~~Offset[0.27999999999999997`], {~~
−	~~Offset[0.5599999999999999]},~~
−	~~Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {~~
−	~~Offset[0.2], {~~
−	~~Offset[0.4]},~~
−	~~Offset[0.2]}, "RowsIndexed" -> {}}],~~
−	~~Column], "", ")"}],~~
−	~~Function[BoxForm`e$,~~
−	~~MatrixForm[BoxForm`e$]]]\)~~
−	~~</pre>~~
−
−
−	~~Solving for the ellipse parameter given the angle and the corresponding X'' and Y'' components~~
−
−
−	~~<pre>~~
−	~~X = 1.6739625828969429`;~~
−	~~Y = 0.17594055713873974`;~~
−	~~\[Theta] = 40;~~
−	~~t = t /. Solve[~~
−	~~X^2 (Cos[6 \[Degree]])^2 +~~
−	~~Y^2 (Sin[6 \[Degree]])^2 + \[CapitalDelta]a^2 +~~
−	~~2 X Y Cos[6 \[Degree]] Sin[6 \[Degree]] +~~
−	~~2 X \[CapitalDelta]a Cos[6 \[Degree]] +~~
−	~~2 Y \[CapitalDelta]a Sin[6 \[Degree]] +~~
−	~~X^2 (Sin[6 \[Degree]])^2 + Y^2 (Cos[6 \[Degree]])^2 -~~
−	~~2 X Y Sin[6 \[Degree]] Cos[6 \[Degree]] ==~~
−	~~a^2 (Cos[t])^2 (Cos[6 \[Degree]])^2 +~~
−	~~b^2 (Sin[t])^2 (Sin[6 \[Degree]])^2 + \[CapitalDelta]a^2 +~~
−	~~2 a b Cos[t] Cos[6 \[Degree]] Sin[t] Sin[6 \[Degree]] +~~
−	~~2 a \[CapitalDelta]a Cos[t] Cos[6 \[Degree]] +~~
−	~~2 b \[CapitalDelta]a Sin[t] Sin[6 \[Degree]] +~~
−	~~a^2 (Cos[t])^2 (Sin[6 \[Degree]])^2 +~~
−	~~b^2 (Sin[t])^2 (Cos[6 \[Degree]])^2 -~~
−	~~2 a b Cos[t] Sin[6 \[Degree]] Cos[6 \[Degree]] Sin[t], t]~~
−	~~ClearAll[X, Y, \[Theta]];~~
−	~~</pre>~~
−
−
−	~~<pre>{-1.31406, 1.4676, 3.06482 - 1.66742 I, 3.06482 + 1.66742 I}</pre>~~
−
−	We can see that at the vertex position, the parameter reaches it's minimum at 1.4676. This point reflects the right and left sides of the ellipse and the corresponding decreases in wire number as the parameter is increased.