Difference between revisions of "Parameterizing the Ellipse Equation"

@@ 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 +
-X Y Cos[6 \[Degree]] Sin[6 \[Degree]] +
-X \[CapitalDelta]a Cos[6 \[Degree]] +
-Y \[CapitalDelta]a Sin[6 \[Degree]] +
-     X^2 (Sin[6 \[Degree]])^2 + Y^2 (Cos[6 \[Degree]])^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 +
-a b Cos[t] Cos[6 \[Degree]] Sin[t] Sin[6 \[Degree]] +
-a \[CapitalDelta]a Cos[t] Cos[6 \[Degree]] +
-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 -
-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.

Difference between revisions of "Parameterizing the Ellipse Equation"

Revision as of 21:12, 3 May 2017

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