Parameterizing the Ellipse Equation

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} // MatrixForm

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

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

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]];

{-1.31406, 1.4676, 3.06482 - 1.66742 I, 3.06482 + 1.66742 I}

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.