Difference between revisions of "Parameterizing the Ellipse Equation"

From New IAC Wiki
Jump to navigation Jump to search
(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…")
 
Line 1: Line 1:
[[File:part1d1.pdf]]
+
[[File:part1d1.png]]
  
  

Revision as of 21:12, 3 May 2017

Part1d1.png


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


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.