Difference between revisions of "Left Hand Wall"

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(Created page with "<center><math>x=-y\ cot\ 29.5^{\circ}+0.09156</math></center> Parameterizing this <center><math>r\mapsto {-y\ cot\ 29.5^{\circ}+0.09156,y,0}</math></center> <center><math>t\ma…")
 
 
(19 intermediate revisions by the same user not shown)
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<center><math>\underline{\textbf{Navigation}}</math>
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[[Right_Hand_Wall|<math>\vartriangleleft </math>]]
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[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]]
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[[The_Ellipse|<math>\vartriangleright </math>]]
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</center>
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<center><math>x=-y\ cot\ 29.5^{\circ}+0.09156</math></center>
 
<center><math>x=-y\ cot\ 29.5^{\circ}+0.09156</math></center>
  
 
Parameterizing this
 
Parameterizing this
  
−
<center><math>r\mapsto {-y\ cot\ 29.5^{\circ}+0.09156,y,0}</math></center>
+
<center><math>r\mapsto \{-y\ cot\ 29.5^{\circ}+0.09156,y,0 \}</math></center>
 +
 
 +
 
 +
<center><math>t\mapsto \{t\ cos\ 29.5^{\circ}+0.09156,-t\ sin\ 29.5^{\circ},0 \}</math></center>
  
−
<center><math>t\mapsto {t\ cos\ 29.5^{\circ}+0.09156,-t\ sin\ 29.5^{\circ},0}</math></center>
 
  
 
where the negative sign is applied to the sine function by the even odd relationships of cosine and sine, i.e. ( sin(-t)=-sin(t), cos(-t)=cos(t)) and the fact that the y'' component is in the 4th quadrant.   
 
where the negative sign is applied to the sine function by the even odd relationships of cosine and sine, i.e. ( sin(-t)=-sin(t), cos(-t)=cos(t)) and the fact that the y'' component is in the 4th quadrant.   
  
−
(x''
+
<center><math>
−
y''
+
\begin{bmatrix}
 +
x'' \\
 +
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)=(cos 6\[Degree] -sin 6\[Degree] 0
+
\begin{bmatrix}
−
sin 6\[Degree] cos 6\[Degree] 0
+
cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\
−
0 0 1
+
sin\ 6^{\circ} & cos\ 6^{\circ} & 0 \\
−
 
+
0 & 0 & 1
−
) . (x'
+
\end{bmatrix} \cdot
−
y'
+
\begin{bmatrix}
 +
x' \\
 +
y' \\
 
z'
 
z'
 +
\end{bmatrix}</math></center>
 +
  
−
)
+
<center><math>
−
 
+
\begin{bmatrix}
−
(x''
+
x'' \\
−
y''
+
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)=(cos 6\[Degree] -sin 6\[Degree] 0
+
\begin{bmatrix}
−
sin 6\[Degree] cos 6\[Degree] 0
+
cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\
−
0 0 1
+
sin\ 6^{\circ} & cos\ 6^{\circ} & 0 \\
−
 
+
0 & 0 & 1
−
) . (t cos (29.5\[Degree])+0.09156
+
\end{bmatrix} \cdot
−
-t sin (29.5\[Degree])
+
\begin{bmatrix}
 +
t\ cos\ 29.5^{\circ}+0.09156 \\
 +
-t\ sin\ 29.5^{\circ} \\
 
0
 
0
 +
\end{bmatrix}</math></center>
  
−
)
 
  
−
(x''
+
<center><math>
−
y''
+
\begin{bmatrix}
 +
x'' \\
 +
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)= (0.09156cos 6 \[Degree]+t cos 6 \[Degree]cos (29.5\[Degree])+t sin 6 \[Degree]sin (29.5\[Degree])
+
\begin{bmatrix}
−
-t cos 6 \[Degree]sin (29.5\[Degree])+0.09156 sin 6 \[Degree]+t cos (29.5\[Degree])sin 6 \[Degree]
+
0.09156\ cos\ 6^{\circ}+t\ cos\ 6^{\circ}cos\ 29.5^{\circ}+t\ sin\ 6^{\circ}sin\ 29.5^{\circ}  \\
 +
-t\ cos\ 6^{\circ}sin\ 29.5^{\circ}+0.09156\ sin\ 6^{\circ}+t\ cos\ 29.5^{\circ}sin\ 6^{\circ} \\
 
0
 
0
 +
\end{bmatrix}</math></center>
  
−
)
 
  
−
(x''
+
<center><math>
−
y''
+
\begin{bmatrix}
 +
x'' \\
 +
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)= (0.09156cos 6 \[Degree]+t (cos 6 \[Degree]cos(29.5\[Degree])+ sin 6 \[Degree]sin (29.5\[Degree]))
+
\begin{bmatrix}
−
0.09156  sin 6 \[Degree]-t (cos 6 \[Degree]sin (29.5\[Degree])-sin 6 \[Degree] cos (29.5\[Degree]))
+
0.09156\ cos\ 6^{\circ}+t\ (cos\ 6^{\circ}cos\ 29.5^{\circ}+ sin\ 6^{\circ}sin\ 29.5^{\circ}) \\
 +
0.09156\ sin\ 6^{\circ}-t\ (cos\ 6^{\circ}sin\ 29.5^{\circ}-sin\ 6^{\circ} cos\ 29.5^{\circ}) \\
 
0
 
0
 +
\end{bmatrix}</math></center>
  
−
)
 
  
−
(x''
+
<center><math>
−
y''
+
\begin{bmatrix}
 +
x'' \\
 +
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)= (0.09156cos 6 \[Degree]+t cos (6\[Degree] -29.5\[Degree])
+
\begin{bmatrix}
−
0.09156  sin 6 \[Degree]+t sin (6 \[Degree]-29.5\[Degree])
+
0.09156\ cos\ 6^{\circ}+t\ cos\ (6^{\circ} -29.5^{\circ}) \\
 +
0.09156\ sin\ 6^{\circ}+t\ sin\ (6^{\circ}-29.5^{\circ}) \\
 
0
 
0
 +
\end{bmatrix}</math></center>
  
−
)
 
  
−
(x''
+
<center><math>
−
y''
+
\begin{bmatrix}
 +
x'' \\
 +
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)= (0.09156cos 6 \[Degree]+t cos (-23.5\[Degree])
+
\begin{bmatrix}
−
0.09156  sin 6 \[Degree]+t sin (-23.5\[Degree])
+
0.09156\ cos\ 6^{\circ}+t\ cos\ (-23.5^{\circ}) \\
 +
0.09156\ sin\ 6^{\circ}+t\ sin\ (-23.5^{\circ}) \\
 
0
 
0
 +
\end{bmatrix}</math></center>
  
−
)
 
  
−
(x''
+
<center><math>
−
y''
+
\begin{bmatrix}
 +
x'' \\
 +
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)= (0.09156cos 6 \[Degree]+t cos (23.5\[Degree])
+
\begin{bmatrix}
−
0.09156  sin 6 \[Degree]-t sin (-23.5\[Degree])
+
0.09156\ cos\ 6^{\circ}+t\ cos\ 23.5^{\circ} \\
 +
0.09156\ sin\ 6^{\circ}-t\ sin\ 23.5^{\circ} \\
 
0
 
0
−
 
+
\end{bmatrix}</math></center>
−
)
 
  
  
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<center><math>y''=0.09156\  sin\ 6^{\circ}-t\ sin\  23.5^{\circ} \Rightarrow t=\frac{-(y''-0.09156  sin 6^{\circ})}{sin 23.5^{\circ}}</math></center>
 
<center><math>y''=0.09156\  sin\ 6^{\circ}-t\ sin\  23.5^{\circ} \Rightarrow t=\frac{-(y''-0.09156  sin 6^{\circ})}{sin 23.5^{\circ}}</math></center>
 +
  
 
Substituting this into the expression for x''
 
Substituting this into the expression for x''
  
 
<center><math>x''=0.09156\ cos\ 6^{\circ}+t\ cos\ 23.5^{\circ}</math></center>
 
<center><math>x''=0.09156\ cos\ 6^{\circ}+t\ cos\ 23.5^{\circ}</math></center>
 +
  
 
<center><math>x''=0.09156\ cos\ 6 ^{\circ}+\frac{-(y''-0.09156  sin 6 ^{\circ})}{sin 23.5^{\circ}}(cos 23.5^{\circ})</math></center>
 
<center><math>x''=0.09156\ cos\ 6 ^{\circ}+\frac{-(y''-0.09156  sin 6 ^{\circ})}{sin 23.5^{\circ}}(cos 23.5^{\circ})</math></center>
 +
  
 
<center><math>x''=0.091058+\frac{y''-.0095706 }{-0.398749} (.917060)</math></center>
 
<center><math>x''=0.091058+\frac{y''-.0095706 }{-0.398749} (.917060)</math></center>
 +
  
 
<center><math>x''=0.091058+(y''-.0095706 ) (-2.299843)</math></center>
 
<center><math>x''=0.091058+(y''-.0095706 ) (-2.299843)</math></center>
 +
  
 
<center><math>x''=-2.299843\ y''+.022011+.091058</math></center>
 
<center><math>x''=-2.299843\ y''+.022011+.091058</math></center>
 +
  
 
<center><math>x''=-2.299843\ y''+.113069</math></center>
 
<center><math>x''=-2.299843\ y''+.113069</math></center>
 +
 +
 +
 +
[[File:leftwall.png]]
 +
 +
 +
----
 +
 +
 +
<center><math>\underline{\textbf{Navigation}}</math>
 +
 +
[[Right_Hand_Wall|<math>\vartriangleleft </math>]]
 +
[[VanWasshenova_Thesis#Determining_wire-theta_correspondence|<math>\triangle </math>]]
 +
[[The_Ellipse|<math>\vartriangleright </math>]]
 +
 +
</center>

Latest revision as of 20:33, 15 May 2018

[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]


[math]x=-y\ cot\ 29.5^{\circ}+0.09156[/math]

Parameterizing this

[math]r\mapsto \{-y\ cot\ 29.5^{\circ}+0.09156,y,0 \}[/math]


[math]t\mapsto \{t\ cos\ 29.5^{\circ}+0.09156,-t\ sin\ 29.5^{\circ},0 \}[/math]


where the negative sign is applied to the sine function by the even odd relationships of cosine and sine, i.e. ( sin(-t)=-sin(t), cos(-t)=cos(t)) and the fact that the y component is in the 4th quadrant.

[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\ sin\ 6^{\circ} & cos\ 6^{\circ} & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} cos\ 6^{\circ} & -sin\ 6^{\circ} & 0 \\ sin\ 6^{\circ} & cos\ 6^{\circ} & 0 \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} t\ cos\ 29.5^{\circ}+0.09156 \\ -t\ sin\ 29.5^{\circ} \\ 0 \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} 0.09156\ cos\ 6^{\circ}+t\ cos\ 6^{\circ}cos\ 29.5^{\circ}+t\ sin\ 6^{\circ}sin\ 29.5^{\circ} \\ -t\ cos\ 6^{\circ}sin\ 29.5^{\circ}+0.09156\ sin\ 6^{\circ}+t\ cos\ 29.5^{\circ}sin\ 6^{\circ} \\ 0 \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} 0.09156\ cos\ 6^{\circ}+t\ (cos\ 6^{\circ}cos\ 29.5^{\circ}+ sin\ 6^{\circ}sin\ 29.5^{\circ}) \\ 0.09156\ sin\ 6^{\circ}-t\ (cos\ 6^{\circ}sin\ 29.5^{\circ}-sin\ 6^{\circ} cos\ 29.5^{\circ}) \\ 0 \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} 0.09156\ cos\ 6^{\circ}+t\ cos\ (6^{\circ} -29.5^{\circ}) \\ 0.09156\ sin\ 6^{\circ}+t\ sin\ (6^{\circ}-29.5^{\circ}) \\ 0 \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} 0.09156\ cos\ 6^{\circ}+t\ cos\ (-23.5^{\circ}) \\ 0.09156\ sin\ 6^{\circ}+t\ sin\ (-23.5^{\circ}) \\ 0 \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} 0.09156\ cos\ 6^{\circ}+t\ cos\ 23.5^{\circ} \\ 0.09156\ sin\ 6^{\circ}-t\ sin\ 23.5^{\circ} \\ 0 \end{bmatrix}[/math]


Using the equation for y we can solve for t

[math]y''=0.09156\ sin\ 6^{\circ}-t\ sin\ 23.5^{\circ} \Rightarrow t=\frac{-(y''-0.09156 sin 6^{\circ})}{sin 23.5^{\circ}}[/math]


Substituting this into the expression for x

[math]x''=0.09156\ cos\ 6^{\circ}+t\ cos\ 23.5^{\circ}[/math]


[math]x''=0.09156\ cos\ 6 ^{\circ}+\frac{-(y''-0.09156 sin 6 ^{\circ})}{sin 23.5^{\circ}}(cos 23.5^{\circ})[/math]


[math]x''=0.091058+\frac{y''-.0095706 }{-0.398749} (.917060)[/math]


[math]x''=0.091058+(y''-.0095706 ) (-2.299843)[/math]


[math]x''=-2.299843\ y''+.022011+.091058[/math]


[math]x''=-2.299843\ y''+.113069[/math]


Leftwall.png



[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]

Retrieved from "https://wiki.iac.isu.edu/index.php?title=Left_Hand_Wall&oldid=122817"