Difference between revisions of "The Wires"

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<center><math>x' = y\ sin\ 6^{\circ}+x_0</math></center>
 
<center><math>x' = y\ sin\ 6^{\circ}+x_0</math></center>
 +
 +
 
<center><math>y' = y\ cos\ 6^{\circ}</math></center>
 
<center><math>y' = y\ cos\ 6^{\circ}</math></center>
 
 
 
The parameterization has reduced two equations with two variables, to two equations which depend on one variable.  Working in the y-x plane, we will undergo a positive rotation,
 
The parameterization has reduced two equations with two variables, to two equations which depend on one variable.  Working in the y-x plane, we will undergo a positive rotation,
  
−
R(Subscript[\[Theta], yx])=(cos 6\[Degree] -sin 6\[Degree] 0
 
−
sin 6\[Degree] cos 6\[Degree] 0
 
−
0 0 1
 
  
−
)
+
<center><math>R(\theta_{yx})=\begin{bmatrix}
 +
cos\ 6^{\circ} &-sin\ 6^{\circ} & 0 \\
 +
sin\ 6^{\circ} & cos\ 6^{\circ} &0 \\
 +
0 &0 & 1
 +
\end{bmatrix}</math></center>
  
  
−
(Components of
+
\begin{bmatrix}
−
same vector
+
Components of \\
 +
same vector \\
 
in new system
 
in new system
−
 
+
\end{bmatrix}<
−
)=(Passive
+
)=\begin{bmatrix}
−
transformation
+
Passive \\
 +
transformation \\
 
matrix
 
matrix
−
 
+
\end{bmatrix}\cdot
−
) . (Components of
+
\begin{bmatrix}
−
vector in
+
Components of \\
 +
vector in \\
 
original system
 
original system
 +
\end{bmatrix}</math></center>
  
−
) (New
 
−
basis
 
−
vectors
 
  
−
)=(Active
 
−
transformation
 
−
matrix
 
  
−
) . (original
 
−
basis
 
−
vectors
 
  
−
)
+
<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
−
) . (x'
+
\end{bmatrix}\cdot
−
y'
+
\begin{bmatrix}
 +
x' \\
 +
y' \\
 
z'
 
z'
 +
\end{bmatrix}</math></center>
 +
  
−
) (x'
+
<center><math>
−
y'
+
\begin{bmatrix}
−
z'
+
x'' \\
−
 
+
y'' \\
−
)=(cos 6\[Degree] sin 6\[Degree] 0
 
−
-sin 6\[Degree] cos 6 \[Degree] 0
 
−
0 0 1
 
−
 
 
−
) . (x''
 
−
y''
 
−
z''
 
−
 
 
−
)
 
−
 
−
(x''
 
−
y''
 
 
z''
 
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}
 +
y'\ sin\ 6^{\circ}+x_0 \\
 +
y'\ cos\ 6^{\circ} \\
 +
0
 +
\end{bmatrix}</math></center>
  
−
)=(cos 6\[Degree] -sin 6\[Degree] 0
 
−
sin 6\[Degree] cos 6\[Degree] 0
 
−
0 0 1
 
  
−
) . ( y sin 6\[Degree]+Subscript[x, 0]
 
−
y cos 6\[Degree]
 
−
0
 
  
−
)
+
<center><math>
−
+
\begin{bmatrix}
−
(x''
+
x'' \\
−
y''
+
y'' \\
 
z''
 
z''
−
 
+
\end{bmatrix}=
−
)= (-y cos 6 \[Degree] sin 6 \[Degree]+Subscript[x, 0]cos 6 \[Degree] +y cos 6 \[Degree]sin 6 \[Degree]
+
\begin{bmatrix}
−
y cos^2 6 \[Degree]+Subscript[x, 0]sin 6 \[Degree]+y sin^2 6 \[Degree]
+
-y'\ cos\ 6^{\circ}sin\ 6^{\circ}+x_0\ cos\ 6^{\circ} +y'\ cos\ 6^{\circ}sin\ 6^{\circ}\\
 +
y'\ cos^2 6^{\circ}+x_0\sin\ 6^{\circ}+y sin^2 6^{\circ} \\
 
0
 
0
 +
\end{bmatrix}</math></center>
 +
  
−
) (x'
 
−
y'
 
−
z'
 
  
−
)= (x'' cos 6\[Degree]+y " sin 6\[Degree]
+
<center><math>
−
-x'' sin 6 \[Degree]+y " cos 6\[Degree]
+
\begin{bmatrix}
 +
x'' \\
 +
y'' \\
 +
z''
 +
\end{bmatrix}=
 +
\begin{bmatrix}
 +
x_0\ cos\ 6^{\circ}\\
 +
y'\ cos^2 6^{\circ}+x_0\sin\ 6^{\circ}+y sin^2 6^{\circ} \\
 
0
 
0
 +
\end{bmatrix}</math></center>
  
−
)
 
−
 
−
  (x''
 
−
y''
 
−
z''
 
−
 
−
)= (Subscript[x, 0]cos 6 \[Degree]
 
−
y +Subscript[x, 0]sin 6 \[Degree]
 
−
0
 
  
−
)
+
This relationship shows us that x'' is a constant in this frame while y'' can have any value, which is the horizontal line with respect to the y axis as expected.

Revision as of 16:22, 1 May 2017

We can parametrize the equations for the wires and wire midpoints to express the equation in vector form. In the y'-x' plane the general equation follows the relationship:

[math]x'=y'\ tan\ 6^{\circ}+x_0[/math]

where [math]x_0[/math] is the point where the line crosses the x axis.

[math]y' \Rightarrow {y\ tan\ 6^{\circ}+x_0, y, 0}[/math]


In this form we can easily see that the components of x and y , in the y'-x' plane are

[math]x' = y\ sin\ 6^{\circ}+x_0[/math]


[math]y' = y\ cos\ 6^{\circ}[/math]

The parameterization has reduced two equations with two variables, to two equations which depend on one variable. Working in the y-x plane, we will undergo a positive rotation,


[math]R(\theta_{yx})=\begin{bmatrix} cos\ 6^{\circ} &-sin\ 6^{\circ} & 0 \\ sin\ 6^{\circ} & cos\ 6^{\circ} &0 \\ 0 &0 & 1 \end{bmatrix}[/math]


\begin{bmatrix} Components of \\ same vector \\ in new system \end{bmatrix}< )=\begin{bmatrix} Passive \\ transformation \\ matrix \end{bmatrix}\cdot \begin{bmatrix} Components of \\ vector in \\ original system

\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} 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} y'\ sin\ 6^{\circ}+x_0 \\ y'\ cos\ 6^{\circ} \\ 0 \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} -y'\ cos\ 6^{\circ}sin\ 6^{\circ}+x_0\ cos\ 6^{\circ} +y'\ cos\ 6^{\circ}sin\ 6^{\circ}\\ y'\ cos^2 6^{\circ}+x_0\sin\ 6^{\circ}+y sin^2 6^{\circ} \\ 0 \end{bmatrix}[/math]


[math] \begin{bmatrix} x'' \\ y'' \\ z'' \end{bmatrix}= \begin{bmatrix} x_0\ cos\ 6^{\circ}\\ y'\ cos^2 6^{\circ}+x_0\sin\ 6^{\circ}+y sin^2 6^{\circ} \\ 0 \end{bmatrix}[/math]


This relationship shows us that x is a constant in this frame while y can have any value, which is the horizontal line with respect to the y axis as expected.

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