Editing Appendix/Mathematics/EulerAngles (section)

==A Sequence of Rotations==

It is quite generally true that we can transition/map/migrate from one set of orthogonal unit vectors &#8212; such as the <math>(\vec{e}_X, \vec{e}_Y, \vec{e}_Z)</math> ''inertial/laboratory'' frame illustrated by the black arrows in the left panel of Figure 1 &#8212; to any other set of orthogonal unit vectors &#8212; such as the <math>(\vec{e}_1, \vec{e}_2, \vec{e}_3)</math> ''body'' frame illustrated by the black arrows in the right panel of Figure 1 &#8212; by carrying out three rotations.  These are traditionally referred to as ''Euler angles''. As is [[#The_Order_of_Rotations|emphasized below]], care must be taken when choosing the order in which rotations are carried out.  One fairly standard sequence of rotations is illustrated in Figure 2:

<table border="1" align="center" cellpadding="3">
<tr><td align="center" colspan="3">'''Figure 2'''</td></tr>
<tr>
  <td align="center">
[[File:BerciuFig2aCorrect.png|250px|Berciu Figure 2a]]
  </td>
  <td align="center">
[[File:BerciuFig2bCorrect.png|250px|Berciu Figure 2b]]
  </td>
  <td align="center">
[[File:BerciuFig2cCorrect.png|250px|Berciu Figure 2c]]
  </td>
</tr>

<tr>
  <td align="center">
Rotation #1
  </td>
  <td align="center">
Rotation #2
  </td>
  <td align="center">
Rotation #3
  </td>
</tr>
</table>

===First Rotation===
As depicted in the left-most panel of Figure 2, rotate the triplet of unit vectors about the <math>Z</math> (''i.e.,'' <math>\vec{e}_Z</math>) axis by an angle, <math>\phi</math>. The result is the ''green'' coordinate system labeled, <math>(1', 2', 3')</math>.  Note: (a) The 3' axis remains aligned with the inertial-frame Z-axis; and (b) we will refer to the 1' axis as the ''line of nodes''.  Given that the 3' axis is aligned with the Z-axis &#8212; that is, <math>\vec{e}_{3'} = \vec{e}_Z</math> &#8212; it is easy to recognize that the other two mappings are:  
<div align="center"><math>\vec{e}_{1'} = \vec{e}_X \cos\phi + \vec{e}_Y \sin\phi \, ,</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\vec{e}_{2'} = \vec{e}_Y \cos\phi - \vec{e}_X \sin\phi</math>.</div>
Hence, the corresponding rotation matrix is,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\hat{R}_{3}(\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
\vec{e}_{1'} \cdot\vec{e}_X & \vec{e}_{1'} \cdot\vec{e}_Y & \vec{e}_{1'} \cdot\vec{e}_Z  \\
\vec{e}_{2'} \cdot\vec{e}_X & \vec{e}_{2'} \cdot\vec{e}_Y & \vec{e}_{2'} \cdot\vec{e}_Z \\
\vec{e}_{3'} \cdot\vec{e}_X & \vec{e}_{3'} \cdot\vec{e}_Y & \vec{e}_{3'} \cdot\vec{e}_Z 
\end{bmatrix}}
=
{\begin{bmatrix}
\cos\phi & \sin\phi & 0  \\
-\sin\phi & \cos\phi & 0 \\
0 & 0 & 1 
\end{bmatrix}} \, .
</math>
  </td>
</tr>
</table>
Note that the subscript, 3, has been attached to <math>\hat{R}</math> in order to indicate that rotation was about the "third" axis.

===Second Rotation===
As depicted in the center panel of Figure 2, rotate the triplet of unit vectors about the ''line of nodes'' by an angle, <math>\theta</math>.  The result is the ''light-blue'' coordinate system labeled, (1", 2", 3").  Note:  (a) The 1" axis is aligned with the 1' axis (''line of nodes''); and (b) the values of the first pair of rotation angles, <math>(\phi, \theta)</math>, have been chosen, here, to ensure that the 3" axis aligns with the <math>\vec{e}_3</math> unit vector.  Given that, <math>\vec{e}_{1''} = \vec{e}_{1'}</math>, the other two mappings are,
<div align="center">
<math>\vec{e}_{2''} = \vec{e}_{2'} \cos\theta + \vec{e}_{3'} \sin\theta </math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\vec{e}_{3''} = \vec{e}_{3'} \cos\theta - \vec{e}_{2'} \sin\theta</math>.
</div>
The corresponding rotation matrix is, then,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\hat{R}_{1}(\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
\vec{e}_{1''} \cdot\vec{e}_{1'} & \vec{e}_{1''} \cdot\vec{e}_{2'} & \vec{e}_{1''} \cdot\vec{e}_{3'}  \\
\vec{e}_{2''} \cdot\vec{e}_{1'} & \vec{e}_{2''} \cdot\vec{e}_{2'} & \vec{e}_{2''} \cdot\vec{e}_{3'} \\
\vec{e}_{3''} \cdot\vec{e}_{1'} & \vec{e}_{3''} \cdot\vec{e}_{2'} & \vec{e}_{3''} \cdot\vec{e}_{3'} 
\end{bmatrix}}
=
{\begin{bmatrix}
1 & 0 & 0  \\
0 & \cos\theta & \sin\theta \\
0 & -\sin\theta & \cos\theta 
\end{bmatrix}} \, .
</math>
  </td>
</tr>
</table>
In this case the subscript, 1, has been attached to <math>\hat{R}</math> in order to indicate that rotation was about the "first" axis.

===Third Rotation===
As depicted in the right-most panel of Figure 2, rotate the triplet of unit vectors about the 3" axis by an angle, <math>\psi</math>.  The result is the desired black coordinate system labeled, <math>(\vec{e}_1, \vec{e}_2, \vec{e}_3)</math>.  Given that this step &#8212; like the first &#8212; calls for rotation about the "third" axis, we can immediately deduce that the relevant rotation matrix is,
<table border="0" align="center" cellpadding="8">
<tr>
  <td align="right">
<math>\hat{R}_{3}(\psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
\cos\psi & \sin\psi & 0  \\
-\sin\psi & \cos\psi & 0 \\
0 & 0 & 1 
\end{bmatrix}} \, .
</math>
  </td>
</tr>
</table>

===Other Properties of Particular Note===
In her [https://phas.ubc.ca/~berciu/TEACHING/PHYS206/LECTURES/FILES/euler.pdf online class notes], Professor Berciu points out:
<ol>
<li><math>\hat{R}_3(\phi_1)\cdot \hat{R}_3(\phi_2) = \hat{R}_3(\phi_1 + \phi_2)</math>. This means that if you rotate, first, by angle <math>\phi_2</math> followed by a rotation by angle <math>\phi_1</math> '''about the same axis(!)''', it is as if you carry out a single rotation by the angle, <math>(\phi_1 + \phi_2)</math>.</li>
<li>In flipping the angle of rotation from positive to negative, the rotation matrix flips to its transpose.  That is to say, for example, <math>\hat{R}_3(-\phi) = \hat{R}_3^{-1}(\phi) = \hat{R}^{T}_3(\phi)</math>.</li>
</ol>

===Simple Numerical Example===

Suppose we set <math>(\phi, \theta, \psi) = (30^\circ, 25^\circ, 15^\circ)</math>, which are roughly consistent with the trio of rotation angles displayed in Figure 2.  The corresponding trio of rotation matrices are:
<table border="0" align="center" cellpadding="2">
<tr>
  <td align="right">
<math>\hat{R}_{3}(\phi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
0.8660 & 0.5000 & 0  \\
-0.5000 & 0.8660 & 0 \\
0 & 0 & 1 
\end{bmatrix}}
</math>
  </td>
<td align="center">, &nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>\hat{R}_{1}(\theta)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
1 & 0 & 0  \\
0 & 0.9063 & 0.4226 \\
0 & -0.4226 & 0.9063 
\end{bmatrix}}
</math>
  </td>
<td align="center">, &nbsp; &nbsp; &nbsp;</td>

  <td align="right">
<math>\hat{R}_{3}(\psi)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
{\begin{bmatrix}
0.9659 & 0.2588 & 0  \\
-0.2588 & 0.9659 & 0 \\
0 & 0 & 1 
\end{bmatrix}} \, .
</math>
  </td>
</tr>
</table>