Editing Orbital elements (section)

== Euler angle transformations ==
The angles {{math|Ω}}, ''{{mvar|i}}'', ''{{mvar|ω}}'' are the [[Euler angles]] (corresponding to ''{{mvar|α}}'', ''{{mvar|β}}'', ''{{mvar|γ}}'' in the notation used in that article) characterizing the orientation of the coordinate system
{{block indent|em=1.5|text='''{{math|x̂}}''', '''{{math|ŷ}}''', '''{{math|ẑ}}''' from the inertial coordinate frame '''{{math|Î}}''', '''{{math|Ĵ}}''', '''{{math|K̂}}'''}}
where:

* '''{{math|Î}}''', '''{{math|Ĵ}}''' is in the equatorial plane of the central body. '''{{math|Î}}''' is in the direction of the vernal equinox. '''{{math|Ĵ}}''' is perpendicular to '''{{math|Î}}''' and with '''{{math|Î}}''' defines the reference plane. '''{{math|K̂}}''' is perpendicular to the reference plane. Orbital elements of bodies (planets, comets, asteroids, ...) in the Solar System usually the [[ecliptic]] as that plane.
* '''{{math|x̂}}''', '''{{math|ŷ}}''' are in the orbital plane and with '''{{math|x̂}}''' in the direction to the [[pericenter]] ([[periapsis]]). '''{{math|ẑ}}''' is perpendicular to the plane of the orbit. '''{{math|ŷ}}''' is mutually perpendicular to '''{{math|x̂}}''' and '''{{math|ẑ}}'''.

Then, the transformation from the '''{{math|Î}}''', '''{{math|Ĵ}}''', '''{{math|K̂}}''' coordinate frame to the '''{{math|x̂}}''', '''{{math|ŷ}}''', '''{{math|ẑ}}''' frame with the Euler angles {{math|Ω}}, ''{{mvar|i}}'', ''{{mvar|ω}}'' is: <math display="block">\begin{align}
x_1 &= \cos \Omega \cdot \cos \omega - \sin \Omega \cdot \cos i \cdot \sin \omega\ ;\\
x_2 &= \sin \Omega \cdot \cos \omega + \cos \Omega \cdot \cos i \cdot \sin \omega\ ;\\
x_3 &= \sin i      \cdot \sin \omega ;\\
\, \\
y_1 &=-\cos \Omega \cdot \sin \omega - \sin \Omega  \cdot \cos i \cdot \cos \omega\ ;\\
y_2 &=-\sin \Omega \cdot \sin \omega + \cos \Omega  \cdot \cos i \cdot \cos \omega\ ;\\
y_3 &= \sin i      \cdot \cos \omega\ ;\\
\, \\
z_1 &= \sin i      \cdot \sin \Omega\ ;\\
z_2 &=-\sin i      \cdot \cos \Omega\ ;\\
z_3 &= \cos i\ ;\\
\end{align}</math> <math display="block">\begin{bmatrix}
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3 \\
z_1 & z_2 & z_3
\end{bmatrix}
 =
\begin{bmatrix}
\cos\omega & \sin\omega & 0 \\
-\sin\omega & \cos\omega& 0 \\
0 & 0 & 1
\end{bmatrix}
\,
\begin{bmatrix}
1 & 0 &0 \\
0 & \cos i & \sin i\\
0 & -\sin i & \cos i
\end{bmatrix}
\,
\begin{bmatrix}
\cos\Omega & \sin\Omega & 0 \\
-\sin\Omega & \cos\Omega& 0 \\
0 & 0 & 1
\end{bmatrix}\,; </math> where <math display="block">\begin{align}
\mathbf\hat{x} &= x_1\mathbf\hat{I} + x_2\mathbf\hat{J} + x_3\mathbf\hat{K} ~;\\
\mathbf\hat{y} &= y_1\mathbf\hat{I} + y_2\mathbf\hat{J} + y_3\mathbf\hat{K} ~;\\
\mathbf\hat{z} &= z_1\mathbf\hat{I} + z_2\mathbf\hat{J} + z_3\mathbf\hat{K} ~.\\
 \end{align}</math>

The inverse transformation, which computes the 3 coordinates in the I-J-K system given the 3 (or 2) coordinates in the x-y-z system, is represented by the inverse matrix. According to the rules of [[Invertible matrix|matrix algebra]], the inverse matrix of the product of the 3 rotation matrices is obtained by inverting the order of the three matrices and switching the signs of the three Euler angles.

That is,

<math display="block">\begin{bmatrix}
i_1 & i_2 & i_3 \\
j_1 & j_2 & j_3 \\
k_1 & k_2 & k_3
\end{bmatrix}
 =
\begin{bmatrix}
\cos\Omega & -\sin\Omega & 0 \\
\sin\Omega & \cos\Omega& 0 \\
0 & 0 & 1
\end{bmatrix}
\,
\begin{bmatrix}
1 & 0 &0 \\
0 & \cos i & -\sin i\\
0 & \sin i & \cos i
\end{bmatrix}
\,
\begin{bmatrix}
\cos\omega & -\sin\omega & 0 \\
\sin\omega & \cos\omega& 0 \\
0 & 0 & 1
\end{bmatrix}\,;
 </math> where <math display="block">\begin{align}
\mathbf\hat{I} &= i_1\mathbf\hat{x} + i_2\mathbf\hat{y} + i_3\mathbf\hat{z} ~;\\
\mathbf\hat{J} &= j_1\mathbf\hat{x} + j_2\mathbf\hat{y} + j_3\mathbf\hat{z} ~;\\
\mathbf\hat{K} &= k_1\mathbf\hat{x} + k_2\mathbf\hat{y} + k_3\mathbf\hat{z} ~.\\
 \end{align}</math>

The transformation from '''{{math|x̂}}''', '''{{math|ŷ}}''', '''{{math|ẑ}}''' to Euler angles {{math|Ω}}, ''{{mvar|i}}'', ''{{mvar|ω}}'' is: <math display="block">\begin{align}
\Omega &= \operatorname{arg}\left( -z_2, z_1 \right)\\
i &= \operatorname{arg}\left( z_3, \sqrt{{z_1}^2 + {z_2}^2} \right)\\
\omega &= \operatorname{arg}\left( y_3, x_3 \right)\\
 \end{align}</math> where {{math|arg(''x'',''y'')}} signifies the polar argument that can be computed with the standard function {{mono|[[atan2|atan2(y,x)]]}} available in many programming languages.