Euler angles: sequence composition and the twelve standard sequences
EulerAnglesSequenceCompositionAndTheTwelveStandardSequences
2026-08-28 23:15:57
2026-08-28 23:15:57
Definition
Improved the figure for readability
Euler angles
Euler sequences
Tait Bryan angles
proper Euler angles
passive rotation matrices
direction cosine matrix
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\section*{Euler Angles: Sequence Composition and the Twelve Standard Sequences}
Once the elementary axis rotations and the intrinsic/extrinsic distinction are
understood, all standard Euler sequences can be organized by one composition
rule.
There are exactly twelve standard three angle sequences:
\begin{itemize}
\item six Tait Bryan sequences, in which all three axis labels are different;
\item six proper Euler sequences, in which the first and third axis labels are
the same.
\end{itemize}
This article derives the counting argument, gives the passive intrinsic product
for every sequence, records the expanded direction cosine matrices, and shows
how the two sequence families differ in their middle angle singularity.
It is intended to serve as the main sequence reference for the remaining Euler
angle articles.
\section{Convention recap}
PhysicsLibrary uses passive coordinate maps between right handed orthonormal
frames:
\begin{equation}
{}^B\mathbf v
=
{}^BC_A\,{}^A\mathbf v.
\end{equation}
For a generic intrinsic $i$-$j$-$k$ sequence with first, second, and third
angles $(\alpha,\beta,\gamma)$,
\begin{equation}
{}^BC_A
=
C_k(\gamma)
C_j(\beta)
C_i(\alpha).
\end{equation}
The elementary passive matrices are
\begin{equation}
C_1(\lambda)
=
\begin{bmatrix}
1&0&0\\
0&\cos\lambda&\sin\lambda\\
0&-\sin\lambda&\cos\lambda
\end{bmatrix},
\end{equation}
\begin{equation}
C_2(\lambda)
=
\begin{bmatrix}
\cos\lambda&0&-\sin\lambda\\
0&1&0\\
\sin\lambda&0&\cos\lambda
\end{bmatrix},
\end{equation}
and
\begin{equation}
C_3(\lambda)
=
\begin{bmatrix}
\cos\lambda&\sin\lambda&0\\
-\sin\lambda&\cos\lambda&0\\
0&0&1
\end{bmatrix}.
\end{equation}
\section{Deriving the intrinsic composition rule}
Let the intermediate frames be
\[
A_0=A,\qquad A_1,\qquad A_2,\qquad A_3=B.
\]
For intrinsic $i$-$j$-$k$,
\begin{equation}
{}^{A_1}C_A
=
C_i(\alpha),
\end{equation}
\begin{equation}
{}^{A_2}C_{A_1}
=
C_j(\beta),
\end{equation}
and
\begin{equation}
{}^BC_{A_2}
=
C_k(\gamma).
\end{equation}
Coordinate maps compose by matching adjacent frame labels:
\begin{equation}
{}^BC_A
=
{}^BC_{A_2}
\,{}^{A_2}C_{A_1}
\,{}^{A_1}C_A.
\end{equation}
Therefore
\begin{equation}
{}^BC_A
=
C_k(\gamma)
C_j(\beta)
C_i(\alpha).
\end{equation}
\begin{center}
\includegraphics[width=0.94\textwidth]{EA04_intrinsic_sequence_composition.png}
\vspace{0.45em}
\textbf{Figure.}
Generic intrinsic sequence composition. The chronological frame rotations
progress from $A_0$ to $A_3$, while the corresponding passive coordinate maps
multiply in the frame-chain order shown.
\end{center}
The rightmost matrix acts first on a coordinate column.
\section{Why there are exactly twelve standard sequences}
The first rotation axis has three possible choices.
The second axis must differ from the first, leaving two choices.
The third axis must differ from the second. There are then two standard
possibilities:
\begin{enumerate}
\item use the remaining axis, producing a Tait Bryan sequence;
\item return to the first axis, producing a proper Euler sequence.
\end{enumerate}
Therefore
\begin{equation}
3\times2\times2=12.
\end{equation}
\begin{center}
\includegraphics[width=0.88\textwidth]{EA04_twelve_standard_sequences.png}
\vspace{0.45em}
\textbf{Figure.}
Classification of the twelve standard intrinsic Euler sequences. Six use
three distinct axes; six repeat the first axis label as the third.
\end{center}
\section{The six Tait Bryan sequences}
The Tait Bryan sequences are
\begin{equation}
123,\quad132,\quad213,\quad231,\quad312,\quad321.
\end{equation}
Their intrinsic passive products are:
\begin{center}
\begin{tabular}{|c|c|}
\hline
Sequence & Passive intrinsic product\\
\hline
$1$-$2$-$3$ & $C_3(\gamma)C_2(\beta)C_1(\alpha)$\\
\hline
$1$-$3$-$2$ & $C_2(\gamma)C_3(\beta)C_1(\alpha)$\\
\hline
$2$-$1$-$3$ & $C_3(\gamma)C_1(\beta)C_2(\alpha)$\\
\hline
$2$-$3$-$1$ & $C_1(\gamma)C_3(\beta)C_2(\alpha)$\\
\hline
$3$-$1$-$2$ & $C_2(\gamma)C_1(\beta)C_3(\alpha)$\\
\hline
$3$-$2$-$1$ & $C_1(\gamma)C_2(\beta)C_3(\alpha)$\\
\hline
\end{tabular}
\end{center}
For a common principal branch,
\begin{equation}
-\frac{\pi}{2}
\leq
\beta
\leq
\frac{\pi}{2}.
\end{equation}
The generic Tait Bryan singularity occurs when
\begin{equation}
\cos\beta=0,
\end{equation}
or
\begin{equation}
\beta=\pm\frac{\pi}{2}.
\end{equation}
At that configuration the first and third rotation axes become aligned and the
outer angles lose independent meaning.
\section{The six proper Euler sequences}
The proper Euler sequences are
\begin{equation}
121,\quad131,\quad212,\quad232,\quad313,\quad323.
\end{equation}
Their intrinsic passive products are:
\begin{center}
\begin{tabular}{|c|c|}
\hline
Sequence & Passive intrinsic product\\
\hline
$1$-$2$-$1$ & $C_1(\gamma)C_2(\beta)C_1(\alpha)$\\
\hline
$1$-$3$-$1$ & $C_1(\gamma)C_3(\beta)C_1(\alpha)$\\
\hline
$2$-$1$-$2$ & $C_2(\gamma)C_1(\beta)C_2(\alpha)$\\
\hline
$2$-$3$-$2$ & $C_2(\gamma)C_3(\beta)C_2(\alpha)$\\
\hline
$3$-$1$-$3$ & $C_3(\gamma)C_1(\beta)C_3(\alpha)$\\
\hline
$3$-$2$-$3$ & $C_3(\gamma)C_2(\beta)C_3(\alpha)$\\
\hline
\end{tabular}
\end{center}
A common proper Euler principal choice is
\begin{equation}
0\leq\beta\leq\pi.
\end{equation}
The generic proper Euler singularity occurs when
\begin{equation}
\sin\beta=0,
\end{equation}
that is,
\begin{equation}
\beta=0
\qquad\hbox{or}\qquad
\beta=\pi.
\end{equation}
Again, the physical orientation remains valid. Only the Euler coordinate
chart becomes singular.
\section{Notation for expanded matrices}
For compactness, define
\begin{equation}
c_\alpha=\cos\alpha,
\qquad
s_\alpha=\sin\alpha,
\end{equation}
\begin{equation}
c_\beta=\cos\beta,
\qquad
s_\beta=\sin\beta,
\end{equation}
and
\begin{equation}
c_\gamma=\cos\gamma,
\qquad
s_\gamma=\sin\gamma.
\end{equation}
The following matrices all map coordinates from the initial frame $A$ into the
final frame $B$ under the PhysicsLibrary passive intrinsic convention.
\section{Expanded Tait Bryan reference matrices}
\subsection*{Intrinsic $1-2-3$}
This is a Tait Bryan sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_3(\gamma)C_2(\beta)C_1(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\beta c_\gamma & c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta & - c_\alpha c_\gamma s_\beta + s_\alpha s_\gamma \\
- c_\beta s_\gamma & c_\alpha c_\gamma - s_\alpha s_\beta s_\gamma & c_\alpha s_\beta s_\gamma + c_\gamma s_\alpha \\
s_\beta & - c_\beta s_\alpha & c_\alpha c_\beta
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $1-3-2$}
This is a Tait Bryan sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_2(\gamma)C_3(\beta)C_1(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\beta c_\gamma & c_\alpha c_\gamma s_\beta + s_\alpha s_\gamma & - c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta \\
- s_\beta & c_\alpha c_\beta & c_\beta s_\alpha \\
c_\beta s_\gamma & c_\alpha s_\beta s_\gamma - c_\gamma s_\alpha & c_\alpha c_\gamma + s_\alpha s_\beta s_\gamma
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $2-1-3$}
This is a Tait Bryan sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_3(\gamma)C_1(\beta)C_2(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\gamma + s_\alpha s_\beta s_\gamma & c_\beta s_\gamma & c_\alpha s_\beta s_\gamma - c_\gamma s_\alpha \\
- c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta & c_\beta c_\gamma & c_\alpha c_\gamma s_\beta + s_\alpha s_\gamma \\
c_\beta s_\alpha & - s_\beta & c_\alpha c_\beta
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $2-3-1$}
This is a Tait Bryan sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_1(\gamma)C_3(\beta)C_2(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\beta & s_\beta & - c_\beta s_\alpha \\
- c_\alpha c_\gamma s_\beta + s_\alpha s_\gamma & c_\beta c_\gamma & c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta \\
c_\alpha s_\beta s_\gamma + c_\gamma s_\alpha & - c_\beta s_\gamma & c_\alpha c_\gamma - s_\alpha s_\beta s_\gamma
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $3-1-2$}
This is a Tait Bryan sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_2(\gamma)C_1(\beta)C_3(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\gamma - s_\alpha s_\beta s_\gamma & c_\alpha s_\beta s_\gamma + c_\gamma s_\alpha & - c_\beta s_\gamma \\
- c_\beta s_\alpha & c_\alpha c_\beta & s_\beta \\
c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta & - c_\alpha c_\gamma s_\beta + s_\alpha s_\gamma & c_\beta c_\gamma
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $3-2-1$}
This is a Tait Bryan sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_1(\gamma)C_2(\beta)C_3(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\beta & c_\beta s_\alpha & - s_\beta \\
c_\alpha s_\beta s_\gamma - c_\gamma s_\alpha & c_\alpha c_\gamma + s_\alpha s_\beta s_\gamma & c_\beta s_\gamma \\
c_\alpha c_\gamma s_\beta + s_\alpha s_\gamma & - c_\alpha s_\gamma + c_\gamma s_\alpha s_\beta & c_\beta c_\gamma
\end{bmatrix}.
\end{equation}
\section{Expanded proper Euler reference matrices}
\subsection*{Intrinsic $1-2-1$}
This is a proper Euler sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_1(\gamma)C_2(\beta)C_1(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\beta & s_\alpha s_\beta & - c_\alpha s_\beta \\
s_\beta s_\gamma & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma & c_\alpha c_\beta s_\gamma + c_\gamma s_\alpha \\
c_\gamma s_\beta & - c_\alpha s_\gamma - c_\beta c_\gamma s_\alpha & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $1-3-1$}
This is a proper Euler sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_1(\gamma)C_3(\beta)C_1(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\beta & c_\alpha s_\beta & s_\alpha s_\beta \\
- c_\gamma s_\beta & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma & c_\alpha s_\gamma + c_\beta c_\gamma s_\alpha \\
s_\beta s_\gamma & - c_\alpha c_\beta s_\gamma - c_\gamma s_\alpha & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $2-1-2$}
This is a proper Euler sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_2(\gamma)C_1(\beta)C_2(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma & s_\beta s_\gamma & - c_\alpha c_\beta s_\gamma - c_\gamma s_\alpha \\
s_\alpha s_\beta & c_\beta & c_\alpha s_\beta \\
c_\alpha s_\gamma + c_\beta c_\gamma s_\alpha & - c_\gamma s_\beta & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $2-3-2$}
This is a proper Euler sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_2(\gamma)C_3(\beta)C_2(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma & c_\gamma s_\beta & - c_\alpha s_\gamma - c_\beta c_\gamma s_\alpha \\
- c_\alpha s_\beta & c_\beta & s_\alpha s_\beta \\
c_\alpha c_\beta s_\gamma + c_\gamma s_\alpha & s_\beta s_\gamma & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $3-1-3$}
This is a proper Euler sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_3(\gamma)C_1(\beta)C_3(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma & c_\alpha c_\beta s_\gamma + c_\gamma s_\alpha & s_\beta s_\gamma \\
- c_\alpha s_\gamma - c_\beta c_\gamma s_\alpha & c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma & c_\gamma s_\beta \\
s_\alpha s_\beta & - c_\alpha s_\beta & c_\beta
\end{bmatrix}.
\end{equation}
\subsection*{Intrinsic $3-2-3$}
This is a proper Euler sequence. Its passive intrinsic product is
\begin{equation}
{}^BC_A
=
C_3(\gamma)C_2(\beta)C_3(\alpha).
\end{equation}
With the shorthand introduced above, the expanded matrix is
\begin{equation}
{}^BC_A
=
\begin{bmatrix}
c_\alpha c_\beta c_\gamma - s_\alpha s_\gamma & c_\alpha s_\gamma + c_\beta c_\gamma s_\alpha & - c_\gamma s_\beta \\
- c_\alpha c_\beta s_\gamma - c_\gamma s_\alpha & c_\alpha c_\gamma - c_\beta s_\alpha s_\gamma & s_\beta s_\gamma \\
c_\alpha s_\beta & s_\alpha s_\beta & c_\beta
\end{bmatrix}.
\end{equation}
\section{Aerospace \(3\)-\(2\)-\(1\) specialization}
For intrinsic $3$-$2$-$1$ yaw pitch roll,
\begin{equation}
\alpha=\psi,
\qquad
\beta=\theta,
\qquad
\gamma=\phi.
\end{equation}
Therefore
\begin{equation}
{}^BC_A
=
C_1(\phi)
C_2(\theta)
C_3(\psi).
\end{equation}
This is the flagship Tait Bryan sequence used throughout the PhysicsLibrary
Euler and quaternion series.
\section{The flagship proper Euler \(3\)-\(1\)-\(3\) sequence}
For intrinsic $3$-$1$-$3$,
\begin{equation}
{}^BC_A
=
C_3(\gamma)
C_1(\beta)
C_3(\alpha).
\end{equation}
This sequence is used as the flagship proper Euler example later in the
series.
Because its sequence label is a palindrome, its equivalent extrinsic sequence
also has the label $3$-$1$-$3$, but the chronological angle association
reverses:
\begin{equation}
\text{intrinsic }3\text{-}1\text{-}3
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }3\text{-}1\text{-}3
(\gamma,\beta,\alpha).
\end{equation}
\section{Intrinsic and extrinsic reference rule}
For completeness, the passive extrinsic $i$-$j$-$k$ rule is
\begin{equation}
{}^BC_A
=
C_i(\alpha)
C_j(\beta)
C_k(\gamma).
\end{equation}
Therefore
\begin{equation}
\text{intrinsic }i\text{-}j\text{-}k
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }k\text{-}j\text{-}i
(\gamma,\beta,\alpha).
\end{equation}
This equivalence should be used when translating sequence descriptions between
moving-axis and fixed-axis sources.
\section{Symmetry across the twelve sequences}
The twelve expanded matrices are not twelve unrelated formulas.
They are generated from the same three elementary matrices and the same
composition rule.
Several useful symmetry observations follow:
\begin{enumerate}
\item Every matrix is proper orthogonal:
\begin{equation}
CC^T=I,
\qquad
\det C=1.
\end{equation}
\item Every Tait Bryan sequence has the same generic middle-angle singularity
condition:
\[
\cos\beta=0.
\]
\item Every proper Euler sequence has the same generic middle-angle
singularity condition:
\[
\sin\beta=0.
\]
\item Relabeling the coordinate axes maps one member of a sequence family into
another.
\item Reversing the intrinsic sequence produces the equivalent extrinsic
description when the angle association is reversed as well.
\end{enumerate}
These symmetries are more useful than memorizing twelve independent matrices.
\section{Verification tests for every sequence}
Each sequence matrix should pass the following tests.
\subsection*{Identity}
\begin{equation}
\alpha=\beta=\gamma=0
\quad\Longrightarrow\quad
{}^BC_A=I.
\end{equation}
\subsection*{First-angle reduction}
Set
\[
\beta=\gamma=0.
\]
Then
\begin{equation}
{}^BC_A=C_i(\alpha).
\end{equation}
\subsection*{Second-angle reduction}
Set
\[
\alpha=\gamma=0.
\]
Then
\begin{equation}
{}^BC_A=C_j(\beta).
\end{equation}
\subsection*{Third-angle reduction}
Set
\[
\alpha=\beta=0.
\]
Then
\begin{equation}
{}^BC_A=C_k(\gamma).
\end{equation}
\subsection*{Reverse coordinate map}
\begin{equation}
{}^AC_B
=
({}^BC_A)^T.
\end{equation}
\subsection*{Quaternion agreement}
For the migrated passive PhysicsLibrary quaternion convention,
\begin{equation}
{}^Bq_A
=
q_k^P(\gamma)
q_j^P(\beta)
q_i^P(\alpha),
\end{equation}
and therefore
\begin{equation}
{}^BC_A
=
C({}^Bq_A).
\end{equation}
\section{Why the twelve legacy sequence pages remain useful}
PhysicsLibrary already has canonical entries for the individual sequences:
\begin{enumerate}
\item Euler 121 sequence;
\item Euler 123 sequence;
\item Euler 131 sequence;
\item Euler 132 sequence;
\item Euler 212 sequence;
\item Euler 213 sequence;
\item Euler 231 sequence;
\item Euler 232 sequence;
\item Euler 312 sequence;
\item Euler 313 sequence;
\item Euler 321 sequence;
\item Euler 323 sequence.
\end{enumerate}
Those pages remain valuable as sequence-specific reference entries.
This EA04 article supplies the common convention and composition framework so
that each legacy sequence page can be modernized without repeating the full
twelve-sequence theory.
\section{Common mistakes}
\begin{enumerate}
\item Treating all twelve sequences as unrelated formulas instead of products
of three elementary matrices.
\item Mixing generic first-second-third angles with aerospace roll-pitch-yaw
names.
\item Forgetting that the rightmost matrix acts first on a coordinate column.
\item Using an extrinsic product while calling the sequence intrinsic.
\item Assuming a repeated first and third axis in a proper Euler sequence means
the two rotations occur about the same physical axis.
\item Using the Tait Bryan singular condition for a proper Euler sequence, or
vice versa.
\item Comparing two expanded matrices before confirming active/passive and
map-direction conventions.
\end{enumerate}
\section{Summary}
There are exactly twelve standard Euler sequences.
The six Tait Bryan sequences are
\begin{equation}
123,\quad132,\quad213,\quad231,\quad312,\quad321.
\end{equation}
The six proper Euler sequences are
\begin{equation}
121,\quad131,\quad212,\quad232,\quad313,\quad323.
\end{equation}
For every intrinsic $i$-$j$-$k$ sequence,
\begin{equation}
{}^BC_A
=
C_k(\gamma)
C_j(\beta)
C_i(\alpha).
\end{equation}
The two sequence families differ in their generic middle-angle singularity:
\begin{equation}
\cos\beta=0
\qquad
\hbox{for Tait Bryan sequences},
\end{equation}
and
\begin{equation}
\sin\beta=0
\qquad
\hbox{for proper Euler sequences}.
\end{equation}
The expanded matrices in this article form the common passive intrinsic
reference set for the remainder of the PhysicsLibrary Euler angle series.
\section{References and further reading}
Henderson provides the classic NASA engineering tabulation of the twelve Euler
sequences and transformation-matrix relationships.
Diebel gives a compact unified treatment of Euler sequences, DCMs, quaternions,
and rotation vectors.
Moore develops orientation through successive reference-frame transformations
and is especially useful for interpreting the frame-chain composition rule.
SciPy and SymPy provide modern software examples in which intrinsic and
extrinsic sequence conventions are explicitly distinguished.
\begin{thebibliography}{9}
\bibitem{Henderson1977}
D. M. Henderson,
\emph{Euler Angles, Quaternions, and Transformation Matrices:
Working Relationships},
JSC-12960,
NASA Johnson Space Center,
1977.
\PMlinkexternal{NASA Technical Reports Server}
{https://ntrs.nasa.gov/citations/19770024290}
\bibitem{Diebel2006}
J. Diebel,
``Representing Attitude:
Euler Angles, Unit Quaternions, and Rotation Vectors,''
Stanford University,
2006.
\PMlinkexternal{Online PDF}
{https://www.astro.rug.nl/software/kapteyn-beta/_downloads/attitude.pdf}
\bibitem{Moore2026}
J. K. Moore,
\emph{Learn Multibody Dynamics},
chapter ``Orientation of Reference Frames,''
2026 edition.
Licensed CC BY 4.0.
\PMlinkexternal{Orientation of Reference Frames}
{https://moorepants.github.io/learn-multibody-dynamics/orientation.html}
\bibitem{SciPyEuler}
SciPy Developers,
``Rotation.from\_euler,''
SciPy documentation.
\PMlinkexternal{SciPy Euler rotation documentation}
{https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.transform.Rotation.from_euler.html}
\bibitem{SymPyReferenceFrames}
SymPy Development Team,
``ReferenceFrame orientation methods,''
SymPy documentation.
\PMlinkexternal{SymPy ReferenceFrame documentation}
{https://docs.sympy.org/latest/modules/physics/vector/api/classes.html}
\end{thebibliography}
\section*{License}
Unless otherwise noted, this PhysicsLibrary entry is intended for release
under the Creative Commons Attribution ShareAlike 4.0 International license.