example of Euler angles: sequence composition and the twelve standard sequences
ExampleOfEulerAnglesSequenceCompositionAndTheTwelveStandardSequences
2026-08-29 03:30:12
2026-08-29 03:30:12
Example
Euler angles: sequence composition and the twelve standard sequences
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
Examples, Exercises, and Solutions}
This entry is the self study companion to
\emph{Euler angles: sequence composition and the twelve standard sequences}.
The goal is to make sequence classification and composition routine enough
that later Euler formulas can be derived rather than memorized.
All exercises are stated first. Complete worked solutions follow afterward.
\section{Convention summary}
PhysicsLibrary uses passive coordinate maps:
\begin{equation}
{}^B\mathbf v
=
{}^BC_A\,{}^A\mathbf v.
\end{equation}
For intrinsic $i$-$j$-$k$,
\begin{equation}
{}^BC_A
=
C_k(\gamma)C_j(\beta)C_i(\alpha).
\end{equation}
For extrinsic $i$-$j$-$k$,
\begin{equation}
{}^BC_A
=
C_i(\alpha)C_j(\beta)C_k(\gamma).
\end{equation}
The reverse-order equivalence is
\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}
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}
\section{Visual reference}
\begin{center}
\includegraphics[width=0.94\textwidth]{EA04_intrinsic_sequence_composition.png}
\vspace{0.35em}
\textbf{Figure.}
Generic intrinsic frame-chain composition.
\end{center}
\begin{center}
\includegraphics[width=0.88\textwidth]{EA04_twelve_standard_sequences.png}
\vspace{0.35em}
\textbf{Figure.}
The twelve standard intrinsic Euler sequences separated into Tait Bryan and
proper Euler families.
\end{center}
\section{Exercises}
\begin{enumerate}
\item \textbf{Derive the count of twelve.}
Explain why the number of standard three-angle Euler sequences is
\[
3\times2\times2=12.
\]
State what each factor counts.
\item \textbf{Classify the sequence family.}
Classify each sequence as Tait Bryan or proper Euler:
\[
123,\quad121,\quad312,\quad313,\quad231,\quad232.
\]
State the family criterion.
\item \textbf{List all six Tait Bryan sequences.}
Construct all six standard sequences with three distinct axis labels.
\item \textbf{List all six proper Euler sequences.}
Construct all six standard sequences in which the first and third axis labels
match.
\item \textbf{Write an intrinsic product.}
Write the passive intrinsic product for sequence $1$-$3$-$2$ with angles
$(\alpha,\beta,\gamma)$.
\item \textbf{Write a proper Euler product.}
Write the passive intrinsic product for $3$-$1$-$3$.
Why can the first and third rotations not generally be combined?
\item \textbf{Frame-chain derivation.}
For intrinsic $2$-$3$-$1$, introduce intermediate frames
$A_0,A_1,A_2,A_3$ and derive the complete passive map ${}^BC_A$.
\item \textbf{Rightmost factor diagnostic.}
Explain why
\[
{}^BC_A=C_1(\gamma)C_3(\beta)C_2(\alpha)
\]
still describes chronological intrinsic axis order $2$-$3$-$1$.
\item \textbf{Single-angle reductions.}
For intrinsic $3$-$1$-$2$,
\[
{}^BC_A=C_2(\gamma)C_1(\beta)C_3(\alpha).
\]
Find the remaining elementary matrix when:
\begin{enumerate}
\item $\beta=\gamma=0$;
\item $\alpha=\gamma=0$;
\item $\alpha=\beta=0$.
\end{enumerate}
\item \textbf{Intrinsic/extrinsic conversion.}
Find the extrinsic sequence and chronological angle order equivalent to
\[
\text{intrinsic }2\text{-}3\text{-}1(\alpha,\beta,\gamma).
\]
\item \textbf{Palindrome sequence subtlety.}
Find the extrinsic description equivalent to intrinsic
\[
3\text{-}2\text{-}3(\alpha,\beta,\gamma).
\]
Why is this conversion easy to misread?
\item \textbf{Tait Bryan singularity.}
State the generic middle-angle singularity condition for every Tait Bryan
sequence and its values on the common principal branch.
\item \textbf{Proper Euler singularity.}
State the generic middle-angle singularity condition for every proper Euler
sequence and its values on the common principal branch.
\item \textbf{Aerospace 3-2-1 specialization.}
Identify $(\alpha,\beta,\gamma)$ with $(\psi,\theta,\phi)$ and write the
PhysicsLibrary passive yaw pitch roll product.
\item \textbf{Numerical 3-2-1 composition.}
Using
\[
\psi=30^\circ,\qquad
\theta=20^\circ,\qquad
\phi=10^\circ,
\]
compute
\[
{}^BC_A=C_1(\phi)C_2(\theta)C_3(\psi).
\]
\item \textbf{Compare two sequence orders.}
Using
\[
\alpha=30^\circ,\qquad
\beta=20^\circ,\qquad
\gamma=10^\circ,
\]
compare intrinsic $3$-$2$-$1$ with intrinsic $1$-$2$-$3$.
\item \textbf{Recognize the family from the singularity.}
A source states that its chart is singular when $\sin\beta=0$.
Which sequence family is it using?
A second source gives $\cos\beta=0$. Which family is that?
\item \textbf{Quaternion counterpart.}
Write the passive quaternion product for intrinsic $2$-$1$-$3$ and state the
DCM consistency relation.
\item \textbf{Debug a wrong product.}
A student claims intrinsic $1$-$2$-$3$ is
\[
{}^BC_A=C_1(\alpha)C_2(\beta)C_3(\gamma).
\]
Identify the mistake and write the correct product.
\item \textbf{Recover a sequence from a product.}
A passive intrinsic DCM is
\[
{}^BC_A=C_2(\gamma)C_3(\beta)C_1(\alpha).
\]
Identify the chronological sequence and its family.
\item \textbf{Recover a proper Euler sequence from a product.}
A passive intrinsic DCM is
\[
{}^BC_A=C_3(\gamma)C_1(\beta)C_3(\alpha).
\]
Identify the sequence and family.
\item \textbf{Verification battery.}
List at least five checks that should be applied to a derived Euler sequence
matrix before using it in software.
\end{enumerate}
\section{Solutions}
\subsection*{Solution 1: derive the count of twelve}
There are three choices for the first axis.
The second axis must differ from the first, leaving two choices.
The third axis has two standard choices: the remaining unused axis, producing
a Tait Bryan sequence, or the first axis again, producing a proper Euler
sequence.
Therefore
\begin{equation}
3\times2\times2=12.
\end{equation}
\subsection*{Solution 2: classify the sequence family}
Tait Bryan means all three axis labels are different. Proper Euler means the
first and third labels match.
Thus
\[
123,\quad312,\quad231
\]
are Tait Bryan, while
\[
121,\quad313,\quad232
\]
are proper Euler.
\subsection*{Solution 3: list all six Tait Bryan sequences}
The six are
\begin{equation}
123,\quad132,\quad213,\quad231,\quad312,\quad321.
\end{equation}
\subsection*{Solution 4: list all six proper Euler sequences}
The six are
\begin{equation}
121,\quad131,\quad212,\quad232,\quad313,\quad323.
\end{equation}
\subsection*{Solution 5: write an intrinsic product}
For $1$-$3$-$2$,
\[
i=1,\qquad j=3,\qquad k=2.
\]
Therefore
\begin{equation}
{}^BC_A=C_2(\gamma)C_3(\beta)C_1(\alpha).
\end{equation}
\subsection*{Solution 6: write a proper Euler product}
For intrinsic $3$-$1$-$3$,
\begin{equation}
{}^BC_A=C_3(\gamma)C_1(\beta)C_3(\alpha).
\end{equation}
The first rotation is about axis $3$ of the initial frame. After the middle
rotation, the third rotation is about axis $3$ of a different intermediate
frame. The two axis labels match, but the physical axes generally do not.
\subsection*{Solution 7: frame-chain derivation}
For intrinsic $2$-$3$-$1$,
\begin{equation}
{}^{A_1}C_A=C_2(\alpha),
\end{equation}
\begin{equation}
{}^{A_2}C_{A_1}=C_3(\beta),
\end{equation}
and
\begin{equation}
{}^BC_{A_2}=C_1(\gamma).
\end{equation}
Therefore
\begin{equation}
{}^BC_A
=
{}^BC_{A_2}\,{}^{A_2}C_{A_1}\,{}^{A_1}C_A
=
C_1(\gamma)C_3(\beta)C_2(\alpha).
\end{equation}
\subsection*{Solution 8: rightmost factor diagnostic}
Coordinate columns are multiplied from the left. Hence $C_2(\alpha)$ acts on
${}^A\mathbf v$ first, followed by $C_3(\beta)$ and then $C_1(\gamma)$.
The chronological intrinsic order is therefore
\[
2\longrightarrow3\longrightarrow1.
\]
\subsection*{Solution 9: single-angle reductions}
From
\[
{}^BC_A=C_2(\gamma)C_1(\beta)C_3(\alpha),
\]
we obtain
\begin{equation}
\beta=\gamma=0
\quad\Longrightarrow\quad
{}^BC_A=C_3(\alpha),
\end{equation}
\begin{equation}
\alpha=\gamma=0
\quad\Longrightarrow\quad
{}^BC_A=C_1(\beta),
\end{equation}
and
\begin{equation}
\alpha=\beta=0
\quad\Longrightarrow\quad
{}^BC_A=C_2(\gamma).
\end{equation}
\subsection*{Solution 10: intrinsic/extrinsic conversion}
Reverse both axis order and associated angle order:
\begin{equation}
\text{intrinsic }2\text{-}3\text{-}1
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }1\text{-}3\text{-}2
(\gamma,\beta,\alpha).
\end{equation}
\subsection*{Solution 11: palindrome sequence subtlety}
Reversing $3$-$2$-$3$ leaves the same digit string, so
\begin{equation}
\text{intrinsic }3\text{-}2\text{-}3
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }3\text{-}2\text{-}3
(\gamma,\beta,\alpha).
\end{equation}
The axis labels look unchanged, but the angle association still reverses.
\subsection*{Solution 12: Tait Bryan singularity}
The generic singularity is
\begin{equation}
\cos\beta=0.
\end{equation}
On
\[
-\frac{\pi}{2}\leq\beta\leq\frac{\pi}{2},
\]
the singular values are
\begin{equation}
\beta=\pm\frac{\pi}{2}.
\end{equation}
\subsection*{Solution 13: proper Euler singularity}
The generic singularity is
\begin{equation}
\sin\beta=0.
\end{equation}
On
\[
0\leq\beta\leq\pi,
\]
the singular values are
\begin{equation}
\beta=0
\qquad\hbox{or}\qquad
\beta=\pi.
\end{equation}
\subsection*{Solution 14: aerospace 3-2-1 specialization}
PhysicsLibrary uses
\[
\alpha=\psi,\qquad
\beta=\theta,\qquad
\gamma=\phi.
\]
Thus
\begin{equation}
{}^BC_A=C_1(\phi)C_2(\theta)C_3(\psi).
\end{equation}
\subsection*{Solution 15: numerical 3-2-1 composition}
Using the elementary passive matrices gives
\begin{equation}
{}^BC_A
\approx
\begin{bmatrix}
0.81380&0.46985&-0.34202\\
-0.44097&0.88256&0.16318\\
0.37852&0.01803&0.92542
\end{bmatrix}.
\end{equation}
This matrix is orthogonal and has determinant $+1$ to numerical precision.
\subsection*{Solution 16: compare two sequence orders}
For intrinsic $3$-$2$-$1$,
\begin{equation}
C_{321}
\approx
\begin{bmatrix}
0.81380&0.46985&-0.34202\\
-0.44097&0.88256&0.16318\\
0.37852&0.01803&0.92542
\end{bmatrix}.
\end{equation}
For intrinsic $1$-$2$-$3$,
\begin{equation}
C_{123}
\approx
\begin{bmatrix}
0.92542&0.31880&-0.20487\\
-0.16318&0.82317&0.54384\\
0.34202&-0.46985&0.81380
\end{bmatrix}.
\end{equation}
They are different, demonstrating sequence dependence and noncommutativity.
\subsection*{Solution 17: recognize the family from the singularity}
If
\[
\sin\beta=0,
\]
the sequence is proper Euler.
If
\[
\cos\beta=0,
\]
the sequence is Tait Bryan.
\subsection*{Solution 18: quaternion counterpart}
For intrinsic $2$-$1$-$3$,
\begin{equation}
{}^Bq_A=q_3^P(\gamma)q_1^P(\beta)q_2^P(\alpha).
\end{equation}
The passive quaternion and DCM must satisfy
\begin{equation}
{}^BC_A=C({}^Bq_A).
\end{equation}
\subsection*{Solution 19: debug a wrong product}
The student wrote the matrices in chronological left-to-right order.
The correct passive intrinsic product is
\begin{equation}
{}^BC_A=C_3(\gamma)C_2(\beta)C_1(\alpha).
\end{equation}
\subsection*{Solution 20: recover a sequence from a product}
Compare
\[
C_2(\gamma)C_3(\beta)C_1(\alpha)
\]
with
\[
C_k(\gamma)C_j(\beta)C_i(\alpha).
\]
Thus
\[
i=1,\qquad j=3,\qquad k=2.
\]
The sequence is
\begin{equation}
1\text{-}3\text{-}2,
\end{equation}
which is Tait Bryan.
\subsection*{Solution 21: recover a proper Euler sequence from a product}
The product
\[
C_3(\gamma)C_1(\beta)C_3(\alpha)
\]
has
\[
i=3,\qquad j=1,\qquad k=3.
\]
Therefore the sequence is
\begin{equation}
3\text{-}1\text{-}3,
\end{equation}
which is proper Euler.
\subsection*{Solution 22: verification battery}
Useful checks include:
\begin{enumerate}
\item zero angles give the identity;
\item each single-angle reduction gives the correct elementary passive matrix;
\item $CC^T=I$;
\item $\det C=1$;
\item ${}^AC_B=({}^BC_A)^T$;
\item known $90^\circ$ sign tests agree with the passive convention;
\item quaternion agreement:
\[
{}^BC_A=C({}^Bq_A);
\]
\item Euler to DCM to Euler round trips recover the selected principal branch
away from singularities.
\end{enumerate}
\section{Compact review}
For every intrinsic sequence,
\begin{equation}
{}^BC_A=C_k(\gamma)C_j(\beta)C_i(\alpha).
\end{equation}
There are six Tait Bryan sequences:
\begin{equation}
123,\quad132,\quad213,\quad231,\quad312,\quad321.
\end{equation}
There are six proper Euler sequences:
\begin{equation}
121,\quad131,\quad212,\quad232,\quad313,\quad323.
\end{equation}
Their generic singularity conditions are
\begin{equation}
\cos\beta=0
\qquad
\hbox{for Tait Bryan},
\end{equation}
and
\begin{equation}
\sin\beta=0
\qquad
\hbox{for proper Euler}.
\end{equation}
\section{Sources and exercise provenance}
The exercises and worked solutions in this companion are newly written for
PhysicsLibrary to reinforce the sequence framework developed in
\emph{Euler angles: sequence composition and the twelve standard sequences}.
\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}
\end{thebibliography}
\section*{License}
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under the Creative Commons Attribution ShareAlike 4.0 International license.