example of Euler angles: intrinsic and extrinsic rotations
ExampleOfEulerAnglesIntrinsicAndExtrinsicRotations
2026-08-28 22:58:47
2026-08-29 22:15:06
Example
Euler angles: intrinsic and extrinsic rotations
updated the equivalence figure
Euler angles
intrinsic rotations
extrinsic rotations
body fixed rotations
space fixed rotations
moving axes
fixed axes
passive coordinate transformation
exercises
worked solutions
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\section*{Euler Angles: Intrinsic and Extrinsic Rotations
Examples, Exercises, and Solutions}
This entry is the self study companion to
\emph{Euler angles: intrinsic and extrinsic rotations}.
The exercises are designed to make the axis geometry and matrix ordering
automatic 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 an intrinsic $i$-$j$-$k$ sequence,
\begin{equation}
{}^BC_A
=
C_k(\gamma)
C_j(\beta)
C_i(\alpha).
\end{equation}
For an extrinsic $i$-$j$-$k$ sequence,
\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}
Intrinsic means that later rotations use axes of the current intermediate
frame.
Extrinsic means that every rotation axis remains attached to the original
reference frame.
\section{Visual reference}
\begin{center}
\includegraphics[width=0.88\textwidth]{Intrinsic_moving_axis_rotations.png}
\vspace{0.35em}
\textbf{Figure.}
Intrinsic construction: the second rotation uses the moved axis of the
intermediate frame.
\end{center}
\begin{center}
\includegraphics[width=0.88\textwidth]{extrinsic_moving_axis_rotations.png}
\vspace{0.35em}
\textbf{Figure.}
Extrinsic construction: the second rotation uses an axis fixed in the original
reference frame.
\end{center}
\begin{center}
\includegraphics[width=0.94\textwidth]{EA03_intrinsic_extrinsic_equivalence.png}
\vspace{0.35em}
\textbf{Figure.}
Reverse-order equivalence between intrinsic and extrinsic sequence
descriptions.
\end{center}
\section{Exercises}
\begin{enumerate}
\item \textbf{Intrinsic or extrinsic?}
A rotation description says:
\begin{quote}
Rotate by $\alpha$ about $z_A$. Then rotate by $\beta$ about the new $y$
axis. Finally rotate by $\gamma$ about the newest $x$ axis.
\end{quote}
Is the sequence intrinsic or extrinsic?
What is its axis sequence?
\item \textbf{Fixed-axis interpretation.}
A second description says:
\begin{quote}
Rotate by $\alpha$ about $z_A$. Then rotate by $\beta$ about $y_A$.
Finally rotate by $\gamma$ about $x_A$.
\end{quote}
Is the sequence intrinsic or extrinsic?
What is its axis sequence?
\item \textbf{Intrinsic matrix product.}
Write the passive DCM for intrinsic $2$-$3$-$1$ with angles
$(\alpha,\beta,\gamma)$.
\item \textbf{Extrinsic matrix product.}
Write the passive DCM for extrinsic $2$-$3$-$1$ with angles
$(\alpha,\beta,\gamma)$.
\item \textbf{Same labels, different interpretation.}
Explain why intrinsic $2$-$3$-$1$ and extrinsic $2$-$3$-$1$ with the same
angles are generally different rotations.
Write both matrix products side by side.
\item \textbf{Reverse-order equivalent sequence.}
Find the extrinsic sequence and angle order equivalent to intrinsic
\[
3\text{-}2\text{-}1
\]
with angles
\[
(\alpha,\beta,\gamma).
\]
\item \textbf{Reverse the other direction.}
Find the intrinsic sequence equivalent to extrinsic
\[
1\text{-}3\text{-}2
\]
with chronological angles
\[
(\mu,\nu,\rho).
\]
State the corresponding intrinsic angle order.
\item \textbf{Aerospace yaw pitch roll.}
PhysicsLibrary uses intrinsic $3$-$2$-$1$ with
\[
\alpha=\psi,
\qquad
\beta=\theta,
\qquad
\gamma=\phi.
\]
Write the passive DCM product.
Then give the equivalent extrinsic axis sequence and chronological angle list.
\item \textbf{Numerical difference for the same sequence labels.}
Let
\[
\alpha=30^\circ,
\qquad
\beta=20^\circ,
\qquad
\gamma=10^\circ.
\]
Compute the passive intrinsic $3$-$2$-$1$ matrix and the passive extrinsic
$3$-$2$-$1$ matrix.
Verify that they are not equal.
Use
\[
C_1(\lambda)
=
\begin{bmatrix}
1&0&0\\
0&\cos\lambda&\sin\lambda\\
0&-\sin\lambda&\cos\lambda
\end{bmatrix},
\]
\[
C_2(\lambda)
=
\begin{bmatrix}
\cos\lambda&0&-\sin\lambda\\
0&1&0\\
\sin\lambda&0&\cos\lambda
\end{bmatrix},
\]
and
\[
C_3(\lambda)
=
\begin{bmatrix}
\cos\lambda&\sin\lambda&0\\
-\sin\lambda&\cos\lambda&0\\
0&0&1
\end{bmatrix}.
\]
\item \textbf{Numerical reverse-order equivalence.}
Using the same angles as Exercise 9, compare:
\[
\text{intrinsic }3\text{-}2\text{-}1
(\alpha,\beta,\gamma)
\]
with
\[
\text{extrinsic }1\text{-}2\text{-}3
(\gamma,\beta,\alpha).
\]
Verify that their passive DCMs are identical.
\item \textbf{Frame-label derivation.}
For intrinsic $i$-$j$-$k$, let the intermediate frames be
\[
A_0=A,\qquad A_1,\qquad A_2,\qquad A_3=B.
\]
Write the three elementary frame-labeled coordinate maps and derive
\[
{}^BC_A
=
C_k(\gamma)C_j(\beta)C_i(\alpha).
\]
\item \textbf{Single-angle diagnostic.}
For intrinsic $3$-$2$-$1$, set
\[
\beta=0,
\qquad
\gamma=0.
\]
What remains?
Do the same for the equivalent extrinsic $1$-$2$-$3$ representation.
Explain why the two reductions agree.
\item \textbf{Proper Euler subtlety.}
Consider intrinsic $3$-$1$-$3$ with angles
$(\alpha,\beta,\gamma)$.
What extrinsic sequence is equivalent?
What are its chronological angles?
Why is this case potentially confusing even though the reversed sequence still
has the label $3$-$1$-$3$?
\item \textbf{Body fixed and space fixed terminology.}
Match each phrase to intrinsic or extrinsic:
\begin{enumerate}
\item body fixed;
\item space fixed.
\end{enumerate}
Explain what the word ``fixed'' refers to in each case.
\item \textbf{Active/passive independence.}
A student says:
\begin{quote}
Intrinsic means passive and extrinsic means active.
\end{quote}
Explain why this statement is incorrect.
List the four logical combinations of axis convention and transformation type.
\item \textbf{Convention audit.}
A textbook gives the formula
\[
C
=
C_3(\psi)C_2(\theta)C_1(\phi)
\]
and calls it ``$3$-$2$-$1$ yaw pitch roll.''
Give at least four questions that must be answered before deciding whether
this formula agrees with the PhysicsLibrary convention.
\item \textbf{Software interface audit.}
A software interface documents uppercase sequence strings as intrinsic and
lowercase strings as extrinsic.
Which sequence strings would you expect to use for:
\begin{enumerate}
\item intrinsic $3$-$2$-$1$;
\item extrinsic $3$-$2$-$1$;
\item the extrinsic sequence equivalent to intrinsic $3$-$2$-$1$?
\end{enumerate}
Why must you still check the software's active/passive and map-direction
semantics?
\item \textbf{Correct an incorrect equivalence statement.}
A student writes
\[
\text{intrinsic }1\text{-}2\text{-}3
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }3\text{-}2\text{-}1
(\alpha,\beta,\gamma).
\]
Identify the error and write the correct equivalence.
\end{enumerate}
\section{Solutions}
\subsection*{Solution 1: intrinsic or extrinsic?}
The second rotation is about the \emph{new} $y$ axis, and the third is about
the \emph{newest} $x$ axis.
Therefore the axes move with the intermediate frame.
The sequence is intrinsic
\[
3\text{-}2\text{-}1.
\]
\subsection*{Solution 2: fixed axis interpretation}
Every rotation is explicitly about an axis of the original frame $A$:
\[
z_A,\qquad y_A,\qquad x_A.
\]
Therefore the construction is extrinsic
\[
3\text{-}2\text{-}1.
\]
\subsection*{Solution 3: intrinsic matrix product}
For intrinsic $2$-$3$-$1$,
\[
i=2,\qquad j=3,\qquad k=1.
\]
Hence
\begin{equation}
{}^BC_A
=
C_1(\gamma)
C_3(\beta)
C_2(\alpha).
\end{equation}
\subsection*{Solution 4: extrinsic matrix product}
For extrinsic $2$-$3$-$1$, the generic passive rule is
\[
{}^BC_A
=
C_i(\alpha)
C_j(\beta)
C_k(\gamma).
\]
Therefore
\begin{equation}
{}^BC_A
=
C_2(\alpha)
C_3(\beta)
C_1(\gamma).
\end{equation}
\subsection*{Solution 5: same labels, different interpretation}
Intrinsic $2$-$3$-$1$ gives
\begin{equation}
C_{\rm int}
=
C_1(\gamma)
C_3(\beta)
C_2(\alpha).
\end{equation}
Extrinsic $2$-$3$-$1$ gives
\begin{equation}
C_{\rm ext}
=
C_2(\alpha)
C_3(\beta)
C_1(\gamma).
\end{equation}
The products have different factor order.
Since finite rotations about different axes generally do not commute,
\[
C_{\rm int}\neq C_{\rm ext}
\]
in general.
\subsection*{Solution 6: reverse-order equivalent sequence}
The general equivalence is
\[
\text{intrinsic }i\text{-}j\text{-}k
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }k\text{-}j\text{-}i
(\gamma,\beta,\alpha).
\]
Thus
\begin{equation}
\text{intrinsic }3\text{-}2\text{-}1
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }1\text{-}2\text{-}3
(\gamma,\beta,\alpha).
\end{equation}
\subsection*{Solution 7: reverse the other direction}
The extrinsic sequence is
\[
1\text{-}3\text{-}2
\]
with chronological angles
\[
(\mu,\nu,\rho).
\]
Reverse the axis order and reverse the associated angle order.
Therefore
\begin{equation}
\text{extrinsic }1\text{-}3\text{-}2
(\mu,\nu,\rho)
\equiv
\text{intrinsic }2\text{-}3\text{-}1
(\rho,\nu,\mu).
\end{equation}
\subsection*{Solution 8: aerospace yaw pitch roll}
For PhysicsLibrary intrinsic $3$-$2$-$1$,
\[
\alpha=\psi,\qquad
\beta=\theta,\qquad
\gamma=\phi.
\]
Hence
\begin{equation}
{}^BC_A
=
C_1(\phi)
C_2(\theta)
C_3(\psi).
\end{equation}
The equivalent extrinsic sequence is
\[
1\text{-}2\text{-}3
\]
with chronological angle list
\begin{equation}
(\phi,\theta,\psi).
\end{equation}
Thus the equivalent fixed axis description is roll about $1_A$, then pitch
about $2_A$, then yaw about $3_A$.
\subsection*{Solution 9: numerical difference for the same sequence labels}
For intrinsic $3$-$2$-$1$,
\[
C_{\rm int}
=
C_1(10^\circ)
C_2(20^\circ)
C_3(30^\circ).
\]
Numerically,
\begin{equation}
C_{\rm int}
\approx
\begin{bmatrix}
0.8138&0.4698&-0.3420\\
-0.4410&0.8826&0.1632\\
0.3785&0.0180&0.9254
\end{bmatrix}.
\end{equation}
For extrinsic $3$-$2$-$1$,
\[
C_{\rm ext}
=
C_3(30^\circ)
C_2(20^\circ)
C_1(10^\circ).
\]
Numerically,
\begin{equation}
C_{\rm ext}
\approx
\begin{bmatrix}
0.8138&0.5438&-0.2049\\
-0.4698&0.8232&0.3188\\
0.3420&-0.1632&0.9254
\end{bmatrix}.
\end{equation}
The matrices are not equal.
Thus identical sequence labels do not imply identical orientation when one
description is intrinsic and the other extrinsic.
\subsection*{Solution 10: numerical reverse-order equivalence}
Intrinsic $3$-$2$-$1$ with
$(\alpha,\beta,\gamma)$ gives
\[
C_{\rm int}
=
C_1(\gamma)
C_2(\beta)
C_3(\alpha).
\]
Extrinsic $1$-$2$-$3$ with
$(\gamma,\beta,\alpha)$ gives
\[
C_{\rm ext}
=
C_1(\gamma)
C_2(\beta)
C_3(\alpha).
\]
The products are symbolically identical.
For the numerical angles of Exercise 9, both give
\begin{equation}
C
\approx
\begin{bmatrix}
0.8138&0.4698&-0.3420\\
-0.4410&0.8826&0.1632\\
0.3785&0.0180&0.9254
\end{bmatrix}.
\end{equation}
\subsection*{Solution 11: frame-label derivation}
The three intrinsic coordinate maps are
\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}
Compose the maps by matching adjacent frame labels:
\[
{}^BC_A
=
{}^BC_{A_2}
{}^{A_2}C_{A_1}
{}^{A_1}C_A.
\]
Therefore
\begin{equation}
{}^BC_A
=
C_k(\gamma)
C_j(\beta)
C_i(\alpha).
\end{equation}
\subsection*{Solution 12: single-angle diagnostic}
For intrinsic $3$-$2$-$1$,
\[
{}^BC_A
=
C_1(\gamma)
C_2(\beta)
C_3(\alpha).
\]
Set
\[
\beta=0,\qquad \gamma=0.
\]
Then
\begin{equation}
{}^BC_A
=
C_3(\alpha).
\end{equation}
The equivalent extrinsic representation is $1$-$2$-$3$ with angle list
$(\gamma,\beta,\alpha)$.
Setting $\gamma=\beta=0$ leaves the final chronological extrinsic rotation
about fixed axis $3_A$ through $\alpha$, whose passive product is also
\begin{equation}
{}^BC_A
=
C_3(\alpha).
\end{equation}
Thus the single-angle reduction agrees.
\subsection*{Solution 13: proper Euler subtlety}
Intrinsic $3$-$1$-$3$ with angles
$(\alpha,\beta,\gamma)$ is equivalent to extrinsic
\[
3\text{-}1\text{-}3
\]
with chronological angles
\[
(\gamma,\beta,\alpha).
\]
So
\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}
The sequence label is a palindrome, so reversing the axis order does not
visibly change the digits.
The angle association still reverses.
That is why proper Euler sequences such as $3$-$1$-$3$ can be especially easy
to misinterpret.
\subsection*{Solution 14: body fixed and space fixed terminology}
Body fixed corresponds to intrinsic.
Each new rotation axis belongs to the current intermediate frame and therefore
moves with the body or moving frame.
Space fixed corresponds to extrinsic.
Each rotation axis remains attached to the original reference frame.
The word ``fixed'' therefore refers to which frame owns the successive
rotation axes, not to whether the mathematical transformation is active or
passive.
\subsection*{Solution 15: active/passive independence}
The statement is incorrect because two independent choices are being mixed.
Intrinsic versus extrinsic specifies the axis behavior.
Active versus passive specifies whether the physical vector or coordinate
frame is being transformed.
The four logical combinations are:
\begin{enumerate}
\item intrinsic passive;
\item extrinsic passive;
\item intrinsic active;
\item extrinsic active.
\end{enumerate}
PhysicsLibrary uses intrinsic passive as its default Euler convention, but the
other three combinations are mathematically valid.
\subsection*{Solution 16: convention audit}
Before comparing
\[
C=C_3(\psi)C_2(\theta)C_1(\phi)
\]
with PhysicsLibrary, one should ask at least:
\begin{enumerate}
\item Is the transformation active or passive?
\item What coordinate map direction is represented?
\item Is the sequence intrinsic or extrinsic?
\item Does ``$3$-$2$-$1$'' describe chronological axis order or written
matrix order?
\item Are $\phi,\theta,\psi$ roll, pitch, and yaw, or merely first, second,
and third sequence angles?
\item Does the source use the same right hand positive-angle convention?
\end{enumerate}
Without these answers, matrix order alone is not enough to compare
conventions.
\subsection*{Solution 17: software interface audit}
Under the stated software convention:
\begin{enumerate}
\item intrinsic $3$-$2$-$1$ would use uppercase
\[
\texttt{ZYX};
\]
\item extrinsic $3$-$2$-$1$ would use lowercase
\[
\texttt{zyx};
\]
\item the extrinsic sequence equivalent to intrinsic $3$-$2$-$1$ is
$1$-$2$-$3$, so it would use
\[
\texttt{xyz}.
\]
\end{enumerate}
One must still verify whether the software rotation object represents an active
vector rotation or a passive coordinate transformation, and which frame-map
direction its matrix represents.
Intrinsic/extrinsic notation alone does not determine those choices.
\subsection*{Solution 18: correct an incorrect equivalence statement}
The student's axis reversal is correct, but the angle order was not reversed.
The correct statement is
\begin{equation}
\text{intrinsic }1\text{-}2\text{-}3
(\alpha,\beta,\gamma)
\equiv
\text{extrinsic }3\text{-}2\text{-}1
(\gamma,\beta,\alpha).
\end{equation}
The axes and their associated angles must be reversed together.
\section{Compact review}
The two passive sequence rules are
\begin{equation}
C_{\rm intrinsic}
=
C_k(\gamma)
C_j(\beta)
C_i(\alpha),
\end{equation}
and
\begin{equation}
C_{\rm extrinsic}
=
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 words intrinsic and extrinsic describe the axes used for successive
rotations.
They do not, by themselves, specify active versus passive transformation.
\section{Sources and exercise provenance}
The exercises and worked solutions in this companion are newly written for
PhysicsLibrary to reinforce the convention and frame geometry developed in
\emph{Euler angles: intrinsic and extrinsic rotations}.
Henderson provides the classic NASA engineering background for Euler sequence
relationships.
Moore develops successive reference-frame orientation with explicit
intermediate frames.
SymPy distinguishes body fixed and space fixed reference-frame sequences in
its mechanics API.
SciPy explicitly distinguishes intrinsic and extrinsic Euler sequence strings.
\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{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{SymPyReferenceFrames}
SymPy Development Team,
``ReferenceFrame orientation methods,''
SymPy documentation.
\PMlinkexternal{SymPy ReferenceFrame documentation}
{https://docs.sympy.org/latest/modules/physics/vector/api/classes.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}
\end{thebibliography}
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