Euler Angles: Intrinsic and Extrinsic Rotations Examples, Exercises, and Solutions

This entry is the self study companion to 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.

Convention summary

PhysicsLibrary uses passive coordinate maps:

$\displaystyle {}^B\mathbf v = {}^BC_A\,{}^A\mathbf v.$

(1)

For an intrinsic $i$ - $j$ - $k$ sequence,

$\displaystyle {}^BC_A = C_k(\gamma) C_j(\beta) C_i(\alpha).$

(2)

For an extrinsic $i$ - $j$ - $k$ sequence,

$\displaystyle {}^BC_A = C_i(\alpha) C_j(\beta) C_k(\gamma).$

(3)

Therefore

intrinsic $\displaystyle i$ - $\displaystyle j$ - $\displaystyle k (\alpha,\beta,\gamma) \equiv$ extrinsic $\displaystyle k$ - $\displaystyle j$ - $\displaystyle i (\gamma,\beta,\alpha).$

(4)

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.

Visual reference

Image EA03_intrinsic_moving_axis_second_rotation

Figure. Intrinsic construction: the second rotation uses the moved axis of the intermediate frame.

Image EA03_extrinsic_fixed_axis_second_rotation

Figure. Extrinsic construction: the second rotation uses an axis fixed in the original reference frame.

Image EA03_intrinsic_extrinsic_equivalence

Figure. Reverse-order equivalence between intrinsic and extrinsic sequence descriptions.

Exercises

Intrinsic or extrinsic?
A rotation description says:

Rotate by $\alpha$ about . Then rotate by $\beta$ about the new axis. Finally rotate by $\gamma$ about the newest axis.

Is the sequence intrinsic or extrinsic?

What is its axis sequence?
Fixed-axis interpretation.
A second description says:

Rotate by $\alpha$ about . Then rotate by $\beta$ about . Finally rotate by $\gamma$ about .

Is the sequence intrinsic or extrinsic?

What is its axis sequence?
Intrinsic matrix product.
Write the passive DCM for intrinsic -- with angles $(\alpha,\beta,\gamma)$ .
Extrinsic matrix product.
Write the passive DCM for extrinsic -- with angles $(\alpha,\beta,\gamma)$ .
Same labels, different interpretation.
Explain why intrinsic -- and extrinsic -- with the same angles are generally different rotations.

Write both matrix products side by side.
Reverse-order equivalent sequence.
Find the extrinsic sequence and angle order equivalent to intrinsic

$\displaystyle 3$ - $\displaystyle 2$ - $\displaystyle 1$

with angles

$\displaystyle (\alpha,\beta,\gamma).$
Reverse the other direction.
Find the intrinsic sequence equivalent to extrinsic

$\displaystyle 1$ - $\displaystyle 3$ - $\displaystyle 2$

with chronological angles

$\displaystyle (\mu,\nu,\rho).$

State the corresponding intrinsic angle order.
Aerospace yaw pitch roll.
PhysicsLibrary uses intrinsic -- with

$\displaystyle \alpha=\psi, \qquad \beta=\theta, \qquad \gamma=\phi.$

Write the passive DCM product.

Then give the equivalent extrinsic axis sequence and chronological angle list.
Numerical difference for the same sequence labels.
Let

$\displaystyle \alpha=30^\circ, \qquad \beta=20^\circ, \qquad \gamma=10^\circ.$

Compute the passive intrinsic -- matrix and the passive extrinsic -- matrix.

Verify that they are not equal.

Use

$\displaystyle C_1(\lambda) = \begin{bmatrix} 1&0&0\ 0&\cos\lambda&\sin\lambda\ 0&-\sin\lambda&\cos\lambda \end{bmatrix},$

$\displaystyle C_2(\lambda) = \begin{bmatrix} \cos\lambda&0&-\sin\lambda\ 0&1&0\ \sin\lambda&0&\cos\lambda \end{bmatrix},$

and

$\displaystyle C_3(\lambda) = \begin{bmatrix} \cos\lambda&\sin\lambda&0\ -\sin\lambda&\cos\lambda&0\ 0&0&1 \end{bmatrix}.$
Numerical reverse-order equivalence.
Using the same angles as Exercise 9, compare:

intrinsic $\displaystyle 3$ - $\displaystyle 2$ - $\displaystyle 1 (\alpha,\beta,\gamma)$

with

extrinsic $\displaystyle 1$ - $\displaystyle 2$ - $\displaystyle 3 (\gamma,\beta,\alpha).$

Verify that their passive DCMs are identical.
Frame-label derivation.
For intrinsic --, let the intermediate frames be

$\displaystyle A_0=A,\qquad A_1,\qquad A_2,\qquad A_3=B.$

Write the three elementary frame-labeled coordinate maps and derive

$\displaystyle {}^BC_A = C_k(\gamma)C_j(\beta)C_i(\alpha).$
Single-angle diagnostic.
For intrinsic --, set

$\displaystyle \beta=0, \qquad \gamma=0.$

What remains?

Do the same for the equivalent extrinsic -- representation.

Explain why the two reductions agree.
Proper Euler subtlety.
Consider intrinsic -- 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 --?
Body fixed and space fixed terminology.
Match each phrase to intrinsic or extrinsic:
1. body fixed;
2. space fixed.
Explain what the word “fixed” refers to in each case.
Active/passive independence.
A student says:

Intrinsic means passive and extrinsic means active.

Explain why this statement is incorrect.

List the four logical combinations of axis convention and transformation type.
Convention audit.
A textbook gives the formula

$\displaystyle C = C_3(\psi)C_2(\theta)C_1(\phi)$

and calls it “-- yaw pitch roll.”

Give at least four questions that must be answered before deciding whether this formula agrees with the PhysicsLibrary convention.
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:
1. intrinsic --;
2. extrinsic --;
3. the extrinsic sequence equivalent to intrinsic --?
Why must you still check the software's active/passive and map-direction semantics?
Correct an incorrect equivalence statement.
A student writes

intrinsic $\displaystyle 1$ - $\displaystyle 2$ - $\displaystyle 3 (\alpha,\beta,\gamma) \equiv$ extrinsic $\displaystyle 3$ - $\displaystyle 2$ - $\displaystyle 1 (\alpha,\beta,\gamma).$

Identify the error and write the correct equivalence.

Solutions

Solution 1: intrinsic or extrinsic?

The second rotation is about the new $y$ axis, and the third is about the newest $x$ axis.

Therefore the axes move with the intermediate frame.

The sequence is intrinsic

$\displaystyle 3$ - $\displaystyle 2$ - $\displaystyle 1.$

Solution 2: fixed axis interpretation

Every rotation is explicitly about an axis of the original frame $A$ :

$\displaystyle z_A,\qquad y_A,\qquad x_A.$

Therefore the construction is extrinsic

$\displaystyle 3$ - $\displaystyle 2$ - $\displaystyle 1.$

Solution 3: intrinsic matrix product

For intrinsic $2$ - $3$ - $1$ ,

$\displaystyle i=2,\qquad j=3,\qquad k=1.$

Hence

$\displaystyle {}^BC_A = C_1(\gamma) C_3(\beta) C_2(\alpha).$

(5)

Solution 4: extrinsic matrix product

For extrinsic $2$ - $3$ - $1$ , the generic passive rule is

$\displaystyle {}^BC_A = C_i(\alpha) C_j(\beta) C_k(\gamma).$

Therefore

$\displaystyle {}^BC_A = C_2(\alpha) C_3(\beta) C_1(\gamma).$

(6)

Solution 5: same labels, different interpretation

Intrinsic $2$ - $3$ - $1$ gives

$\displaystyle C_{\rm int} = C_1(\gamma) C_3(\beta) C_2(\alpha).$

(7)

Extrinsic $2$ - $3$ - $1$ gives

$\displaystyle C_{\rm ext} = C_2(\alpha) C_3(\beta) C_1(\gamma).$

(8)

The products have different factor order.

Since finite rotations about different axes generally do not commute,

$\displaystyle C_{\rm int}\neq C_{\rm ext}$

in general.

Solution 6: reverse-order equivalent sequence

The general equivalence is

intrinsic $\displaystyle i$ - $\displaystyle j$ - $\displaystyle k (\alpha,\beta,\gamma) \equiv$ extrinsic $\displaystyle k$ - $\displaystyle j$ - $\displaystyle i (\gamma,\beta,\alpha).$

Thus

intrinsic $\displaystyle 3$ - $\displaystyle 2$ - $\displaystyle 1 (\alpha,\beta,\gamma) \equiv$ extrinsic $\displaystyle 1$ - $\displaystyle 2$ - $\displaystyle 3 (\gamma,\beta,\alpha).$

(9)

Solution 7: reverse the other direction

The extrinsic sequence is

$\displaystyle 1$ - $\displaystyle 3$ - $\displaystyle 2$

with chronological angles

$\displaystyle (\mu,\nu,\rho).$

Reverse the axis order and reverse the associated angle order.

Therefore

extrinsic $\displaystyle 1$ - $\displaystyle 3$ - $\displaystyle 2 (\mu,\nu,\rho) \equiv$ intrinsic $\displaystyle 2$ - $\displaystyle 3$ - $\displaystyle 1 (\rho,\nu,\mu).$

(10)

Solution 8: aerospace yaw pitch roll

For PhysicsLibrary intrinsic $3$ - $2$ - $1$ ,

$\displaystyle \alpha=\psi,\qquad \beta=\theta,\qquad \gamma=\phi.$

Hence

$\displaystyle {}^BC_A = C_1(\phi) C_2(\theta) C_3(\psi).$

(11)

The equivalent extrinsic sequence is

$\displaystyle 1$ - $\displaystyle 2$ - $\displaystyle 3$

with chronological angle list

$\displaystyle (\phi,\theta,\psi).$

(12)

Thus the equivalent fixed axis description is roll about $1_A$ , then pitch about $2_A$ , then yaw about $3_A$ .

Solution 9: numerical difference for the same sequence labels

For intrinsic $3$ - $2$ - $1$ ,

$\displaystyle C_{\rm int} = C_1(10^\circ) C_2(20^\circ) C_3(30^\circ).$

Numerically,

$\displaystyle C_{\rm int} \approx \begin{bmatrix} 0.8138&0.4698&-0.3420\ -0.4410&0.8826&0.1632\ 0.3785&0.0180&0.9254 \end{bmatrix}.$

(13)

For extrinsic $3$ - $2$ - $1$ ,

$\displaystyle C_{\rm ext} = C_3(30^\circ) C_2(20^\circ) C_1(10^\circ).$

Numerically,

$\displaystyle C_{\rm ext} \approx \begin{bmatrix} 0.8138&0.5438&-0.2049\ -0.4698&0.8232&0.3188\ 0.3420&-0.1632&0.9254 \end{bmatrix}.$

(14)

The matrices are not equal.

Thus identical sequence labels do not imply identical orientation when one description is intrinsic and the other extrinsic.

Solution 10: numerical reverse-order equivalence

Intrinsic $3$ - $2$ - $1$ with $(\alpha,\beta,\gamma)$ gives

$\displaystyle C_{\rm int} = C_1(\gamma) C_2(\beta) C_3(\alpha).$

Extrinsic $1$ - $2$ - $3$ with $(\gamma,\beta,\alpha)$ gives

$\displaystyle C_{\rm ext} = C_1(\gamma) C_2(\beta) C_3(\alpha).$

The products are symbolically identical.

For the numerical angles of Exercise 9, both give

$\displaystyle C \approx \begin{bmatrix} 0.8138&0.4698&-0.3420\ -0.4410&0.8826&0.1632\ 0.3785&0.0180&0.9254 \end{bmatrix}.$

(15)

Solution 11: frame-label derivation

The three intrinsic coordinate maps are

$\displaystyle {}^{A_1}C_A = C_i(\alpha),$

(16)

$\displaystyle {}^{A_2}C_{A_1} = C_j(\beta),$

(17)

and

$\displaystyle {}^BC_{A_2} = C_k(\gamma).$

(18)

Compose the maps by matching adjacent frame labels:

$\displaystyle {}^BC_A = {}^BC_{A_2} {}^{A_2}C_{A_1} {}^{A_1}C_A.$

Therefore

$\displaystyle {}^BC_A = C_k(\gamma) C_j(\beta) C_i(\alpha).$

(19)

Solution 12: single-angle diagnostic

For intrinsic $3$ - $2$ - $1$ ,

$\displaystyle {}^BC_A = C_1(\gamma) C_2(\beta) C_3(\alpha).$

Set

$\displaystyle \beta=0,\qquad \gamma=0.$

Then

$\displaystyle {}^BC_A = C_3(\alpha).$

(20)

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

$\displaystyle {}^BC_A = C_3(\alpha).$

(21)

Thus the single-angle reduction agrees.

Solution 13: proper Euler subtlety

Intrinsic $3$ - $1$ - $3$ with angles $(\alpha,\beta,\gamma)$ is equivalent to extrinsic

$\displaystyle 3$ - $\displaystyle 1$ - $\displaystyle 3$

with chronological angles

$\displaystyle (\gamma,\beta,\alpha).$

intrinsic $\displaystyle 3$ - $\displaystyle 1$ - $\displaystyle 3 (\alpha,\beta,\gamma) \equiv$ extrinsic $\displaystyle 3$ - $\displaystyle 1$ - $\displaystyle 3 (\gamma,\beta,\alpha).$

(22)

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.

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.

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:

intrinsic passive;
extrinsic passive;
intrinsic active;
extrinsic active.

PhysicsLibrary uses intrinsic passive as its default Euler convention, but the other three combinations are mathematically valid.

Solution 16: convention audit

Before comparing

$\displaystyle C=C_3(\psi)C_2(\theta)C_1(\phi)$

with PhysicsLibrary, one should ask at least:

Is the transformation active or passive?
What coordinate map direction is represented?
Is the sequence intrinsic or extrinsic?
Does “--” describe chronological axis order or written matrix order?
Are $\phi,\theta,\psi$ roll, pitch, and yaw, or merely first, second, and third sequence angles?
Does the source use the same right hand positive-angle convention?

Without these answers, matrix order alone is not enough to compare conventions.

Solution 17: software interface audit

Under the stated software convention:

intrinsic -- would use uppercase

$\displaystyle \texttt{ZYX};$
extrinsic -- would use lowercase

$\displaystyle \texttt{zyx};$
the extrinsic sequence equivalent to intrinsic -- is --, so it would use

$\displaystyle \texttt{xyz}.$

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.

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

intrinsic $\displaystyle 1$ - $\displaystyle 2$ - $\displaystyle 3 (\alpha,\beta,\gamma) \equiv$ extrinsic $\displaystyle 3$ - $\displaystyle 2$ - $\displaystyle 1 (\gamma,\beta,\alpha).$

(23)

The axes and their associated angles must be reversed together.

Compact review

The two passive sequence rules are

$\displaystyle C_{\rm intrinsic} = C_k(\gamma) C_j(\beta) C_i(\alpha),$

(24)

and

$\displaystyle C_{\rm extrinsic} = C_i(\alpha) C_j(\beta) C_k(\gamma).$

(25)

The reverse-order equivalence is

intrinsic $\displaystyle i$ - $\displaystyle j$ - $\displaystyle k (\alpha,\beta,\gamma) \equiv$ extrinsic $\displaystyle k$ - $\displaystyle j$ - $\displaystyle i (\gamma,\beta,\alpha).$

(26)

The words intrinsic and extrinsic describe the axes used for successive rotations.

They do not, by themselves, specify active versus passive transformation.

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 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.

Bibliography

1: D. M. Henderson, Euler Angles, Quaternions, and Transformation Matrices: Working Relationships, JSC-12960, NASA Johnson Space Center, 1977. NASA Technical Reports Server
2: J. K. Moore, Learn Multibody Dynamics, chapter “Orientation of Reference Frames,” 2026 edition. Licensed CC BY 4.0. Orientation of Reference Frames
3: SymPy Development Team, “ReferenceFrame orientation methods,” SymPy documentation. SymPy ReferenceFrame documentation
4: SciPy Developers, “Rotation.from_euler,” SciPy documentation. SciPy Euler rotation documentation

License

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