Euler Angles: Definition and Basic Geometry Examples, Exercises, and Solutions

This entry is the self study companion to Euler Angles: definition and basic geometry.

The purpose is to build geometric fluency before introducing large sequence-specific formula tables.

All exercises are stated first. Complete worked solutions follow afterward.

Convention summary

PhysicsLibrary uses right handed orthonormal frames and passive coordinate transformations.

If the same physical vector has coordinate columns ${}^{A}\mathbf v$ and ${}^{B}\mathbf v$ , then

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

(1)

Generic Euler sequences are intrinsic moving axis sequences.

For intrinsic $i$ - $j$ - $k$ with first, second, and third angles $\alpha,\beta,\gamma$ ,

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

(2)

The chronological sequence is

$\displaystyle i \longrightarrow j \longrightarrow k,$

while the rightmost matrix acts first on a coordinate column.

The six Tait Bryan sequences are

$\displaystyle 123,\quad 132,\quad 213,\quad 231,\quad 312,\quad 321.$

(3)

The six proper Euler sequences are

$\displaystyle 121,\quad 131,\quad 212,\quad 232,\quad 313,\quad 323.$

(4)

Exercises

Frame-map interpretation.
A matrix is labeled

$\displaystyle {}^{B}C_A.$

State exactly what coordinate map it performs.

If

$\displaystyle {}^{A}\mathbf v = \begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix},$

write the equation that gives ${}^{B}\mathbf v$ .
Reverse map.
Given a proper direction cosine matrix

$\displaystyle {}^{B}C_A,$

derive the matrix for the reverse coordinate map

$\displaystyle B\rightarrow A.$

Why is the transpose sufficient?
Interpret an intrinsic sequence.
Describe in words the intrinsic sequence

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

with angles

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

Be explicit about which frame each second and third axis belongs to.
Write the passive matrix product.
For intrinsic sequence

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

write the complete passive coordinate transformation ${}^{B}C_A$ in terms of elementary matrices.
Rightmost matrix acts first.
Explain why

$\displaystyle {}^{B}C_A = C_1(\gamma) C_3(\beta) C_2(\alpha)$

still describes the chronological intrinsic sequence

$\displaystyle 2 \longrightarrow 3 \longrightarrow 1.$
Classify the sequence family.
Classify each sequence as Tait Bryan or proper Euler:

$\displaystyle 321,\quad 313,\quad 123,\quad 232,\quad 132,\quad 121.$

State the criterion you used.
Why adjacent axes cannot repeat.
Show that

$\displaystyle C_1(\beta) C_1(\alpha) = C_1(\alpha+\beta).$ (5)

Explain why a nominal sequence such as

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

is not a standard three-angle Euler sequence.
Why there are twelve standard sequences.
Use a counting argument to derive the number of admissible standard three-angle Euler sequences.

Your reasoning should explain the factor

$\displaystyle 3\times2\times2.$
Noncommutativity.
Using positive $90^\circ$ passive frame rotations about and , compare

$\displaystyle C_2\left(\frac{\pi}{2}\right) C_1\left(\frac{\pi}{2}\right)$

with

$\displaystyle C_1\left(\frac{\pi}{2}\right) C_2\left(\frac{\pi}{2}\right).$

Show that the two final orientations differ.
Single-axis reduction.
For intrinsic --,

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

Set

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

What transformation remains?

Interpret the result geometrically.
Aerospace notation specialization.
For intrinsic --, identify the correspondence between

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

and

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

Then write the PhysicsLibrary passive yaw pitch roll DCM product.
Euler triples are not vectors.
Explain why

$\displaystyle (\alpha_2-\alpha_1,\beta_2-\beta_1,\gamma_2-\gamma_1)$

is not generally the exact relative orientation between two finite Euler attitudes.

Give one geometric reason.
Intrinsic versus extrinsic.
An orientation is described intrinsically by

$\displaystyle 3 \longrightarrow 2 \longrightarrow 1.$

State the corresponding extrinsic axis order that can describe the same final orientation when the angle association is handled consistently.

Does this equivalence eliminate the need to declare intrinsic versus extrinsic?
Proper Euler first and third axes.
In an intrinsic -- sequence, why can the first and third rotations not generally be combined into one rotation about axis ?
Nonuniqueness from periodicity.
Show that

$\displaystyle C_i(\lambda+2\pi n) = C_i(\lambda), \qquad n\in\mathbb{Z}.$ (6)

Use this result to explain why Euler angle coordinates are not globally unique.
Coordinate singularity concept.
For a Tait Bryan sequence, the generic singular condition is

$\displaystyle \cos\beta=0.$

For a proper Euler sequence, the generic singular condition is

$\displaystyle \sin\beta=0.$

State the corresponding principal singular values of $\beta$ for each family.

Does the physical orientation itself become undefined at those values?

Solutions

Solution 1: frame-map interpretation

The notation

$\displaystyle {}^{B}C_A$

means that the matrix maps coordinate columns from frame $A$ into frame $B$ .

Thus

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

(7)

For the supplied vector,

$\displaystyle {}^{B}\mathbf v = {}^{B}C_A \begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}.$

(8)

The physical vector itself is unchanged. Only its coordinate representation changes.

Solution 2: reverse map

A valid DCM is orthogonal, so

$\displaystyle ({}^{B}C_A)^{-1} = ({}^{B}C_A)^T.$

The reverse map is therefore

$\displaystyle {}^{A}C_B = ({}^{B}C_A)^T.$

(9)

This works because an orthonormal change of basis has inverse equal to transpose.

Solution 3: interpret an intrinsic sequence

Intrinsic $3$ - $2$ - $1$ means:

rotate the original frame through $\alpha$ about axis of the original frame;
rotate the resulting intermediate frame through $\beta$ about axis of that intermediate frame;
rotate the next intermediate frame through $\gamma$ about axis of that newest frame.

The second and third axes are moving axes because they belong to frames created by earlier rotations.

Solution 4: write the passive matrix product

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

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

Therefore

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

(10)

Solution 5: rightmost matrix acts first

A coordinate column is multiplied as

$\displaystyle {}^{B}\mathbf v = C_1(\gamma) C_3(\beta) C_2(\alpha) {}^{A}\mathbf v.$

The first multiplication applied to ${}^{A}\mathbf v$ is therefore

$\displaystyle C_2(\alpha).$

The result is then multiplied by

$\displaystyle C_3(\beta),$

and finally by

$\displaystyle C_1(\gamma).$

Thus the chronological intrinsic order is

$\displaystyle 2 \longrightarrow 3 \longrightarrow 1.$

The written matrix order is ordinary composition of linear maps.

Solution 6: classify the sequence family

A Tait Bryan sequence uses three distinct axes.

A proper Euler sequence repeats the first axis as the third axis.

Therefore:

$\displaystyle 321 \quad\hbox{is Tait Bryan},$

$\displaystyle 313 \quad\hbox{is proper Euler},$

$\displaystyle 123 \quad\hbox{is Tait Bryan},$

$\displaystyle 232 \quad\hbox{is proper Euler},$

$\displaystyle 132 \quad\hbox{is Tait Bryan},$

and

$\displaystyle 121 \quad\hbox{is proper Euler}.$

Solution 7: why adjacent axes cannot repeat

Elementary rotations about the same axis form an ordinary one-parameter rotation group.

For axis $1$ ,

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

Direct multiplication gives

$\displaystyle C_1(\beta)C_1(\alpha) = \begin{bmatrix} 1&0&0\ 0&\cos(\alpha+\b... ...&\sin(\alpha+\beta)\ 0&-\sin(\alpha+\beta)&\cos(\alpha+\beta) \end{bmatrix}.$

Hence

$\displaystyle C_1(\beta) C_1(\alpha) = C_1(\alpha+\beta).$

(11)

Two consecutive rotations about the same current axis collapse into one rotation.

Thus a nominal sequence such as $1$ - $1$ - $3$ does not supply three independent orientation coordinates.

Solution 8: why there are twelve standard sequences

There are three choices for the first axis.

Once the first axis is selected, the second axis must differ from the first, leaving two choices.

For the third axis there are two standard admissible possibilities:

choose the remaining unused axis, producing a Tait Bryan sequence;
return to the first axis, producing a proper Euler sequence.

Therefore

$\displaystyle 3\times2\times2=12.$

(12)

Solution 9: noncommutativity

The passive elementary matrices are

$\displaystyle C_1\left(\frac{\pi}{2}\right) = \begin{bmatrix} 1&0&0\ 0&0&1\ 0&-1&0 \end{bmatrix}$

and

$\displaystyle C_2\left(\frac{\pi}{2}\right) = \begin{bmatrix} 0&0&-1\ 0&1&0\ 1&0&0 \end{bmatrix}.$

First,

$\displaystyle C_2\left(\frac{\pi}{2}\right) C_1\left(\frac{\pi}{2}\right) = \begin{bmatrix} 0&1&0\ 0&0&1\ 1&0&0 \end{bmatrix}.$

(13)

Reversing the order gives

$\displaystyle C_1\left(\frac{\pi}{2}\right) C_2\left(\frac{\pi}{2}\right) = \begin{bmatrix} 0&0&-1\ -1&0&0\ 0&1&0 \end{bmatrix}.$

(14)

The matrices are different, so the final orientations are different.

Finite rotations in three dimensions do not generally commute.

Solution 10: single-axis reduction

Starting from

$\displaystyle {}^{B}C_A = C_1(\gamma) C_2(\beta) C_3(\alpha),$

set

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

Since

$\displaystyle C_1(0)=I, \qquad C_2(0)=I,$

we obtain

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

(15)

Thus only the first intrinsic frame rotation remains: a rotation through $\alpha$ about axis $3$ .

Solution 11: aerospace notation specialization

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

$\displaystyle \alpha = \psi =$ yaw $\displaystyle ,$

$\displaystyle \beta = \theta =$ pitch $\displaystyle ,$

and

$\displaystyle \gamma = \phi =$ roll $\displaystyle .$

Therefore

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

(16)

Solution 12: Euler triples are not vectors

Euler angles are nonlinear sequence coordinates.

The second and third angles are defined about intermediate moving axes, so their geometric meaning depends on earlier rotations.

Therefore coordinate subtraction,

$\displaystyle (\alpha_2-\alpha_1, \beta_2-\beta_1, \gamma_2-\gamma_1),$

does not generally reproduce the finite relative orientation.

The correct relative orientation must be obtained by composing or inverting the corresponding rotation maps.

Solution 13: intrinsic versus extrinsic

An intrinsic sequence

$\displaystyle 3 \longrightarrow 2 \longrightarrow 1$

can be represented by a corresponding extrinsic sequence using the reverse fixed axis order

$\displaystyle 1 \longrightarrow 2 \longrightarrow 3,$

with the corresponding angle association handled consistently.

This equivalence does not eliminate the need to specify intrinsic versus extrinsic.

The two descriptions use different geometric constructions even though they can produce the same final orientation.

Solution 14: proper Euler first and third axes

In intrinsic $3$ - $1$ - $3$ , the first rotation is about axis $3$ of the initial frame.

The second rotation changes the orientation of the frame.

The third rotation is about axis $3$ of the newest intermediate frame.

That third axis is generally not aligned with the original axis $3$ .

Therefore the first and third rotations are not generally rotations about the same physical direction and cannot be combined.

Solution 15: nonuniqueness from periodicity

Elementary rotation matrices depend on sine and cosine.

Because

$\displaystyle \cos(\lambda+2\pi n)=\cos\lambda$

and

$\displaystyle \sin(\lambda+2\pi n)=\sin\lambda,$

we have

$\displaystyle C_i(\lambda+2\pi n) = C_i(\lambda).$

(17)

Thus adding integer multiples of $2\pi$ to an Euler coordinate may leave the orientation unchanged.

Euler coordinates are therefore not globally unique.

Additional nonuniqueness arises from alternate branches and singular configurations.

Solution 16: coordinate singularity concept

For a Tait Bryan sequence,

$\displaystyle \cos\beta=0$

when

$\displaystyle \beta = \pm\frac{\pi}{2}$

(18)

on the usual principal interval.

For a proper Euler sequence,

$\displaystyle \sin\beta=0$

when

$\displaystyle \beta=0 \qquad\hbox{or}\qquad \beta=\pi$

(19)

on the usual principal interval.

The physical orientation remains completely well defined.

Only the chosen Euler coordinate chart loses rank.

Compact review

The principal geometric facts reinforced by this companion are:

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

(20)

$\displaystyle {}^{B}C_A = C_k(\gamma) C_j(\beta) C_i(\alpha),$

(21)

$\displaystyle {}^{A}C_B = ({}^{B}C_A)^T,$

(22)

and

$\displaystyle 3\times2\times2=12$

(23)

standard sequence choices.

The six Tait Bryan sequences use three distinct axes.

The six proper Euler sequences repeat the first axis as the third axis.

Euler triples are local orientation coordinates, not ordinary three component vectors.

Sources and exercise provenance

The exercises and solutions in this companion are newly written for PhysicsLibrary to reinforce the geometric definitions developed in Euler angles: definition and basic geometry.

Henderson provides a classic engineering treatment of Euler sequences and transformation matrices.

Diebel presents a modern comparison of Euler angles, matrices, quaternions, and rotation vectors.

Moore develops orientation from reference-frame basis vectors and passive direction cosine matrices.

Goldstein, Poole, and Safko provide the classical rigid-body mechanics context for proper Euler angles.

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. Diebel, “Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors,” Stanford University, 2006. Online PDF
3: J. K. Moore, Learn Multibody Dynamics, chapter “Orientation of Reference Frames,” 2026 edition. Licensed CC BY 4.0. Orientation of Reference Frames
4: H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison Wesley, 2002. Publisher search

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