Euler Angles: Proper Euler Angles

Proper Euler Angles are Euler Angle coordinates in which the first and third rotation axis labels are the same and the middle axis is different.

For an intrinsic sequence,

$\displaystyle i$ - $\displaystyle j$ - $\displaystyle i,$

the defining conditions are

$\displaystyle i\neq j, \qquad k=i.$

(1)

The first and third labels are the same, but the corresponding physical axes are generally not the same direction because the middle rotation changes the orientation of the intermediate frame.

Proper Euler angles are especially common in classical rigid body mechanics, spacecraft attitude descriptions, orbital orientation, and historical treatments of Euler's rotation coordinates.

The flagship proper Euler sequence in this PhysicsLibrary series is intrinsic $3$ - $1$ - $3$ .

Passive intrinsic convention

PhysicsLibrary uses passive coordinate transformations.

For a fixed physical vector,

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

(2)

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

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

(3)

For a proper Euler sequence, $k=i$ , so

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

(4)

The rightmost matrix acts first on a coordinate column.

Why there are six proper Euler sequences

There are three choices for the repeated first and third axis.

Once that outer axis is chosen, there are two possible choices for the different middle axis.

Therefore

$\displaystyle 3\times2=6.$

(5)

The six intrinsic proper Euler sequences are

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

(6)

Figure. The six standard intrinsic proper Euler sequences. The first and third axis labels match, while the middle axis is different.

The six passive intrinsic products

Applying the universal intrinsic composition rule gives:

Sequence	Passive intrinsic product
--	$C_1(\gamma)C_2(\beta)C_1(\alpha)$
--	$C_1(\gamma)C_3(\beta)C_1(\alpha)$
--	$C_2(\gamma)C_1(\beta)C_2(\alpha)$
--	$C_2(\gamma)C_3(\beta)C_2(\alpha)$
--	$C_3(\gamma)C_1(\beta)C_3(\alpha)$
--	$C_3(\gamma)C_2(\beta)C_3(\alpha)$

These six products are generated by the same frame-chain rule used for every Euler sequence.

Why the repeated outer axis still gives three coordinates

Consider intrinsic $3$ - $1$ - $3$ .

The first rotation is about axis $3$ of the initial frame $A$ .

The second rotation is about axis $1$ of the new intermediate frame.

That middle rotation generally changes the direction of axis $3$ of the current frame.

The final rotation is therefore about axis $3$ of the newest intermediate frame, not generally about the original physical $z_A$ axis.

Thus the first and third rotations cannot usually be combined.

The repeated label indicates the same coordinate-axis number in two different intermediate frames.

Proper Euler versus Tait Bryan angles

The two Euler families differ in the third axis choice.

For Tait Bryan sequences,

$\displaystyle i,\quad j,\quad k$

are all different.

For proper Euler sequences,

$\displaystyle k=i.$

This distinction controls the middle-angle singularity.

Tait Bryan sequences are singular when

$\displaystyle \cos\beta=0.$

Proper Euler sequences are singular when

$\displaystyle \sin\beta=0.$

Principal angle ranges

A common proper Euler principal branch is

$\displaystyle -\pi<\alpha\leq\pi,$

(7)

$\displaystyle 0\leq\beta\leq\pi,$

(8)

and

$\displaystyle -\pi<\gamma\leq\pi.$

(9)

On this branch,

$\displaystyle \sin\beta\geq0.$

(10)

The universal proper Euler singularity

Every proper Euler sequence becomes singular when

$\displaystyle \sin\beta=0.$

(11)

On the standard principal branch,

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

(12)

At those values, the first and third physical rotation axes become collinear.

Image EA06_proper_euler_middle_angle_singularity_tikz

Figure. For intrinsic $3$

, the third rotation axis $z_2$

is aligned with $z_A$

at $\beta=0$ and anti-aligned with $z_A$

at $\beta=\pi$ . The same mechanism occurs in every proper Euler sequence after relabeling the axes.

The physical orientation remains well defined.

Only the Euler coordinate chart loses local uniqueness.

Outer-angle coupling at $\beta=0$

$\displaystyle \beta=0,$

the middle rotation is the identity.

For intrinsic $i$ - $j$ - $i$ ,

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

(13)

Rotations about the same axis add:

$\displaystyle C_i(\gamma) C_i(\alpha) = C_i(\alpha+\gamma).$

(14)

Therefore only the sum

$\displaystyle \alpha+\gamma$

(15)

is observable from the final orientation.

The two outer angles are not independently identifiable.

Outer-angle coupling at $\beta=\pi$

$\displaystyle \beta=\pi,$

the middle rotation reverses the repeated outer axis.

The first and third physical axes are anti-aligned.

Consequently one difference combination of the outer angles remains observable, while the two angles cannot be recovered independently.

The exact sign of the coupled difference depends on the specific proper Euler sequence and the chosen angle conventions.

The important geometric fact is that the outer axes are again collinear.

Flagship intrinsic 3-1-3 sequence

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

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

(16)

Define

$\displaystyle c_\alpha=\cos\alpha, \qquad s_\alpha=\sin\alpha,$

and similarly for $\beta$ and $\gamma$ .

Multiplication gives

$\displaystyle {}^BC_A = \begin{bmatrix} c_\alpha c_\gamma-s_\alpha c_\beta s_\g... ...\beta c_\gamma\ s_\alpha s_\beta & -c_\alpha s_\beta & c_\beta \end{bmatrix}.$

(17)

Inverse extraction for intrinsic 3-1-3

Let

$\displaystyle C = {}^BC_A = \begin{bmatrix} C_{11}&C_{12}&C_{13}\ C_{21}&C_{22}&C_{23}\ C_{31}&C_{32}&C_{33} \end{bmatrix}.$

On the nonsingular principal branch,

$\displaystyle \beta = \arccos(C_{33}),$

(18)

$\displaystyle \alpha = \operatorname{atan2}(C_{31},-C_{32}),$

(19)

and

$\displaystyle \gamma = \operatorname{atan2}(C_{13},C_{23}).$

(20)

These equations require

$\displaystyle \sin\beta\neq0.$

(21)

Near $\beta=0$ or $\beta=\pi$ , software should use a declared singular-case policy instead of attempting to recover two independently meaningless outer angles.

Equivalent extrinsic description

For any proper Euler sequence,

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

(22)

The digit string is unchanged because proper Euler sequences are palindromes.

The angle association is still reversed.

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

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

(23)

This is a common source of convention mistakes because the axis labels alone do not reveal whether the construction is intrinsic or extrinsic.

Passive quaternion equivalent

For a positive frame rotation about axis $i$ ,

$\displaystyle q_i^P(\lambda) = \cos\frac{\lambda}{2} - \mathbf e_i\sin\frac{\lambda}{2}.$

(24)

A generic intrinsic proper Euler sequence has quaternion product

$\displaystyle {}^Bq_A = q_i^P(\gamma) q_j^P(\beta) q_i^P(\alpha).$

(25)

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

$\displaystyle {}^Bq_A = q_3^P(\gamma) q_1^P(\beta) q_3^P(\alpha).$

(26)

The DCM and quaternion represent the same passive map:

$\displaystyle {}^BC_A = C({}^Bq_A).$

(27)

Numerical 3-1-3 example

Take

$\displaystyle \alpha=30^\circ, \qquad \beta=60^\circ, \qquad \gamma=-20^\circ.$

(28)

Then

$\displaystyle {}^BC_A = C_3(-20^\circ) C_1(60^\circ) C_3(30^\circ)$

(29)

gives approximately

$\displaystyle {}^BC_A \approx \begin{bmatrix} +0.89930 & +0.32175 & -0.29620 \ +0.06127 & +0.57791 & +0.81380 \ +0.43301 & -0.75000 & +0.50000 \end{bmatrix}.$

(30)

The matrix is orthogonal and has determinant $+1$ to numerical precision.

The inverse formulas recover the principal triple

$\displaystyle (\alpha,\beta,\gamma) = (30^\circ,60^\circ,-20^\circ).$

An alternate Euler branch

Away from the singularity, a proper Euler orientation has alternate representations.

One useful equivalent triple is

$\displaystyle (\alpha',\beta',\gamma') = (\alpha+\pi,-\beta,\gamma+\pi),$

(31)

with the outer angles wrapped by integer multiples of $2\pi$ as needed.

The principal condition

$\displaystyle 0\leq\beta\leq\pi$

selects one standard representative.

Proper Euler angles are coordinates, not vectors

A proper Euler triple

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

is a nonlinear coordinate description of orientation.

In general,

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

(32)

is not the exact finite relative orientation between two attitudes.

The correct finite relative attitude should be computed using DCM or quaternion composition.

Applications

Proper Euler sequences appear naturally when an orientation has a meaningful preferred axis before and after a nutation-like middle rotation.

Examples include:

classical rigid body dynamics;
symmetric tops;
spacecraft and orbital-frame descriptions;
precession, nutation, and spin decompositions;
historical formulations of rotational mechanics.

The $3$ - $1$ - $3$ sequence is especially common in classical mechanics.

When another representation is preferable

A quaternion or DCM is generally preferable for internal propagation when:

the motion may approach $\beta=0$ or $\beta=\pi$ ;
repeated finite composition is central;
angular velocity directly drives the attitude state;
global numerical regularity is important;
an estimator or optimizer must remain well conditioned across a wide orientation range.

Proper Euler coordinates remain useful for interpretation and reporting even when the internal state uses another attitude representation.

Common mistakes

Assuming the first and third rotations of a proper Euler sequence are about the same physical axis for every $\beta$ .
Forgetting that the repeated outer axis belongs to different intermediate frames.
Using the Tait Bryan singularity condition $\cos\beta=0$ instead of the proper Euler condition $\sin\beta=0$ .
Assuming intrinsic and extrinsic -- are automatically the same because the sequence digits form a palindrome.
Forgetting that the angle association reverses between equivalent intrinsic and extrinsic descriptions.
Using a -- extraction formula on a -- DCM.
Interpreting the coordinate singularity as a physical singularity.

Verification checks

A proper Euler implementation should pass the following checks.

Zero angles give the identity matrix.
Single-angle reductions reproduce the appropriate passive elementary matrices.
The DCM is orthogonal:

$\displaystyle CC^T=I.$
The determinant is

$\displaystyle \det C=1.$
The reverse map is the transpose.
DCM-to-Euler round trips recover the selected principal branch away from $\sin\beta=0$ .
The DCM agrees with the passive quaternion product.

Summary

The six proper Euler sequences are

$\displaystyle 121,\quad131,\quad212,\quad232,\quad313,\quad323.$

(33)

For intrinsic $i$ - $j$ - $i$ ,

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

(34)

Their universal middle-angle singularity is

$\displaystyle \sin\beta=0.$

(35)

On the common principal branch,

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

(36)

For the flagship intrinsic $3$ - $1$ - $3$ sequence,

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

(37)

Proper Euler angles provide compact and historically important local coordinates for rigid body orientation, but they remain singular coordinate charts rather than global vector representations.

The next article, Euler angles: Euler 321 yaw pitch roll, develops the most widely used Tait Bryan sequence in full engineering detail.

References and further reading

Goldstein, Poole, and Safko provide the classical mechanics context in which proper Euler angles are especially common.

Henderson gives the classic NASA engineering tabulation of all twelve Euler sequences.

Diebel provides a unified comparison of Euler angles, DCMs, quaternions, and rotation vectors.

Landau and Lifshitz provide another standard classical mechanics treatment of rigid body orientation.

Bibliography

1: H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison Wesley, 2002. Publisher search
2: D. M. Henderson, Euler Angles, Quaternions, and Transformation Matrices: Working Relationships, JSC-12960, NASA Johnson Space Center, 1977. NASA Technical Reports Server
3: J. Diebel, “Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors,” Stanford University, 2006. Online PDF
4: L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed., Pergamon Press, 1976. Publisher search

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