Euler Angles: Sequence Composition and the Twelve Standard Sequences

Once the elementary axis rotations and the intrinsic/extrinsic distinction are understood, all standard Euler sequences can be organized by one composition rule.

There are exactly twelve standard three angle sequences:

six Tait Bryan sequences, in which all three axis labels are different;
six proper Euler sequences, in which the first and third axis labels are the same.

This article derives the counting argument, gives the passive intrinsic product for every sequence, records the expanded direction cosine matrices, and shows how the two sequence families differ in their middle angle singularity.

It is intended to serve as the main sequence reference for the remaining Euler Angle articles.

Convention recap

PhysicsLibrary uses passive coordinate maps between right handed orthonormal frames:

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

(1)

For a generic intrinsic $i$ - $j$ - $k$ sequence with first, second, and third angles $(\alpha,\beta,\gamma)$ ,

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

(2)

The elementary passive matrices are

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

(3)

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

(4)

and

$\displaystyle C_3(\lambda) = \begin{bmatrix} \cos\lambda&\sin\lambda&0\ -\sin\lambda&\cos\lambda&0\ 0&0&1 \end{bmatrix}.$

(5)

Deriving the intrinsic composition rule

Let the intermediate frames be

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

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

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

(6)

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

(7)

and

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

(8)

Coordinate maps compose by matching adjacent frame labels:

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

(9)

Therefore

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

(10)

Image EA04_intrinsic_sequence_composition

Figure. Generic intrinsic sequence composition. The chronological frame rotations progress from $A_0$

, while the corresponding passive coordinate maps multiply in the frame-chain order shown.

The rightmost matrix acts first on a coordinate column.

Why there are exactly twelve standard sequences

The first rotation axis has three possible choices.

The second axis must differ from the first, leaving two choices.

The third axis must differ from the second. There are then two standard possibilities:

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

Therefore

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

(11)

Figure. Classification of the twelve standard intrinsic Euler sequences. Six use three distinct axes; six repeat the first axis label as the third.

The six Tait Bryan sequences

The Tait Bryan sequences are

$\displaystyle 123,\quad132,\quad213,\quad231,\quad312,\quad321.$

(12)

Their intrinsic passive products are:

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

For a common principal branch,

$\displaystyle -\frac{\pi}{2} \leq \beta \leq \frac{\pi}{2}.$

(13)

The generic Tait Bryan singularity occurs when

$\displaystyle \cos\beta=0,$

(14)

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

(15)

At that configuration the first and third rotation axes become aligned and the outer angles lose independent meaning.

The six proper Euler sequences

The proper Euler sequences are

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

(16)

Their intrinsic passive products are:

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

A common proper Euler principal choice is

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

(17)

The generic proper Euler singularity occurs when

$\displaystyle \sin\beta=0,$

(18)

that is,

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

(19)

Again, the physical orientation remains valid. Only the Euler coordinate chart becomes singular.

Notation for expanded matrices

For compactness, define

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

(20)

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

(21)

and

$\displaystyle c_\gamma=\cos\gamma, \qquad s_\gamma=\sin\gamma.$

(22)

The following matrices all map coordinates from the initial frame $A$ into the final frame $B$ under the PhysicsLibrary passive intrinsic convention.

Expanded Tait Bryan reference matrices

Intrinsic

This is a Tait Bryan sequence. Its passive intrinsic product is

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

(23)

With the shorthand introduced above, the expanded matrix is

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

(24)

Intrinsic

This is a Tait Bryan sequence. Its passive intrinsic product is

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

(25)

With the shorthand introduced above, the expanded matrix is

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

(26)

Intrinsic

This is a Tait Bryan sequence. Its passive intrinsic product is

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

(27)

With the shorthand introduced above, the expanded matrix is

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

(28)

Intrinsic

This is a Tait Bryan sequence. Its passive intrinsic product is

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

(29)

With the shorthand introduced above, the expanded matrix is

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

(30)

Intrinsic

This is a Tait Bryan sequence. Its passive intrinsic product is

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

(31)

With the shorthand introduced above, the expanded matrix is

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

(32)

Intrinsic

This is a Tait Bryan sequence. Its passive intrinsic product is

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

(33)

With the shorthand introduced above, the expanded matrix is

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

(34)

Expanded proper Euler reference matrices

Intrinsic

This is a proper Euler sequence. Its passive intrinsic product is

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

(35)

With the shorthand introduced above, the expanded matrix is

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

(36)

Intrinsic

This is a proper Euler sequence. Its passive intrinsic product is

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

(37)

With the shorthand introduced above, the expanded matrix is

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

(38)

Intrinsic

This is a proper Euler sequence. Its passive intrinsic product is

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

(39)

With the shorthand introduced above, the expanded matrix is

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

(40)

Intrinsic

This is a proper Euler sequence. Its passive intrinsic product is

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

(41)

With the shorthand introduced above, the expanded matrix is

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

(42)

Intrinsic

This is a proper Euler sequence. Its passive intrinsic product is

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

(43)

With the shorthand introduced above, the expanded matrix is

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

(44)

Intrinsic

This is a proper Euler sequence. Its passive intrinsic product is

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

(45)

With the shorthand introduced above, the expanded matrix is

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

(46)

Aerospace -- specialization

For intrinsic $3$ - $2$ - $1$ yaw pitch roll,

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

(47)

Therefore

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

(48)

This is the flagship Tait Bryan sequence used throughout the PhysicsLibrary Euler and quaternion series.

The flagship proper Euler -- sequence

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

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

(49)

This sequence is used as the flagship proper Euler example later in the series.

Because its sequence label is a palindrome, its equivalent extrinsic sequence also has the label $3$ - $1$ - $3$ , but the chronological angle association reverses:

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

(50)

Intrinsic and extrinsic reference rule

For completeness, the passive extrinsic $i$ - $j$ - $k$ rule is

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

(51)

Therefore

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

(52)

This equivalence should be used when translating sequence descriptions between moving-axis and fixed-axis sources.

Symmetry across the twelve sequences

The twelve expanded matrices are not twelve unrelated formulas.

They are generated from the same three elementary matrices and the same composition rule.

Several useful symmetry observations follow:

Every matrix is proper orthogonal:

$\displaystyle CC^T=I, \qquad \det C=1.$ (53)
Every Tait Bryan sequence has the same generic middle-angle singularity condition:

$\displaystyle \cos\beta=0.$
Every proper Euler sequence has the same generic middle-angle singularity condition:

$\displaystyle \sin\beta=0.$
Relabeling the coordinate axes maps one member of a sequence family into another.
Reversing the intrinsic sequence produces the equivalent extrinsic description when the angle association is reversed as well.

These symmetries are more useful than memorizing twelve independent matrices.

Verification tests for every sequence

Each sequence matrix should pass the following tests.

Identity

$\displaystyle \alpha=\beta=\gamma=0 \quad\Longrightarrow\quad {}^BC_A=I.$

(54)

First-angle reduction

Set

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

Then

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

(55)

Second-angle reduction

Set

$\displaystyle \alpha=\gamma=0.$

Then

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

(56)

Third-angle reduction

Set

$\displaystyle \alpha=\beta=0.$

Then

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

(57)

Reverse coordinate map

$\displaystyle {}^AC_B = ({}^BC_A)^T.$

(58)

Quaternion agreement

For the migrated passive PhysicsLibrary quaternion convention,

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

(59)

and therefore

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

(60)

Why the twelve legacy sequence pages remain useful

PhysicsLibrary already has canonical entries for the individual sequences:

Those pages remain valuable as sequence-specific reference entries.

This EA04 article supplies the common convention and composition framework so that each legacy sequence page can be modernized without repeating the full twelve-sequence theory.

Common mistakes

Treating all twelve sequences as unrelated formulas instead of products of three elementary matrices.
Mixing generic first-second-third angles with aerospace roll-pitch-yaw names.
Forgetting that the rightmost matrix acts first on a coordinate column.
Using an extrinsic product while calling the sequence intrinsic.
Assuming a repeated first and third axis in a proper Euler sequence means the two rotations occur about the same physical axis.
Using the Tait Bryan singular condition for a proper Euler sequence, or vice versa.
Comparing two expanded matrices before confirming active/passive and map-direction conventions.

Summary

There are exactly twelve standard Euler sequences.

The six Tait Bryan sequences are

$\displaystyle 123,\quad132,\quad213,\quad231,\quad312,\quad321.$

(61)

The six proper Euler sequences are

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

(62)

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

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

(63)

The two sequence families differ in their generic middle-angle singularity:

$\displaystyle \cos\beta=0 \qquad \hbox{for Tait Bryan sequences},$

(64)

and

$\displaystyle \sin\beta=0 \qquad \hbox{for proper Euler sequences}.$

(65)

The expanded matrices in this article form the common passive intrinsic reference set for the remainder of the PhysicsLibrary Euler angle series.

References and further reading

Henderson provides the classic NASA engineering tabulation of the twelve Euler sequences and transformation-matrix relationships.

Diebel gives a compact unified treatment of Euler sequences, DCMs, quaternions, and rotation vectors.

Moore develops orientation through successive reference-frame transformations and is especially useful for interpreting the frame-chain composition rule.

SciPy and SymPy provide modern software examples in which intrinsic and extrinsic sequence conventions are explicitly distinguished.

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: SciPy Developers, “Rotation.from_euler,” SciPy documentation. SciPy Euler rotation documentation
5: SymPy Development Team, “ReferenceFrame orientation methods,” SymPy documentation. SymPy ReferenceFrame documentation

License

Unless otherwise noted, this PhysicsLibrary entry is intended for release under the Creative Commons Attribution ShareAlike 4.0 International license.

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