Euler angles: sequence composition and the twelve standard sequences

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.

1 Convention recap

PhysicsLibrary uses passive coordinate maps between right handed orthonormal frames:

B B A v = CA v.

(1)

For a generic intrinsic i-j-k sequence with first, second, and third angles (α,β,γ),

B CA = Ck(γ)Cj (β)Ci(α).

(2)

The elementary passive matrices are

⌊ ⌋ 1 0 0 C1 (λ) = ⌈0 cos λ sin λ⌉ , 0 − sin λ cosλ

(3)

⌊ ⌋ cos λ 0 − sinλ ⌈ ⌉ C2 (λ) = 0 1 0 , sin λ 0 cosλ

(4)

and

⌊ ⌋ cos λ sin λ 0 C3 (λ) = ⌈− sin λ cosλ 0⌉ . 0 0 1

(5)

2 Deriving the intrinsic composition rule

Let the intermediate frames be

A = A, A , A , A = B. 0 1 2 3

For intrinsic i-j-k,

A1 CA = Ci(α),

(6)

A2C = C (β ), A1 j

(7)

and

B CA2 = Ck(γ).

(8)

Coordinate maps compose by matching adjacent frame labels:

BCA = BCA2 A2CA1 A1CA.

(9)

Therefore

BCA = Ck(γ)Cj (β)Ci(α).

(10)

Figure. Generic intrinsic sequence composition. The chronological frame rotations progress from A₀ to A₃, while the corresponding passive coordinate maps multiply in the frame-chain order shown.

The rightmost matrix acts first on a coordinate column.

3 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

3 × 2 × 2 = 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.

4 The six Tait Bryan sequences

The Tait Bryan sequences are

123, 132, 213, 231, 312, 321.

(12)

Their intrinsic passive products are:


Sequence	Passive intrinsic product

1-2-3	C₃(γ)C₂(β)C₁(α)

1-3-2	C₂(γ)C₃(β)C₁(α)

2-1-3	C₃(γ)C₁(β)C₂(α)

2-3-1	C₁(γ)C₃(β)C₂(α)

3-1-2	C₂(γ)C₁(β)C₃(α)

3-2-1	C₁(γ)C₂(β)C₃(α)

For a common principal branch,

− π-≤ β ≤ π-. 2 2

(13)

The generic Tait Bryan singularity occurs when

cosβ = 0,

(14)

β = ± π. 2

(15)

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

5 The six proper Euler sequences

The proper Euler sequences are

121, 131, 212, 232, 313, 323.

(16)

Their intrinsic passive products are:


Sequence	Passive intrinsic product

1-2-1	C₁(γ)C₂(β)C₁(α)

1-3-1	C₁(γ)C₃(β)C₁(α)

2-1-2	C₂(γ)C₁(β)C₂(α)

2-3-2	C₂(γ)C₃(β)C₂(α)

3-1-3	C₃(γ)C₁(β)C₃(α)

3-2-3	C₃(γ)C₂(β)C₃(α)

A common proper Euler principal choice is

0 ≤ β ≤ π.

(17)

The generic proper Euler singularity occurs when

sin β = 0,

(18)

that is,

β = 0 or β = π.

(19)

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

6 Notation for expanded matrices

For compactness, define

cα = cosα, sα = sinα,

(20)

cβ = cosβ, sβ = sinβ,

(21)

and

cγ = cosγ, sγ = sinγ.

(22)

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

7 Expanded Tait Bryan reference matrices

Intrinsic 1 − 2 − 3

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

BC = C (γ)C (β )C (α ). A 3 2 1

(23)

With the shorthand introduced above, the expanded matrix is

⌊ cβcγ cαsγ + cγs αsβ − cαc γsβ + s αsγ⌋ B ⌈ ⌉ CA = − cβsγ cαcγ − sαs βsγ cαsβsγ + cγsα . sβ − cβsα cαcβ

(24)

Intrinsic 1 − 3 − 2

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

BCA = C2(γ)C3 (β )C1(α ).

(25)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ cβcγ cαcγsβ + sαsγ − cαsγ + cγsαsβ BCA = ⌈ − sβ cαcβ cβsα ⌉ . cβsγ cαsβsγ − cγsα cαcγ + sαsβsγ

(26)

Intrinsic 2 − 1 − 3

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

BCA = C3(γ)C1 (β )C2(α ).

(27)

With the shorthand introduced above, the expanded matrix is

⌊ cαcγ + sαs βsγ cβsγ cαsβsγ − cγsα⌋ B ⌈ ⌉ CA = − cαs γ + cγsαs β cβcγ cαcγsβ + sαsγ . cβsα − s β cαcβ

(28)

Intrinsic 2 − 3 − 1

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

BCA = C1(γ)C3 (β )C2(α ).

(29)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ cαcβ sβ − cβsα BCA = ⌈ − cαcγsβ + sαsγ cβcγ cαs γ + cγsαs β⌉ . c s s + c s − c s c c − s s s α β γ γ α β γ α γ α β γ

(30)

Intrinsic 3 − 1 − 2

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

BCA = C2(γ)C1 (β )C3(α ).

(31)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ B cαcγ − sαsβsγ cαsβsγ + cγsα − cβsγ CA = ⌈ − cβsα cαcβ sβ ⌉ . cαsγ + cγsαsβ − cαcγsβ + sαsγ cβcγ

(32)

Intrinsic 3 − 2 − 1

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

B CA = C1(γ)C2 (β )C3(α ).

(33)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ cαcβ cβsα − sβ BCA = ⌈ cαsβsγ − cγsα cαcγ + sαsβsγ cβsγ⌉ . cαcγsβ + sαsγ − cαsγ + cγsαsβ cβcγ

(34)

8 Expanded proper Euler reference matrices

Intrinsic 1 − 2 − 1

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

BCA = C1(γ)C2 (β )C1(α ).

(35)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ cβ sαsβ − cαsβ BCA = ⌈ sβsγ cαcγ − cβsαsγ cαcβsγ + cγsα⌉ . c s − c s − c c s c c c − s s γ β α γ β γ α α β γ α γ

(36)

Intrinsic 1 − 3 − 1

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

BCA = C1(γ)C3 (β )C1(α ).

(37)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ cβ cαsβ sαsβ BCA = ⌈ − cγsβ cαc βcγ − s αsγ cαs γ + cβcγs α⌉ . sβsγ − cαcβs γ − cγsα cαc γ − cβsαs γ

(38)

Intrinsic 2 − 1 − 2

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

BC = C (γ)C (β )C (α ). A 2 1 2

(39)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ cαcγ − cβsαsγ sβsγ − cαc βsγ − cγs α BC = ⌈ s s c c s ⌉ . A α β β α β cαsγ + cβcγsα − cγsβ cαcβcγ − sαsγ

(40)

Intrinsic 2 − 3 − 2

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

B CA = C2(γ)C3 (β )C2(α ).

(41)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ B cαcβcγ − sαsγ cγsβ − cαsγ − cβcγsα CA = ⌈ − cαsβ cβ sαsβ ⌉ . cαcβsγ + cγsα sβsγ cαcγ − cβsαsγ

(42)

Intrinsic 3 − 1 − 3

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

BC = C (γ)C (β )C (α ). A 3 1 3

(43)

With the shorthand introduced above, the expanded matrix is

⌊ cαcγ − cβs αsγ cαcβsγ + cγsα sβsγ⌋ B ⌈ ⌉ CA = − cαs γ − cβcγs α cαcβcγ − s αsγ cγsβ . sαsβ − cαsβ cβ

(44)

Intrinsic 3 − 2 − 3

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

BCA = C3(γ)C2 (β )C3(α ).

(45)

With the shorthand introduced above, the expanded matrix is

⌊ ⌋ cαcβcγ − sαsγ cαsγ + cβcγsα − cγsβ BCA = ⌈ − cαcβsγ − cγsα cαcγ − cβsαsγ sβsγ ⌉ . cαsβ sαsβ cβ

(46)

9 Aerospace $3$ - $2$ - $1$ specialization

For intrinsic 3-2-1 yaw pitch roll,

α = ψ, β = 𝜃, γ = ϕ.

(47)

Therefore

B CA = C1(ϕ)C2 (𝜃)C3(ψ).

(48)

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

10 The flagship proper Euler $3$ - $1$ - $3$ sequence

For intrinsic 3-1-3,

B CA = C3(γ)C1 (β )C3(α ).

(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 3-1-3(α,β,γ ) ≡ extrinsic 3-1- 3 (γ, β,α ).

(50)

11 Intrinsic and extrinsic reference rule

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

B CA = Ci(α)Cj (β )Ck(γ).

(51)

Therefore

intrinsic i-j- k (α,β,γ ) ≡ extrinsic k-j- i(γ,β, α).

(52)

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

12 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:
$CCT = I, detC = 1.$ (53)
Every Tait Bryan sequence has the same generic middle-angle singularity condition:

$cosβ = 0.$
Every proper Euler sequence has the same generic middle-angle singularity condition:

$sin β = 0.$
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.

13 Verification tests for every sequence

Each sequence matrix should pass the following tests.

Identity

α = β = γ = 0 = ⇒ BCA = I.

(54)

First-angle reduction

Set

β = γ = 0.

Then

B CA = Ci(α ).

(55)

Second-angle reduction

Set

α = γ = 0.

Then

B CA = Cj (β ).

(56)

Third-angle reduction

Set

α = β = 0.

Then

B CA = Ck (γ).

(57)

Reverse coordinate map

ACB = (BCA )T .

(58)

Quaternion agreement

For the migrated passive PhysicsLibrary quaternion convention,

BqA = qP (γ)qP(β)qP (α), k j i

(59)

and therefore

BCA = C (BqA ).

(60)

14 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.

15 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.

16 Summary

There are exactly twelve standard Euler sequences.

The six Tait Bryan sequences are

123, 132, 213, 231, 312, 321.

(61)

The six proper Euler sequences are

121, 131, 212, 232, 313, 323.

(62)

For every intrinsic i-j-k sequence,

BC = C (γ)C (β)C (α). A k j i

(63)

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

cos β = 0 for Tait Bryan sequences,

(64)

and

sin β = 0 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.

17 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.

References

[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.

"Euler angles: sequence composition and the twelve standard sequences" is owned by bloftin.
(view preamble)