Euler angles: 3-1-3 proper Euler sequence

Euler Angles: 3-1-3 Proper Euler Sequence

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

Its chronological frame rotations are

rotate through α about axis 3 of the initial frame;
rotate through β about axis 1 of the first intermediate frame;
rotate through γ about axis 3 of the second intermediate frame.

The resulting passive coordinate transformation is

BC = C (γ)C (β )C (α ). A 3 1 3

(1)

The sequence is called a proper Euler sequence because the first and third axis labels are the same while the middle axis is different.

This article develops the moving-axis geometry, full direction cosine matrix, inverse extraction formulas, singularity structure, quaternion equivalent, angular-velocity relationship, and numerical checks for intrinsic 3-1-3.

The separate Euler 313 sequence entry remains the compact sequence reference. EA08 is the deeper derivation and engineering treatment.

1 Frame and transformation convention

Let A be the reference frame and B the final oriented frame.

PhysicsLibrary uses the passive coordinate map

Bv = BCA Av.

(2)

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

Positive frame rotation follows the right hand rule. Passive does not mean that the physical frame rotation is reversed. The opposite sign appears when the same transformation is interpreted as an active rotation of a vector.

For intrinsic Euler rotations, every later rotation axis belongs to the current intermediate frame.

2 Intrinsic 3-1-3 frame construction

Define

A0 = A.

After the first rotation, call the intermediate frame A₁. After the second rotation, call the next frame A₂. After the third rotation,

A3 = B.

Figure. Intrinsic 3-1-3 frame chain. The first rotation is about z_A, the middle rotation is about the moved axis x₁, and the final rotation is about the moved axis z₂.

The three intermediate passive coordinate maps are

A1C = C (α), A 3

(3)

A2C = C (β ), A1 1

(4)

and

B CA2 = C3(γ).

(5)

Matching adjacent frame labels gives

B B A2 A1 CA = CA2 CA1 CA,

(6)

BCA = C3(γ)C1 (β )C3(α ).

(7)

The chronological frame rotations progress from left to right through the intermediate frames, while the rightmost matrix acts first on a coordinate column.

3 Why the repeated axis label does not mean the same physical axis

The defining feature of a proper Euler sequence can be misleading at first. The sequence is 3-1-3, but the two axis-3 rotations are generally not about the same physical line.

The first rotation uses

z = z . A 1

The middle rotation is about x₁. Because a nonzero rotation about x₁ changes the directions of y₁ and z₁, the new axis z₂ is generally tilted away from z_A.

The third rotation is about this new axis z₂.

Figure. Moving-axis interpretation of intrinsic 3-1-3. The middle rotation changes the direction of the current axis 3, so the final +γ rotation is generally not about the original z_A line.

This distinction is the reason the first and third rotations cannot normally be combined into one rotation.

4 Elementary passive matrices

The passive axis-1 and axis-3 matrices used by the sequence are

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

(8)

and

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

(9)

Each matrix maps coordinates from the old frame into coordinates of the new frame after a positive physical frame rotation.

5 Derivation of the full 3-1-3 DCM

Introduce the abbreviations

cα = cosα, sα = sinα,

cβ = cosβ, sβ = sinβ,

and

cγ = cosγ, sγ = sinγ.

First multiply the middle and first transformations:

⌊ ⌋ cα sα 0 C1 (β)C3 (α ) = ⌈− cβsα cβcα sβ⌉ . s s − s c c β α β α β

(10)

Premultiplying by C₃(γ) gives

⌊ c c − c s s c c s + c s s s ⌋ B ⌈ α γ β α γ α β γ γ α β γ⌉ CA = − cαs γ − cβcγs α cαcβcγ − s αsγ cγsβ . sαsβ − cαsβ cβ

(11)

This is the canonical PhysicsLibrary passive intrinsic 3-1-3 DCM.

6 Single-angle reductions

The full matrix should reduce to the elementary transformations when two angles vanish.

β = γ = 0,

then

B CA = C3 (α ).

(12)

α = γ = 0,

then

B CA = C1 (β).

(13)

α = β = 0,

then

BCA = C3 (γ).

(14)

These three tests are among the fastest ways to detect sign or multiplication order errors in an implementation.

7 Meaning of the DCM rows and columns

For

C = BC , A

the matrix elements satisfy

Cij = eB ⋅ eA . i j

(15)

Thus row i is the ith basis vector of frame B expressed in frame A, while column j is the jth basis vector of frame A expressed in frame B.

In particular, the third row is

A B [ ] e3 = s αsβ − cαs β cβ .

(16)

The angle between z_A and z_B is therefore controlled directly by β, because

eA3 ⋅ eB3 = C33 = cosβ.

(17)

For this reason the middle angle is often interpreted as a nutation or tilt angle in classical mechanics applications.

8 Line of nodes interpretation

After the first rotation about z_A, the axis x₁ lies in the plane perpendicular to z_A.

The middle rotation is performed about x₁. Since x₁ is unchanged by that middle rotation,

x2 = x1.

(18)

Geometrically, this axis is the line about which the original z_A direction is tilted into the new z₂ direction.

In many classical rigid-body treatments it is called the line of nodes. The terminology is useful, but the frame labels remain the more reliable way to specify the convention.

9 Principal angle ranges

A common principal branch for proper Euler Angles is

− π < α ≤ π,

(19)

0 ≤ β ≤ π,

(20)

and

− π < γ ≤ π.

(21)

On this branch,

sin β ≥ 0.

(22)

The end points β = 0 and β = π are precisely the singular configurations of the coordinate chart.

10 Inverse extraction from the DCM

Let

⌊C C C ⌋ B ⌈ 11 12 13⌉ C = CA = C21 C22 C23 . C31 C32 C33

Away from the singularity, the principal middle angle follows from

β = arccos(C33).

(23)

Because

C31 = sin α sin β, C32 = − cos α sin β,

we obtain

α = atan2 (C31,− C32).

(24)

Likewise,

C13 = sinβ sinγ, C23 = sinβ cos γ,

γ = atan2(C13, C23).

(25)

The two-argument function atan2 is essential because it retains the correct quadrant of each outer angle.

11 Numerically useful atan2 form for the middle angle

Since

C231 + C232 = sin2 β,

(26)

and sin β ≥ 0 on the principal branch, one may compute

(∘ ---------- ) β = atan2 C2 + C2 ,C . 31 32 33

(27)

The same result follows from

C213 + C223 = sin2 β.

(28)

This form can be preferable in software because it uses both sine-like and cosine-like information.

12 The 3-1-3 singularity

The inverse formulas for the outer angles require

sin β ⁄= 0.

Therefore intrinsic 3-1-3 becomes singular when

sin β = 0.

(29)

On the standard principal branch,

β = 0 or β = π.

(30)

Figure. Proper Euler middle-angle singularity. At β = 0, the first and third axis-3 directions are aligned. At β = π, they are anti-aligned.

The physical orientation remains perfectly valid. Only the Euler coordinate chart loses one independent coordinate direction.

13 Exact outer-angle coupling at beta equal to zero

Set

β = 0.

Then

BC = C (γ)C (α ). A 3 3

(31)

Rotations about the same axis add, giving

B CA = C3 (α + γ).

(32)

Explicitly,

⌊ cos(α + γ) sin(α + γ) 0⌋ B ⌈ ⌉ CA = − sin(α + γ) cos(α + γ ) 0 . 0 0 1

(33)

Only the combination

α + γ

(34)

is observable. The individual outer angles cannot be recovered.

14 Exact outer-angle coupling at beta equal to pi

Set

β = π.

Substituting into the full matrix gives

⌊ ⌋ cos(α − γ) sin(α − γ ) 0 BCA = ⌈sin(α − γ) − cos(α − γ ) 0 ⌉ . 0 0 − 1

(35)

Only the difference

α − γ

(36)

is independently observable.

Thus both singular end points collapse the two outer angles into one observable combination, but the combination changes from a sum to a difference.

15 A practical singular-case extraction policy

A numerical implementation must choose a convention when | sin β| falls below a selected threshold.

One possible policy is to set

γ = 0

at singularity and assign the entire observable outer-angle combination to α.

At β = 0, define

δ0 = atan2 (C12, C11) = α + γ.

(37)

With γ = 0,

α = δ0.

(38)

At β = π, define

δπ = atan2 (C12, C11) = α − γ.

(39)

Again choosing γ = 0 gives

α = δπ.

(40)

A different implementation may hold α fixed and assign the coupled angle to γ. The important requirement is an explicit and consistent policy.

16 Alternate Euler branch

Away from singularity, a nonprincipal triple representing the same orientation is

(α′,β′,γ′) = (α + π,− β,γ + π ),

(41)

with the outer angles wrapped by integer multiples of 2π as needed.

The principal restriction

0 ≤ β ≤ π

selects one representative away from the singular end points.

This is another reminder that Euler triples are coordinates, not unique physical vectors.

17 Equivalent extrinsic description

Intrinsic 3-1-3 with chronological angle list

(α,β,γ )

is equivalent to extrinsic 3-1-3 with the reversed chronological angle list

(γ, β,α).

Therefore

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

(42)

Because the axis sequence is palindromic, the digit string remains 313 even though the chronology and angle association reverse.

18 Active versus passive 3-1-3 matrices

The PhysicsLibrary matrix is passive:

Bv = BC Av. A

The corresponding active rotation matrix is the transpose,

B T Ractive = ( CA ) .

(43)

Consequently, a matrix found in another reference may appear transposed or may use the opposite signs in its elementary factors while describing the same physical orientation under an active-vector convention.

19 Passive quaternion equivalent

For a positive passive frame rotation about axis i,

λ λ qPi (λ) = cos--− eisin -. 2 2

(44)

Therefore intrinsic 3-1-3 has quaternion product

BqA = qP3 (γ)qP1 (β)qP3 (α).

(45)

Under Hamilton multiplication with scalar-first storage, this product reduces to the compact form

⌊ ⌋ β α + γ | cos 2-cos--2--- | || β α − γ || B |− sin --cos------| qA = || 2β α 2− γ || . ||− sin --sin ------|| ⌈ 2β α 2+ γ ⌉ − cos --sin ------ 2 2

(46)

The quaternion and DCM represent the same passive map:

B B CA = C ( qA ).

(47)

The sum and difference of the outer angles already appear naturally in this quaternion expression, foreshadowing the singularity structure of the Euler coordinates.

20 Angular velocity from the Euler angle rates

Let

⌊ ⌋ p BωB ∕A = ⌈ q⌉ r

(48)

be the angular velocity of frame B relative to frame A, expressed in frame B coordinates.

The physical angular velocity is the sum of the three instantaneous frame rotations:

A ˙ A1 A2 ωB ∕A = α˙e3 + β e1 + γ˙e3 .

(49)

Expressing these axes in frame B gives

⌊s s ⌋ B A ⌈ β γ⌉ e 3 = s βcγ , cβ

(50)

⌊ ⌋ cγ BeA11 = ⌈ − sγ⌉ , 0

(51)

and

⌊ ⌋ B A2 0 e3 = ⌈0 ⌉. 1

(52)

Therefore

⌊ ⌋ ⌊ ⌋ ⌊ ⌋ p sβsγ cγ 0 α˙ ⌈ q⌉ = ⌈ sβcγ − sγ 0⌉ ⌈β˙⌉ . r c 0 1 β γ˙

(53)

The determinant of this rate matrix is

− sinβ.

(54)

Thus the rate mapping becomes singular at exactly the same configurations as the Euler coordinate extraction.

21 Euler rates from body angular velocity

Away from sin β = 0, the inverse relationship is

⌊ ⌋ ⌊ ⌋ sγ- -cγ- 0 ⌊ ⌋ ˙α | sβ s β | p ⌈ ˙β⌉ = ⌈ cγ − sγ 0⌉ ⌈q ⌉. ˙γ − s cot β − c cot β 1 r γ γ

(55)

Equivalently,

sγp + cγq α˙= ---------, s β

(56)

˙β = cγp − sγq,

(57)

and

˙γ = r − cot β (sγp + cγq) .

(58)

The blow-up of the coefficients as β approaches 0 or π is a coordinate effect, not a physical divergence of angular velocity.

22 Classical precession, nutation, and spin language

A common classical-mechanics use of 3-1-3 associates the three angles with

α: precession about the reference axis z_A;
β: nutation or tilt about the line of nodes x₁;
γ: spin about the current body-related axis z₂.

This terminology is useful in rigid-body dynamics, but it is not a substitute for stating the axis sequence and active/passive convention. Different texts may assign different symbols to precession, nutation, and spin.

23 Numerical example

Take

α = 20∘, β = 60∘, γ = − 15∘.

(59)

Then

B ∘ ∘ ∘ CA = C3 (− 15 )C1 (60 )C3(20 )

(60)

becomes approximately

⌊+0.95193 +0.20876 − 0.22414⌋ B ⌈ ⌉ CA ≈ +0.07803 +0.54236 +0.83652 . +0.29620 − 0.81380 +0.50000

(61)

The corresponding passive quaternion is

⌊ ⌋ +0.865201 B ||− 0.476858 || qA ≈ ⌈− 0.150353 ⌉ . − 0.037776

(62)

The matrix satisfies

T CC = I,

(63)

and

detC = 1

(64)

to numerical precision.

Applying the inverse formulas gives

(α, β,γ) = (20∘,60 ∘,− 15∘),

(65)

confirming the principal-branch round trip.

24 Relative attitude is not found by subtracting Euler angles

Suppose two attitudes are represented by 3-1-3 triples

(α1, β1,γ1)

and

(α2,β2, γ2).

The exact relative orientation is not generally obtained by subtracting the three coordinates component by component.

B1 B2 C1 = CA, C2 = CA,

then the exact relative passive DCM is

B2CB1 = C2CT1 .

(66)

If a relative 3-1-3 triple is needed, extract it from this relative DCM. Quaternion composition gives the equivalent exact result.

25 Implementation guidance

A robust software implementation should separate three tasks:

build a DCM or quaternion from the Euler coordinates;
propagate orientation with a nonsingular representation when practical;
extract 3-1-3 angles only when the coordinate representation is actually needed.

Near β = 0 or β = π, the extracted outer angles may change rapidly while the underlying DCM or quaternion varies smoothly. This is expected behavior for the coordinate chart.

For numerical extraction, clamp roundoff-sensitive values such as C₃₃ to the interval [−1, 1] before using arccos, and define an explicit singularity threshold for | sin β|.

26 Convention traps in the literature

A statement such as “Euler 313 angles” does not fully determine a matrix. At minimum, identify:

intrinsic or extrinsic axes;
active or passive transformation;
coordinate-map direction;
column-vector or row-vector convention;
positive-angle convention;
chronological rotation order versus written matrix-factor order;
the symbol assignment to first, second, and third angles;
the principal angle ranges used for inverse extraction.

For a palindromic sequence such as 313, intrinsic/extrinsic conversion is especially easy to misread because the digit string itself does not change. The angle chronology must still reverse.

27 Verification battery

A correct PhysicsLibrary 3-1-3 implementation should satisfy:

zero angles give the identity matrix;
β = γ = 0 gives C₃(α);
α = γ = 0 gives C₁(β);
α = β = 0 gives C₃(γ);
CC^T = I;
det C = 1;
the reverse coordinate map is C^T;
DCM-to-Euler round trips recover the principal branch away from sin β = 0;
at β = 0, the DCM depends only on α + γ;
at β = π, the DCM depends only on α − γ;
the DCM agrees with the passive quaternion product;
the Euler-rate matrix has determinant − sin β and becomes singular at the same middle-angle configurations.

28 Summary

The PhysicsLibrary intrinsic 3-1-3 proper Euler sequence is

α : 3A, β : 11, γ : 32.

(67)

Its passive coordinate map is

B CA = C3(γ)C1 (β )C3(α ).

(68)

Away from singularity,

β = arccos(C33),

(69)

α = atan2 (C31,− C32),

(70)

and

γ = atan2(C13, C23).

(71)

The singular configurations are

β = 0 and β = π.

(72)

At the first, only α + γ is observable; at the second, only α − γ is observable.

The equivalent passive quaternion is

B P P P qA = q3 (γ)q1 (β)q3 (α).

(73)

The 3-1-3 sequence is therefore a useful local coordinate system for classical rigid-body orientation, while DCMs and quaternions remain preferable for globally nonsingular numerical attitude propagation.

The next article, Euler angles: DCM to Euler angles, develops systematic inverse extraction across Euler sequences.

29 References and further reading

Henderson provides the classic NASA engineering development and tabulation of Euler angle, quaternion, and transformation-matrix relationships across the twelve standard sequences.

Moore develops reference-frame orientation and body-fixed Euler sequences in a modern mechanics treatment, including angular-velocity formulas and gimbal-lock considerations.

Diebel gives a compact unified comparison of Euler angles, direction cosine matrices, unit quaternions, and rotation vectors.

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. K. Moore, Learn Multibody Dynamics, chapters “Orientation of Reference Frames” and “Angular Kinematics,” 2026 edition. Licensed CC BY 4.0. Learn Multibody Dynamics

[3] J. Diebel, “Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors,” Stanford University, 2006. Online PDF

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