Euler angles: definition and basic geometry

Euler Angles: Definition and Basic Geometry

Euler angles describe the orientation of one three dimensional coordinate frame relative to another by means of three successive one axis rotations.

The phrase “Euler angles” is often used loosely. In precise engineering and mathematical work, however, an Euler angle triple is not defined by three numbers alone. Its meaning also depends on:

the coordinate frames being related;
whether the rotations are intrinsic or extrinsic;
the three rotation axes;
the order in which the rotations occur;
the sign convention for positive rotation;
whether the matrices act actively on physical vectors or passively on coordinate representations.

This article develops the geometry behind Euler angles before specializing to individual sequences such as 3-2-1 yaw pitch roll or 3-1-3 proper Euler angles.

The PhysicsLibrary convention is passive and intrinsic.

1 Orientation of two coordinate frames

Let A and B be right handed orthonormal frames.

A physical vector v may be represented by coordinate columns

Av

and

Bv.

The passive direction cosine matrix from frame A coordinates to frame B coordinates is defined by

Bv = BC Av. A

(1)

The matrix

B CA

therefore describes the orientation of frame B relative to frame A while mapping coordinate descriptions from A into B.

Because both frames are orthonormal,

BCA (BCA )T = I,

(2)

and

B det( CA ) = 1.

(3)

Thus

A B T B −1 CB = ( CA ) = ( CA ) .

(4)

The two matrices represent opposite coordinate map directions for the same relative physical orientation.

2 The basic Euler construction

An Euler angle construction introduces two intermediate frames between the initial and final frames.

Let

A0 = A, A3 = B.

(5)

The sequence is

A −→ A − → A −→ A . 0 1 2 3

Each arrow represents one rotation about one coordinate axis.

For a generic intrinsic sequence i-j-k,

A −→ A − → A −→ A . 0 1 2 3

The first step is a rotation through α about axis i of A₀. The second is a rotation through β about axis j of the moving frame A₁. The third is a rotation through γ about axis k of the moving frame A₂.

The three angles are therefore:

α = first intrinsic rotation angle,

(6)

β = second intrinsic rotation angle,

(7)

and

γ = third intrinsic rotation angle.

(8)

The intermediate frames are not optional bookkeeping devices. They are part of the geometric meaning of an intrinsic Euler sequence.

Figure. Intrinsic Euler angle construction. Frame A₀ = A is rotated through three successive intrinsic rotations to produce A₁, then A₂, and finally A₃ = B. The angles α, β, and γ are applied in that chronological order, while the passive coordinate transformation is written as

BC = C (γ)C (β)C (α). A k j i

The second and third rotations are taken about axes of the intermediate moving frames, which is why intrinsic Euler angles are sequence dependent and should not be treated as components of an ordinary vector.

3 Intrinsic means moving axis

In an intrinsic sequence, each rotation after the first is performed about an axis of the frame produced by the previous rotation.

For intrinsic i-j-k:

rotate frame A₀ through α about axis i of A₀;
rotate the resulting frame A₁ through β about axis j of A₁;
rotate the resulting frame A₂ through γ about axis k of A₂.

The second and third axes have therefore generally moved in physical space.

This is the reason finite Euler angle rotations cannot be interpreted as three independent rotations about the original frame axes.

4 Passive coordinate maps for the intermediate frames

Let

Ci(α)

be the passive elementary coordinate transformation associated with the first intrinsic frame rotation.

Then

A1v = Ci(α )A0v.

(9)

The second step is

A2v = Cj(β )A1v.

(10)

The third step is

A3v = C (γ )A2v. k

(11)

Substitute successively:

A3 A0 v = Ck(γ)Cj (β )Ci(α) v.

Since

A0 = A

and

A3 = B,

the complete intrinsic Euler coordinate transformation is

B CA = Ck(γ)Cj (β)Ci(α).

(12)

This is the central composition rule for the PhysicsLibrary Euler angle series.

5 Why the rightmost matrix acts first

The product

Ck (γ)Cj(β )Ci (α)

is sometimes misread as though the k rotation happened first.

That is not what the matrix equation means.

Coordinate columns are multiplied from the left:

Bv = Ck(γ)Cj (β )Ci(α)Av.

The first operation applied to the column is therefore

C (α). i

The chronological intrinsic sequence is still

i → j → k.

The matrix product writes the composed maps in the usual function composition order.

6 Elementary passive frame rotations

For completeness, the three elementary passive transformations are

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

(13)

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

(14)

and

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

(15)

Each matrix describes the change in coordinates caused by a positive rotation of the coordinate frame.

For example,

⌊ ⌋ (π ) 0 1 0 C3 -- = ⌈ − 1 0 0⌉ . 2 0 0 1

(16)

Thus a fixed physical vector having coordinates

⌊ 1⌋ A ⌈ ⌉ v = 0 0

has coordinates

⌊ ⌋ 0 Bv = ⌈− 1⌉ 0

(17)

after the frame is rotated positively by 90^∘ about +z.

7 Euler angles are not components of a vector

An Euler angle triple is often written as

(α, β,γ).

This notation resembles a vector, but its geometric meaning is completely different.

In general,

(α1,β1,γ1) + (α2,β2,γ2)

(18)

does not represent the composition of two orientations.

Likewise,

(α2 − α1,β2 − β1,γ2 − γ1)

(19)

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

The reason is that the meaning of the second and third Euler angles depends on the intermediate moving frames.

Euler angles are nonlinear coordinates on the orientation manifold, not components of an ordinary geometric vector.

8 Noncommutativity appears immediately

Consider two positive frame rotations:

90 ∘ about + x

and

90∘ about + y.

The corresponding passive matrices are

( ) C1 π- 2

and

( π) C2 -- . 2

In general,

( ) ( ) ( ) ( ) π- π- π- π- C2 2 C1 2 ⁄= C1 2 C2 2 .

(20)

Thus changing the sequence changes the final orientation.

This is not a numerical artifact. Noncommutativity is a fundamental property of finite rotations in three dimensions.

9 Why three rotations are used

The orientation of one rigid orthonormal frame relative to another has three independent degrees of freedom.

Euler angles provide three scalar coordinates for those three degrees of freedom.

The construction uses three successive rotations because two rotations are not sufficient to generate every possible orientation, while three properly chosen rotations are sufficient locally.

The use of three parameters is minimal.

A direction cosine matrix uses nine stored numbers subject to six independent orthonormality constraints.

A unit quaternion uses four stored numbers subject to one unit norm constraint.

Euler angles use exactly three scalar coordinates, but that minimality requires coordinate singularities somewhere in the representation.

10 Why adjacent axes cannot repeat

Suppose two consecutive intrinsic rotations are both about the same current axis.

For example,

A0 −→ A1 − → A2.

Here both consecutive steps are rotations about the current axis 1, with angles α and β respectively.

Because the first rotation leaves its own rotation axis unchanged in the current frame, the second rotation is about the same physical axis.

The two rotations combine:

C1 (β)C1(α ) = C1(α + β ).

(21)

Thus two adjacent rotations about the same axis do not provide two independent orientation coordinates.

A valid standard three angle sequence therefore requires

j ⁄= i

and

k ⁄= j.

The first and third axes may either differ or coincide.

11 Why there are twelve standard sequences

There are three choices for the first axis.

Once the first axis is chosen, there are two choices for the second axis because the second axis must differ from the first.

For the third axis, there are two admissible choices:

use the remaining third axis;
return to the first axis.

Therefore the number of standard sequences is

3 × 2 × 2 = 12.

(22)

These divide naturally into two families.

12 Tait Bryan sequences

A Tait Bryan sequence uses three distinct axes.

The six intrinsic Tait Bryan sequences are

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

(23)

Because all three axes differ, these are also called Cardan sequences in some literature.

Aerospace yaw pitch roll is normally represented by the intrinsic 3-2-1 Tait Bryan sequence.

13 Proper Euler sequences

A proper Euler sequence returns to the first axis on the third rotation.

The six intrinsic proper Euler sequences are

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

(24)

These sequences are common in classical rigid body mechanics, orbital orientation, and historical treatments of Euler’s rotational coordinates.

14 A generic Tait Bryan example

Consider intrinsic 3-2-1.

The frame chain is

A0 −→ A1 − → A2 −→ A3.

The three intrinsic rotations are α about axis 3 of A₀, then β about axis 2 of A₁, then γ about axis 1 of A₂.

The passive coordinate transformation is

BCA = C1(γ)C2 (β )C3(α ).

(25)

In aerospace notation,

α = ψ, β = 𝜃, γ = ϕ.

(26)

Therefore

BCA = C1(ϕ)C2 (𝜃)C3(ψ).

(27)

This is the standard PhysicsLibrary yaw pitch roll specialization.

15 A generic proper Euler example

Consider intrinsic 3-1-3.

The frame chain is

A0 −→ A1 − → A2 −→ A3.

The three intrinsic rotations are α about axis 3 of A₀, then β about axis 1 of A₁, then γ about axis 3 of A₂.

The passive transformation is

BCA = C3(γ)C1 (β )C3(α ).

(28)

Although the first and third axis labels are both 3, they refer to axis 3 of different intermediate frames.

That moving axis distinction is essential.

16 Intrinsic and extrinsic are related but not identical descriptions

An intrinsic sequence uses moving axes.

An extrinsic sequence uses axes fixed in the original reference frame.

A given physical orientation can often be described either way.

An intrinsic i-j-k sequence can be reinterpreted as a corresponding extrinsic sequence with reverse axis order, provided the angle association is handled consistently.

For example, an intrinsic

3 → 2 → 1

description corresponds to a fixed axis description involving

1 → 2 → 3.

This equivalence does not mean the words “intrinsic” and “extrinsic” may be omitted. They describe different geometric constructions that happen to produce the same final orientation under the corresponding reversal rule.

17 Active and passive are independent of intrinsic and extrinsic

The distinction between intrinsic and extrinsic concerns which axes are used.

The distinction between active and passive concerns what is being rotated.

An active transformation rotates a physical vector while holding the coordinate frame fixed.

A passive transformation rotates the coordinate frame while holding the physical vector fixed.

These are independent choices.

The PhysicsLibrary Euler series uses passive coordinate transformations and intrinsic moving axis sequence names.

18 Worked example 1: a single intrinsic rotation

Suppose the sequence is intrinsic 3-2-1, but only the first angle is nonzero:

∘ α = 30 , β = 0, γ = 0.

Then

BCA = C1(0)C2 (0)C3(30∘).

Since

C1(0) = I

and

C (0) = I, 2

the complete transformation reduces to

B ∘ CA = C3 (30 ).

(29)

Thus the generic sequence correctly reduces to the expected single elementary frame rotation.

19 Worked example 2: two sequences with the same angles

Compare

1 → 2

with

2 → 1

using positive 90^∘ angles.

For the first chronological order,

( ) ( ) π- π- C12 = C2 2 C1 2 .

Using the elementary matrices,

⌊ ⌋ 0 1 0 C12 = ⌈ 0 0 1⌉ . 1 0 0

(30)

For the reversed chronological order,

( ) ( ) C = C π- C π- . 21 1 2 2 2

This gives

⌊ ⌋ 0 0 − 1 C = ⌈ − 1 0 0 ⌉ . 21 0 1 0

(31)

Therefore

C12 ⁄= C21.

(32)

The same two angle magnitudes give different final orientations when the sequence is changed.

20 Worked example 3: why the first and third axes may repeat

Consider the proper Euler sequence

3 → 1 → 3.

At first glance it may appear that the first and third rotations are about the same axis and could be combined.

That is generally false.

The first rotation is about axis 3 of frame A₀.

The third rotation is about axis 3 of frame A₂.

After the intermediate rotation about axis 1 of A₁, axis 3 of A₂ is generally not aligned with axis 3 of A₀.

Therefore the first and third rotations are generally about different physical directions even though the sequence labels both axes with the number 3.

This is exactly why proper Euler sequences remain three parameter representations.

21 Nonuniqueness of Euler angle coordinates

A given physical orientation may have multiple Euler angle descriptions.

The simplest source of nonuniqueness is periodicity.

For any integer n,

Ci(λ + 2 πn) = Ci(λ ).

(33)

Thus adding full turns to sequence angles can leave the final orientation unchanged.

More subtle alternate branches occur when extracting angles from a DCM.

At singular configurations, the first and third Euler angles may no longer be independently identifiable.

Therefore Euler angles require declared principal ranges when used as a unique coordinate output.

22 Coordinate singularities

Minimal three parameter orientation coordinates cannot cover all of three dimensional orientation space with one globally nonsingular chart.

For a Tait Bryan sequence, the middle angle becomes singular when the first and third axes align after the middle rotation.

The generic condition is

cosβ = 0.

(34)

For 3-2-1,

β = 𝜃 = ± π. 2

(35)

For a proper Euler sequence, the singularity occurs when

sin β = 0.

(36)

On the usual principal interval,

β = 0 or β = π.

(37)

The physical orientation is not singular at these configurations.

Only the Euler coordinate chart is singular.

23 Euler angles and the orientation manifold

The set of all proper three dimensional rotations is the rotation group

SO (3 ).

Euler angles provide local coordinates on SO(3).

A local coordinate map may be written schematically as

(α,β,γ ) ↦− → BCA.

The map is smooth away from singular configurations.

At a gimbal lock configuration, the local coordinate Jacobian loses rank.

This geometric viewpoint explains several practical facts at once:

Euler angles are not vectors;
finite angle differences are not exact relative rotations;
coordinate singularities are unavoidable;
several triples may represent the same orientation;
alternative attitude representations can remain regular when a chosen Euler chart becomes singular.

24 Relationship to the quaternion series

The migrated PhysicsLibrary quaternion series uses the same passive frame map direction.

For intrinsic i-j-k,

BqA = qPk (γ)qPj (β)qPi (α),

(38)

while

BCA = Ck(γ)Cj (β)Ci(α).

(39)

The representations are connected by

B B CA = C ( qA ).

(40)

This agreement is a useful convention check.

Euler angles supply a minimal human interpretable coordinate chart.

A unit quaternion supplies a four component nonsingular attitude representation subject to a unit norm constraint.

Neither representation changes the underlying physical orientation.

25 What Euler angles are good for

Euler angles are especially useful when:

the orientation naturally has physically meaningful yaw, pitch, and roll coordinates;
the system remains well away from the sequence singularity;
human readability is important;
limits are naturally expressed in angular coordinates;
initial or final boundary conditions are specified in a particular sequence;
visualization or operator displays require intuitive attitude angles.

26 When another representation may be preferable

A DCM or quaternion may be preferable when:

the motion can pass through an Euler singularity;
the attitude must be propagated numerically for long intervals;
angular velocity measurements drive the attitude state;
repeated composition is central;
derivatives and optimization must remain smooth over large orientation ranges;
a global orientation representation is required.

Euler angles are still useful in such systems as display coordinates even when the internal state is represented by a quaternion or DCM.

27 Common conceptual mistakes

Treating (α,β,γ) as a geometric vector.
Forgetting that the second and third intrinsic axes move.
Reading the matrix product from left to right as chronological order.
Assuming a 3-2-1 sequence is automatically yaw pitch roll without declaring intrinsic or extrinsic.
Using ϕ,𝜃,ψ as generic first, second, and third angles while also calling them roll, pitch, and yaw.
Assuming active versus passive is determined by whether a sequence is intrinsic or extrinsic.
Assuming the first and third axes of a proper Euler sequence are the same physical axis.
Subtracting two finite Euler triples to obtain an exact relative orientation.
Ignoring alternate angle branches.
Treating gimbal lock as a physical loss of orientation rather than a coordinate singularity.

28 Checks every Euler sequence should satisfy

Any derived sequence formula should pass the following tests.

Identity

α = β = γ = 0 = ⇒ BCA = I.

(41)

Single angle reductions

For example,

β = γ = 0

must give

B CA = Ci(α ).

(42)

Likewise for the second and third rotations.

Orthogonality

BC (BC )T = I. A A

(43)

Proper determinant

det(BC ) = 1. A

(44)

Reverse map

AC = (BC )T . B A

(45)

Quaternion agreement

BCA = C (BqA ).

(46)

These checks catch many sign and sequence mistakes before a formula is used in software.

29 Summary

Euler angles describe orientation through three ordered one axis frame rotations.

In the PhysicsLibrary convention:

Bv = BCAAv,

(47)

and intrinsic i-j-k means

BCA = Ck(γ)Cj (β)Ci(α).

(48)

The intermediate frames

A0, A1, A2, A3

are part of the definition.

There are twelve standard sequences:

six Tait Bryan sequences with three distinct axes;
six proper Euler sequences with first and third axes equal by label.

Euler angles are minimal and interpretable, but they are nonlinear local coordinates rather than vector components. Their sequence dependence, noncommutativity, nonuniqueness, and singularities are fundamental geometric features, not implementation defects.

The next article in the series, Elementary Axis Rotations and Passive Rotation Matrices, derives the three elementary matrices used throughout all twelve Euler sequences.

30 References and further reading

Henderson provides a classic engineering tabulation of Euler sequences, quaternions, and transformation matrices.

Diebel presents a modern unified treatment of attitude representations.

Moore develops frame orientation from basis vectors and direction cosine matrices in a form especially useful for passive coordinate transformations.

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

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] H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison Wesley, 2002. Publisher search

License

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

"Euler angles: definition and basic geometry" is owned by bloftin.
(view preamble)