Euler Angles: Orientation, Notation, and Conventions

Euler angles describe the relative orientation of two three-dimensional reference frames by means of three successive one axis rotations. They provide a minimal, three parameter description of orientation and are therefore intuitive and economical, but the same economy also produces coordinate singularities and convention ambiguities.

The phrase “Euler angles” is used broadly in engineering for both proper Euler sequences, in which the first and third axes are the same, and Tait–Bryan sequences, in which all three axes are different. Examples are the $3$ - $1$ - $3$ proper Euler sequence and the $3$ - $2$ - $1$ yaw–pitch–roll Tait–Bryan sequence.

A formula involving Euler angles is incomplete unless it states at least:

the handedness of the coordinate frames;
whether the operation is an active vector rotation or a passive coordinate transformation;
whether the three rotations are intrinsic (moving axis) or extrinsic (fixed axis);
the axis sequence;
the direction of the frame to frame coordinate map; and
the meanings and ordering of the three angle symbols.

This entry establishes the convention used by the PhysicsLibrary Euler angle series.

PhysicsLibrary Euler angle convention contract

Unless an article explicitly states otherwise, the PhysicsLibrary Euler angle series uses the following convention.

$\fbox{\begin{minipage}{0.92\linewidth} \textbf{PhysicsLibrary Euler angle conven... ...pitch},\qquad \phi=\text{roll}.} \end{displaymath}\end{enumerate}\end{minipage}}$

This convention is deliberately explicit because many apparently contradictory Euler angle formulas differ only by a transpose, a reversed sequence description, or a relabeling of the three angles.

Passive elementary coordinate transformations

Let frame $B$ be obtained from frame $A$ by a positive right-handed rotation of the axes through angle $\lambda$ about one coordinate axis. The passive coordinate transformation from $A$ to $B$ is written $C_i(\lambda)$ , where $i=1,2,3$ denotes the $x,y,z$ axis, respectively.

The three elementary transformations are

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

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

and

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

These matrices are orthogonal,

$\displaystyle C_i^{-1}=C_i^T,$

and have determinant $+1$ .

They are the transposes of the matrices commonly written for an active rotation of a vector by the same positive angle. The sign difference is a consequence of changing the basis rather than moving the physical vector.

A $90^\circ$ sign check

Suppose frame $B$ is obtained by rotating the axes of frame $A$ through $+90^\circ$ about $+\hat{\mathbf z}$ . A physical vector with coordinates

$\displaystyle {}^{A}\mathbf v= \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}$

has coordinates in frame $B$

$\displaystyle {}^{B}\mathbf v = C_3\left(\frac{\pi}{2}\right){}^{A}\mathbf v = \begin{bmatrix} 0\\ -1\\ 0 \end{bmatrix}.$

The vector itself has not rotated. The new coordinate axes have rotated, so the coordinates change with the opposite sign from an active vector rotation.

Intrinsic Euler sequences

Consider an original frame $A=A_0$ and a final frame $B=A_3$ . An intrinsic $i$ - $j$ - $k$ Euler sequence is constructed as follows:

$\displaystyle A_0=A \ \xrightarrow[$ axis $\displaystyle i]{\alpha} A_1 \ \xrightarrow[$ axis $\displaystyle j$ of $\displaystyle A_1]{\beta} A_2 \ \xrightarrow[$ axis $\displaystyle k$ of $\displaystyle A_2]{\gamma} A_3=B .$

Each transformation maps coordinates into the newly rotated frame,

$\displaystyle {}^{A_1}\mathbf v=C_i(\alpha)\,{}^{A}\mathbf v,$

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

and

$\displaystyle {}^{B}\mathbf v=C_k(\gamma)\,{}^{A_2}\mathbf v.$

Substitution therefore gives the universal PhysicsLibrary rule

$\displaystyle \boxed{ {}^{B}C_A = C_k(\gamma)\,C_j(\beta)\,C_i(\alpha) }$

for an intrinsic $i$ - $j$ - $k$ sequence.

The rightmost matrix acts first on the coordinate column. This is not a reversal of the physical sequence: it is the ordinary rule for composition of linear transformations.

The twelve three angle sequences

A valid three angle sequence cannot use the same axis for two consecutive rotations, because two consecutive rotations about the same current axis would combine into a single rotation. This leaves twelve standard sequences.

Tait–Bryan sequences

The six Tait–Bryan sequences use three distinct axes.

$\begin{displaymath} \begin{array}{c\vert c} \text{intrinsic sequence} & {}^{B}C_... ...ext{-}2\text{-}1 & C_1(\gamma)C_2(\beta)C_3(\alpha) \end{array}\end{displaymath}$

Proper Euler sequences

The six proper Euler sequences repeat the first axis as the third axis.

$\begin{displaymath} \begin{array}{c\vert c} \text{intrinsic sequence} & {}^{B}C_... ...ext{-}2\text{-}3 & C_3(\gamma)C_2(\beta)C_3(\alpha) \end{array}\end{displaymath}$

The expanded matrices are developed in the individual PhysicsLibrary sequence entries: Euler 123 sequence, Euler 132 sequence, Euler 121 sequence, Euler 131 sequence, Euler 213 sequence, Euler 231 sequence, Euler 212 sequence, Euler 232 sequence, Euler 312 sequence, Euler 321 sequence, Euler 313 sequence, and Euler 323 sequence.

Intrinsic versus extrinsic descriptions

An intrinsic rotation sequence uses moving axes. An extrinsic sequence uses axes fixed in the original frame. These are not two different classes of physical orientation; an intrinsic sequence can be re described as an extrinsic sequence if the axis order and interpretation are reversed appropriately.

For example, an intrinsic $3$ - $2$ - $1$ sequence may be described geometrically as:

yaw about the original $\displaystyle 3$ -axis $\displaystyle ,$ pitch about the new $\displaystyle 2$ -axis $\displaystyle ,$ roll about the new $\displaystyle 1$ -axis $\displaystyle .$

The same final orientation can be represented using a corresponding fixed axis description, but PhysicsLibrary names and derives Euler sequences intrinsically. Whenever formulas from another source are imported, the source's intrinsic/extrinsic convention must be checked before comparing matrix products.

Why generic angles are written as $\alpha,\beta,\gamma$

The legacy PhysicsLibrary Euler entries generally used $\phi$ , $\theta$ , and $\psi$ as the first, second, and third rotation parameters independent of which physical axes were used. That is internally workable, but it conflicts with the widespread aerospace convention

$\displaystyle \phi=$ roll $\displaystyle ,\qquad \theta=$ pitch $\displaystyle ,\qquad \psi=$ yaw $\displaystyle .$

In a $3$ - $2$ - $1$ sequence, the first rotation is yaw and the third is roll. Consequently, the legacy notation can make a correct matrix appear to have the yaw and roll symbols interchanged.

PhysicsLibrary therefore uses

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

for generic first/second/third sequence angles and reserves

$\displaystyle (\phi,\theta,\psi)$

for contexts in which the physical meanings roll/pitch/yaw are intended.

Aerospace -- yaw–pitch–roll specialization

For the intrinsic $3$ - $2$ - $1$ sequence,

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

where $\psi$ is yaw, $\theta$ is pitch, and $\phi$ is roll. The passive coordinate transformation from the reference frame $A$ to the final frame $B$ is therefore

$\displaystyle \boxed{ {}^{B}C_A = C_1(\phi)\,C_2(\theta)\,C_3(\psi). }$

Writing

$\displaystyle c_\phi=\cos\phi,\quad s_\phi=\sin\phi, \qquad c_\theta=\cos\theta,\quad s_\theta=\sin\theta, \qquad c_\psi=\cos\psi,\quad s_\psi=\sin\psi,$

gives

$\displaystyle \boxed{ {}^{B}C_A= \begin{bmatrix} c_\theta c_\psi & c_\theta s_\... ..._\psi & c_\phi s_\theta s_\psi-s_\phi c_\psi & c_\phi c_\theta \end{bmatrix}.}$

The dedicated Euler 321 sequence entry derives this matrix step by step and discusses inverse extraction and the singularity at $\theta=\pm\pi/2$ .

Euler angles are coordinates, not vectors

The three numbers in an Euler angle triple are parameters of a nonlinear composition of rotations. In general,

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

does not represent the composition of the corresponding orientations. Finite rotations in three dimensions do not commute, and an Euler angle triple should not be transformed or added as though it were an ordinary geometric vector.

This distinction becomes especially important in attitude estimation and control, where small attitude errors may sometimes be approximated by a threecomponent vector even though the finite Euler angles themselves are not a vector.

Coordinate singularities and nonuniqueness

Euler angles provide only a local coordinate chart on three-dimensional orientation. No three parameter Euler chart is globally nonsingular.

For a Tait–Bryan sequence the singularity occurs when the middle rotation places the first and third rotation axes in alignment. In the $3$ - $2$ - $1$ sequence this occurs at

$\displaystyle \boxed{\theta=\pm\frac{\pi}{2}.}$

This loss of one independent rotational degree of freedom in the coordinates is commonly called gimbal lock. The physical orientation remains perfectly well defined; only the chosen Euler angle coordinates become singular.

Euler triples are also not globally unique. Angle periodicity and the structure of the sequence allow more than one triple to represent the same final orientation. Software interfaces should therefore state the chosen principal ranges for inverse Euler angle extraction.

Relationship to direction cosine matrices

A direction cosine matrix represents the complete orientation without a coordinate singularity, although it uses nine matrix elements subject to six orthonormality constraints.

Under the convention of this article,

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

The inverse transformation is

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

with

$\displaystyle \boxed{ {}^A C_B = ({}^B C_A)^T = ({}^B C_A)^{-1}. }$

A source that writes the transpose of a PhysicsLibrary Euler matrix may therefore be describing the same physical orientation but mapping coordinates in the opposite direction.

Compatibility with the PhysicsLibrary quaternion convention

The PhysicsLibrary quaternion series uses Hamilton multiplication and writes ${}^{A}q_B$ for the orientation of frame $B$ relative to frame $A$ , with the coordinate map

$\displaystyle {}^{A}v = {}^{A}q_B\,{}^{B}v\,({}^{A}q_B)^*.$

If $R({}^{A}q_B)$ denotes the corresponding matrix that maps $B$ coordinates to $A$ coordinates, then

$\displaystyle {}^{A}C_B=R({}^{A}q_B).$

The passive Euler matrix used in this article maps in the opposite coordinate direction:

$\displaystyle \boxed{ {}^{B}C_A = R({}^{A}q_B)^T. }$

Thus the Euler and quaternion conventions describe the same relative orientation without changing Hamilton's quaternion multiplication law. The frame direction is carried explicitly by the notation rather than hidden in a sign convention.

For the $3$ - $2$ - $1$ yaw–pitch–roll orientation,

$\displaystyle {}^{A}q_B = q_3(\psi)\,q_2(\theta)\,q_1(\phi)$

under the PhysicsLibrary/Hamilton quaternion convention, while

$\displaystyle {}^{B}C_A = C_1(\phi)\,C_2(\theta)\,C_3(\psi) = R({}^{A}q_B)^T.$

See Quaternions for Physics and Engineering: Orientation, Notation, and Conventions and quaternions and Euler angleshttps://physicslibrary.org/encyclopedia/QuaternionsAndEulerAngles.html for the quaternion side of the relationship.

A minimum Euler angle convention declaration

Any engineering paper, simulation, software interface, or PhysicsLibrary entry that uses Euler angles should make enough information available to answer the following questions:

Are the coordinate frames right-handed?
What is the positive sense of rotation?
Is the operation active or passive?
What coordinate direction does the matrix map?
Is the sequence intrinsic or extrinsic?
What is the axis order?
Which angle is applied first?
What do the angle symbols mean physically?
What are the principal angle ranges used for inverse extraction?

If these questions are answered, most apparent Euler angle disagreements reduce to a transpose, an inverse, an order reversal, or an angle relabeling.

Common pitfalls

Calling a matrix simply “the rotation matrix.” Without a map such as ${}^{B}C_A$ , the direction of the coordinate transformation is ambiguous.
Mixing active and passive matrices. For the same positive geometric rotation, the active vector rotation matrix and passive coordinate transformation matrix are transposes of one another.
Calling a sequence -- without saying intrinsic or extrinsic. The axis labels alone do not completely specify the verbal operation.
Assuming $\phi,\theta,\psi$ always mean first, second, and third rotations. In aerospace work they normally mean roll, pitch, and yaw, respectively.
Reading the matrix product from left to right as the order of action on a column vector. The rightmost factor acts first.
Treating Euler angles as an ordinary three vector. Finite rotation composition is nonlinear and noncommutative.
Ignoring the middle angle singularity. Inverse formulas fail or become nonunique at gimbal lock.
Copying quaternion Euler formulas from another convention. Quaternion multiplication, frame direction, active/passive interpretation, and Euler sequence must be converted as one consistent set.

Relation to later PhysicsLibrary entries

The present entry establishes the convention contract. The individual sequence entries develop the twelve matrices in detail. Separate entries treat:

each of the twelve intrinsic Euler sequences;
conversion from a direction cosine matrix to Euler angles;
Euler angle rates and angular velocity;
quaternion Euler conversion; and
examples and exercises.

The $3$ - $2$ - $1$ yaw–pitch–roll sequence is treated first because of its importance in aerospace, navigation, robotics, and rigid-body dynamics.

References and convention notes

The matrix convention in this article agrees with the passive direction cosine matrix construction used in modern reference frame mechanics treatments, where the rows contain the new basis vectors resolved in the old basis. Henderson's NASA memorandum is a useful engineering reference because it tabulates all twelve three-angle transformations and emphasizes the importance of transformation direction. Diebel provides a broad comparison of Euler angles, rotation matrices, quaternions, and rotation vectors.

Bibliography

1: D. M. Henderson, Euler Angles, Quaternions, and Transformation Matrices—Working Relationships, JSC-12960, NASA Johnson Space Center, July 1977. NASA Technical Reports Server https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770024290.pdf
2: J. K. Moore, Learn Multibody Dynamics, chapter “Orientation of Reference Frames,” 2026 edition, CC BY 4.0. Orientation of Reference Frames https://moorepants.github.io/learn-multibody-dynamics/orientation.html
3: J. Diebel, “Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors,” Stanford University, 2006. online PDF https://www.astro.rug.nl/software/kapteyn-beta/_downloads/attitude.pdf

License

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

"Notation Euler Angles: Orientation" is owned by bloftin.
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