# Euler angles

Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler.

In common parlance, "rotation" refers to a physical motion of some object about a pivot or axis. In mathematics, the motion aspect is abstracted away, and only the "before" and "after" positions are considered.

To be concrete, consider a brick, set on a table, that is then pivoted about one corner. The pivot maintains contact with the table, in the same position throughout, but the rest of the brick may be lifted and turned. The before and after positions of the brick are characterized by knowledge of the three edges meeting the pivot corner. Replacing the three brick edges with mathematical lines, and the pivot corner with a point, one arrives at two coordinate systems sharing a common origin.

Euler angles are one of several ways of specifying the relative position of two such coordinate systems.

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## Definition

Given two coordinate systems xyz and XYZ with common origin, one can specify the position of the second in terms of the first using three angles α, β, γ in three equivalent, as follows:

• Static The intersection of the xy and the XY coordinate planes is called the line of nodes.
• α is the angle between the x-axis and the line of nodes.
• β is the angle between the z-axis and the Z-axis.
• γ is the angle between the X-axis and the line of nodes.
• Fixed axes of rotation Start with the XYZ system equalling the xyz system.
• Rotate the XYZ-system about the z-axis by α; the xyz-system does not move, now or later.
• Rotate it again about the x-axis by β.
• Rotate it a third time about the z-axis by γ.
(Note that the first and third axes are identical.)
• Moving axes of rotation Start with the XYZ system equalling the xyz system.
• Rotate the XYZ-system about the Z-axis by γ; the xyz-system does not move, now or later.
• Rotate it again about the now rotated X-axis by β.
• Rotate it a third time about the now doubly rotated Z-axis by α.
(Note that the angles are in reverse order.)

These three angles α, β, γ are the Euler angles. The equivalence of these three definitions is verified below.

### Angle ranges

• α and γ range from 0 to 2π radians.
• β ranges from 0 to π radians.

These angles are uniquely determined, with certain exceptions. (This is most easily verified using the static description.)

• With α and γ, 0 and 2π radians give the same 3D rotation.
• With β, 0 and π give the same 3D rotation.

This corresponds to the xy and the XY planes being identical, so the rotation is just a rotation of α+γ about the z-axis. (This last ambiguity is known as gimbal lock in applications.)

### Relation to physical motions

Having emphasized that "physical motions" have been abstracted away, their reappearance in two of the above definitions might seem inconsistent. In fact, these three motions are simply "nomimal". The actual motion of an object may or may not follow the three Euler angles literally.

If, for example, a satellite has spin control in two orthogonal directions, then reorienting the satellite can be accomplished by using the Euler angles directly, in the moving axes definition. But if engineering reasons dictated a different control mechanism, Euler angles will still describe the before and after relative positions.

This is like using ordinary rectangular coordinates. A given x,y specifies x to the right, y forward, which may be used directly, as on street grids, or not.

## Equivalence of the definitions

The static description is usually used in conjunction with spherical trigonometry. It is the only form in older sources. The two rotating axes descriptions are usually used in conjunction with matrices, since 2D coordinate rotations have a simple form. These last two are easily seen to be equivalent, since rotation about a moved axis is the conjugation of the original rotation by the move in question.

To be explicit, in the fixed axes description, let x(φ) and z(φ) denote the rotations of angle φ about the x-axis and z-axis, respectively. In the moving axes description, let Z(φ)=z(φ), X′(φ) be the rotation of angle φ about the once-rotated X-axis, and let Z″(φ) be the rotation of angle φ about the twice-rotated Z-axis. Then

Z″(α)⋅X′(β)⋅Z(γ) = [ (X′(β)z(γ)) ⋅ z(α) ⋅ (X′(β)z(γ))−1 ] ⋅ X′(β) ⋅ z(γ)
= [ {z(γ)x(β)z(−γ) z(γ)} ⋅ z(α) ⋅ {z(−γ) z(γ)x(−β)z(−γ)} ] ⋅ [ z(γ)x(β)z(−γ) ] ⋅ z(γ)
= z(γ)x(β)z(α)x(−β)x(β) = z(γ)x(β)z(α) .

The equivalence of the static description with the rotating axes descriptions can be verified by direct geometric construction, or by showing that the nine direction cosines (between the three xyz axes and the three XYZ axes) form the correct rotation matrix.

## Conventions

There are numerous conventions regarding the Euler angles in use. The above three descriptions are for the z-x-z form. z-y-z is also common, especially in quantum mechanics. One also finds variation over the use of left versus right handed coordinate systems, clockwise versus counterclockwise angles, and active versus passive coordinate transformations.

Even within a given mathematical choice of convention, one frequently finds different and conflicting choice of notation. Authors have been known to use conventions incorrectly.

To add to the confusion, flight and aerospace engineers, when using yaw, pitch, roll (also called heading, attitude, bank) to refer to rotations about the x, y, z axes, respectively, often call these the Euler angles. These x-y-z angles are properly known as the Tait-Bryan angles, also called Cardan angles or nautical angles.

## Properties of Euler angles

The Euler angles form a chart on all of SO(3), the special orthogonal group of rotations in 3D space. The chart is smooth except for a polar coordinate style singularity along β=0. See charts on SO(3) for a more complete treatment.

A similar three angle decomposition applies to SU(2), the special unitary group of rotations in complex 2D space, with the difference that β ranges from 0 to 2π. These are also called Euler angles.

## Applications

Euler angles are used extensively in the classical mechanics of rigid bodies, and in the quantum mechanics of angular momentum.

When studying rigid bodies, one calls the xyz system space coordinates, and the XYZ system body coordinates. The space coordinates are treated as unmoving, while the body coordinates are considered embedded in the moving body. Calculations involving kinetic energy are usually easiest in body coordinates, because the three components of a rigid body's moment of inertia are then constant.

The angular velocity, in body coordinates, of a rigid body takes a simple form using Euler angles:

[itex](\dot\alpha\sin\beta\sin\gamma+\dot\beta\sin\gamma){\bold I}
     +(\dot\alpha\sin\beta\cos\gamma-\dot\beta\sin\gamma){\bold J}
+(\dot\alpha\cos\beta+\dot\gamma){\bold K}[itex],


where IJK are unit vectors for XYZ.

In quantum mechanics, explicit descriptions of the representations of SO(3) are very important for calculations, and almost all the work has been done using Euler angles. In the early history of quantum mechanics, when physicists and chemists had a sharply negative reaction towards abstract group theoretic methods (called the Gruppenpest), reliance on Euler angles was also essential for basic theoretical work.

Haar measure for Euler angles has the simple form sin(β)dαdβdγ, usually normalized by a factor of 1/8π2. For example, to generate uniformly randomized orientations, let α and γ be uniform from 0 to 2π, let z be uniform from -1 to 1, and let β=arccos(z).

Unit quaternions, also known as Euler-Rodrigues parameters, provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description. Quaternions are generally quicker for most calculations, conceptually simpler to interpolate, and are not subject to gimbal lock. Much high speed 3D graphics programming (gaming, for example) uses quaternions because of this.

## References

• L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, MA , 1981.
• Andrew Gray, A Treatise on Gyrostatics and Rotational Motion, MacMillan, London, 1918.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, W. H. Freeman, San Francisco, CA, 1973.
• M. E. Rose, Elementary Theory of Angular Momentum, John Wiley, New York, NY, 1957.

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