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The Euler angles are three angles introduced by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
to describe the
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
of a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
with respect to a fixed
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478
PDF
/ref> They can also represent the orientation of a mobile
frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
in physics or the orientation of a general
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
in 3-dimensional
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
. Alternative forms were later introduced by
Peter Guthrie Tait Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote wi ...
and
George H. Bryan George Hartley Bryan FRS (1 March 1864 – 13 October 1928) was an English applied mathematician who was an authority on thermodynamics and aeronautics. He was born in Cambridge, and was educated at Peterhouse, Cambridge, obtaining his BA in 188 ...
intended for use in aeronautics and engineering.


Chained rotations equivalence

Euler angles can be defined by elemental
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
or by composition of rotations. The geometrical definition demonstrates that three composed ''
elemental rotation In physics and engineering, Davenport chained rotations are three chained intrinsic rotations about body-fixed specific axes. Euler rotations and Tait–Bryan rotations are particular cases of the Davenport general rotation decomposition. The angle ...
s'' (rotations about the axes of a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
) are always sufficient to reach any target frame. The three elemental rotations may be
extrinsic In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass ...
(rotations about the axes ''xyz'' of the original coordinate system, which is assumed to remain motionless), or
intrinsic In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass ...
(rotations about the axes of the rotating coordinate system ''XYZ'', solidary with the moving body, which changes its orientation with respect to the extrinsic frame after each elemental rotation). Euler angles are typically denoted as ''α'', ''β'', ''γ'', or ''ψ'', ''θ'', ''φ''. Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. Therefore, any discussion employing Euler angles should always be preceded by their definition. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: * Proper Euler angles * Tait–Bryan angles . Tait–Bryan angles are also called Cardan angles; nautical angles;
heading Heading can refer to: * Heading (metalworking), a process which incorporates the extruding and upsetting processes * Headline, text at the top of a newspaper article * Heading (navigation), the direction a person or vehicle is facing, usually s ...
, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called "Euler angles". In that case, the sequences of the first group are called ''proper'' or ''classic'' Euler angles.


Proper Euler angles


Geometrical definition

The axes of the original frame are denoted as ''x'', ''y'', ''z'' and the axes of the rotated frame as ''X'', ''Y'', ''Z''. The geometrical definition (sometimes referred to as static) begins by defining the
line of nodes An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes. Planes of reference Common planes of refere ...
(N) as the intersection of the planes ''xy'' and ''XY'' (it can also be defined as the common perpendicular to the axes ''z'' and ''Z'' and then written as the vector product ''N'' = ''z'' \times ''Z''). Using it, the three Euler angles can be defined as follows: * \alpha (or \varphi) is the signed angle between the ''x'' axis and the ''N'' axis (''x''-convention – it could also be defined between ''y'' and ''N'', called ''y''-convention). * \beta (or \theta) is the angle between the ''z'' axis and the ''Z'' axis. * \gamma (or \psi) is the signed angle between the ''N'' axis and the ''X'' axis (''x''-convention). Euler angles between two reference frames are defined only if both frames have the same
handedness In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more Fine motor skill, dextrous. The other hand, comparatively often the weaker, less dextrous or sim ...
.


Conventions by intrinsic rotations

Intrinsic rotations are elemental rotations that occur about the axes of a coordinate system ''XYZ'' attached to a moving body. Therefore, they change their orientation after each elemental rotation. The ''XYZ'' system rotates, while ''xyz'' is fixed. Starting with ''XYZ'' overlapping ''xyz'', a composition of three intrinsic rotations can be used to reach any target orientation for ''XYZ''. Euler angles can be defined by intrinsic rotations. The rotated frame ''XYZ'' may be imagined to be initially aligned with ''xyz'', before undergoing the three elemental rotations represented by Euler angles. Its successive orientations may be denoted as follows: * ''x''-''y''-''z'', or ''x''0-''y''0-''z''0 (initial) * ''x''′-''y''′-''z''′, or ''x''1-''y''1-''z''1 (after first rotation) * ''x''″-''y''″-''z''″, or ''x''2-''y''2-''z''2 (after second rotation) * ''X''-''Y''-''Z'', or ''x''3-''y''3-''z''3 (final) For the above-listed sequence of rotations, the
line of nodes An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes. Planes of reference Common planes of refere ...
''N'' can be simply defined as the orientation of ''X'' after the first elemental rotation. Hence, ''N'' can be simply denoted ''x''′. Moreover, since the third elemental rotation occurs about ''Z'', it does not change the orientation of ''Z''. Hence ''Z'' coincides with ''z''″. This allows us to simplify the definition of the Euler angles as follows: * ''α'' (or \varphi) represents a rotation around the ''z'' axis, * ''β'' (or \theta) represents a rotation around the ''x''′ axis, * ''γ'' (or \psi) represents a rotation around the ''z''″ axis.


Conventions by extrinsic rotations

Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system ''xyz''. The ''XYZ'' system rotates, while ''xyz'' is fixed. Starting with ''XYZ'' overlapping ''xyz'', a composition of three extrinsic rotations can be used to reach any target orientation for ''XYZ''. The Euler or Tait–Bryan angles (''α'', ''β'', ''γ'') are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows (note the reversed order of Euler angle application): * The ''XYZ'' system rotates about the ''z'' axis by ''γ''. The ''X'' axis is now at angle ''γ'' with respect to the ''x'' axis. * The ''XYZ'' system rotates again, but this time about the ''x'' axis by ''β''. The ''Z'' axis is now at angle ''β'' with respect to the ''z'' axis. * The ''XYZ'' system rotates a third time, about the ''z'' axis again, by angle ''α''. In sum, the three elemental rotations occur about ''z'', ''x'' and ''z''. Indeed, this sequence is often denoted ''z''-''x''-''z'' (or 3-1-3). Sets of rotation axes associated with both proper Euler angles and Tait–Bryan angles are commonly named using this notation (see above for details).


Signs, ranges and conventions

Angles are commonly defined according to the
right-hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of th ...
. Namely, they have positive values when they represent a rotation that appears clockwise when looking in the positive direction of the axis, and negative values when the rotation appears counter-clockwise. The opposite convention (left hand rule) is less frequently adopted. About the ranges (using
interval notation In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
): * for ''α'' and ''γ'', the range is defined modulo 2
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s. For instance, a valid range could be . * for ''β'', the range covers radians (but can't be said to be modulo ). For example, it could be or . The angles ''α'', ''β'' and ''γ'' are uniquely determined except for the singular case that the ''xy'' and the ''XY'' planes are identical, i.e. when the ''z'' axis and the ''Z'' axis have the same or opposite directions. Indeed, if the ''z'' axis and the ''Z'' axis are the same, ''β'' = 0 and only (''α'' + ''γ'') is uniquely defined (not the individual values), and, similarly, if the ''z'' axis and the ''Z'' axis are opposite, ''β'' =  and only (''α'' − ''γ'') is uniquely defined (not the individual values). These ambiguities are known as
gimbal lock Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
in applications. There are six possibilities of choosing the rotation axes for proper Euler angles. In all of them, the first and third rotation axes are the same. The six possible sequences are: # ''z''1-''x''′-''z''2″ (intrinsic rotations) or ''z''2-''x''-''z''1 (extrinsic rotations) # ''x''1-''y''′-''x''2″ (intrinsic rotations) or ''x''2-''y''-''x''1 (extrinsic rotations) # ''y''1-''z''′-''y''2″ (intrinsic rotations) or ''y''2-''z''-''y''1 (extrinsic rotations) # ''z''1-''y''′-''z''2″ (intrinsic rotations) or ''z''2-''y''-''z''1 (extrinsic rotations) # ''x''1-''z''′-''x''2″ (intrinsic rotations) or ''x''2-''z''-''x''1 (extrinsic rotations) # ''y''1-''x''′-''y''2″ (intrinsic rotations) or ''y''2-''x''-''y''1 (extrinsic rotations)


Precession, nutation and intrinsic rotation

Precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
,
nutation Nutation () is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame ...
, and intrinsic rotation (spin) are defined as the movements obtained by changing one of the Euler angles while leaving the other two constant. These motions are not expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis ''z'', the second rotates around the line of nodes ''N'' and the third one is an intrinsic rotation around ''Z'', an axis fixed in the body that moves. The static definition implies that: * ''α'' (precession) represents a rotation around the ''z'' axis, * ''β'' (nutation) represents a rotation around the ''N'' or x′ axis, * ''γ'' (intrinsic rotation) represents a rotation around the ''Z'' or z″ axis. If ''β'' is zero, there is no rotation about ''N''. As a consequence, ''Z'' coincides with ''z'', ''α'' and ''γ'' represent rotations about the same axis (''z''), and the final orientation can be obtained with a single rotation about ''z'', by an angle equal to . As an example, consider a
top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will usually wobble for a few ...
. The top spins around its own axis of symmetry; this corresponds to its intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top can wobble up and down; the inclination angle is the nutation angle. The same example can be seen with the movements of the earth. Though all three movements can be represented by a rotation operator with constant coefficients in some frame, they cannot be represented by these operators all at the same time. Given a reference frame, at most one of them will be coefficient-free. Only precession can be expressed in general as a matrix in the basis of the space without dependencies of the other angles. These movements also behave as a gimbal set. If we suppose a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will exist an external fixed frame, one final frame and two frames in the middle, which are called "intermediate frames". The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.


Tait–Bryan angles

The second type of formalism is called Tait–Bryan angles, after
Peter Guthrie Tait Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote wi ...
and
George H. Bryan George Hartley Bryan FRS (1 March 1864 – 13 October 1928) was an English applied mathematician who was an authority on thermodynamics and aeronautics. He was born in Cambridge, and was educated at Peterhouse, Cambridge, obtaining his BA in 188 ...
. It is the convention normally used for aerospace applications, so that zero degrees elevation represents the horizontal attitude. Tait–Bryan angles represent the orientation of the aircraft with respect to the world frame. When dealing with other vehicles, different
axes conventions In ballistics and flight dynamics, axes conventions are standardized ways of establishing the location and orientation of coordinate axes for use as a frame of reference. Mobile objects are normally tracked from an external frame considered fixed. ...
are possible.


Definitions

The definitions and notations used for Tait–Bryan angles are similar to those described above for proper Euler angles ( geometrical definition, intrinsic rotation definition, extrinsic rotation definition). The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. ''x''-''y''-''z'', or ''x''-''y''′-''z''″), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., ''z''-''x''-''z'', or ''z''-''x''′-''z''″). This implies a different definition for the
line of nodes An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes. Planes of reference Common planes of refere ...
in the geometrical construction. In the proper Euler angles case it was defined as the intersection between two homologous Cartesian planes (parallel when Euler angles are zero; e.g. ''xy'' and ''XY''). In the Tait–Bryan angles case, it is defined as the intersection of two non-homologous planes (perpendicular when Euler angles are zero; e.g. ''xy'' and ''YZ'').


Conventions

The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless ( extrinsic rotations), or about the axes of the rotating coordinate system, which changes its orientation after each elemental rotation (
intrinsic rotations The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
). There are six possibilities of choosing the rotation axes for Tait–Bryan angles. The six possible sequences are: * ''x''-''y''′-''z''″ (intrinsic rotations) or ''z''-''y''-''x'' (extrinsic rotations) * ''y''-''z''′-''x''″ (intrinsic rotations) or ''x''-''z''-''y'' (extrinsic rotations) * ''z''-''x''′-''y''″ (intrinsic rotations) or ''y''-''x''-''z'' (extrinsic rotations) * ''x''-''z''′-''y''″ (intrinsic rotations) or ''y''-''z''-''x'' (extrinsic rotations) * ''z''-''y''′-''x''″ (intrinsic rotations) or ''x''-''y''-''z'' (extrinsic rotations): the intrinsic rotations are known as: yaw, pitch and roll * ''y''-''x''′-''z''″ (intrinsic rotations) or ''z''-''x''-''y'' (extrinsic rotations)


Signs and ranges

Tait–Bryan convention is widely used in engineering with different purposes. There are several
axes conventions In ballistics and flight dynamics, axes conventions are standardized ways of establishing the location and orientation of coordinate axes for use as a frame of reference. Mobile objects are normally tracked from an external frame considered fixed. ...
in practice for choosing the mobile and fixed axes, and these conventions determine the signs of the angles. Therefore, signs must be studied in each case carefully. The range for the angles ''ψ'' and ''φ'' covers 2 radians. For ''θ'' the range covers radians.


Alternative names

These angles are normally taken as one in the external reference frame (
heading Heading can refer to: * Heading (metalworking), a process which incorporates the extruding and upsetting processes * Headline, text at the top of a newspaper article * Heading (navigation), the direction a person or vehicle is facing, usually s ...
, bearing), one in the intrinsic moving frame (
bank A bank is a financial institution that accepts deposits from the public and creates a demand deposit while simultaneously making loans. Lending activities can be directly performed by the bank or indirectly through capital markets. Because ...
) and one in a middle frame, representing an
elevation The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Vert ...
or inclination with respect to the horizontal plane, which is equivalent to the line of nodes for this purpose. For an aircraft, they can be obtained with three rotations around its principal axes if done in the proper order. A yaw will obtain the bearing, a pitch will yield the elevation and a roll gives the bank angle. Therefore, in aerospace they are sometimes called yaw, pitch and roll. Notice that this will not work if the rotations are applied in any other order or if the airplane axes start in any position non-equivalent to the reference frame. Tait–Bryan angles, following ''z''-''y''′-''x''″ (intrinsic rotations) convention, are also known as nautical angles, because they can be used to describe the orientation of a ship or aircraft, or Cardan angles, after the Italian mathematician and physicist
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
, who first described in detail the
Cardan suspension A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...
and the
Cardan joint A universal joint (also called a universal coupling or U-joint) is a joint or coupling connecting rigid shafts whose axes are inclined to each other. It is commonly used in shafts that transmit rotary motion. It consists of a pair of hinges lo ...
.


Angles of a given frame

A common problem is to find the Euler angles of a given frame. The fastest way to get them is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix (see later table of matrices). Hence the three Euler Angles can be calculated. Nevertheless, the same result can be reached avoiding matrix algebra and using only elemental geometry. Here we present the results for the two most commonly used conventions: ''ZXZ'' for proper Euler angles and ''ZYX'' for Tait–Bryan. Notice that any other convention can be obtained just changing the name of the axes.


Proper Euler angles

Assuming a frame with
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
s (''X'', ''Y'', ''Z'') given by their coordinates as in the main diagram, it can be seen that: :\cos (\beta) = Z_3. And, since :\sin^2 x = 1 - \cos^2 x, for 0 we have :\sin (\beta) = \sqrt . As Z_2 is the double projection of a unitary vector, :\cos(\alpha) \cdot \sin(\beta) = -Z_2, :\cos(\alpha) = -Z_2 / \sqrt. There is a similar construction for Y_3, projecting it first over the plane defined by the axis ''z'' and the line of nodes. As the angle between the planes is \pi/2 - \beta and \cos(\pi/2 - \beta) = \sin(\beta), this leads to: :\sin(\beta) \cdot \cos(\gamma) = Y_3, :\cos(\gamma) = Y_3 / \sqrt, and finally, using the inverse cosine function, :\alpha = \arccos\left(-Z_2 / \sqrt\right), :\beta = \arccos\left(Z_3\right), :\gamma = \arccos\left(Y_3 / \sqrt\right).


Tait–Bryan angles

Assuming a frame with
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
s (''X'', ''Y'', ''Z'') given by their coordinates as in this new diagram (notice that the angle theta is negative), it can be seen that: :\sin (\theta) = -X_3 As before, :\cos^2 x = 1 - \sin^2 x, for -\pi/2 we have :\cos (\theta) = \sqrt . in a way analogous to the former one: :\sin(\psi) = X_2 / \sqrt. :\sin(\phi) = Y_3 / \sqrt. Looking for similar expressions to the former ones: :\psi = \arcsin\left(X_2 / \sqrt\right), :\theta = \arcsin(-X_3), :\phi = \arcsin\left(Y_3 / \sqrt\right).


Last remarks

Note that the inverse sine and cosine functions yield two possible values for the argument. In this geometrical description, only one of the solutions is valid. When Euler angles are defined as a sequence of rotations, all the solutions can be valid, but there will be only one inside the angle ranges. This is because the sequence of rotations to reach the target frame is not unique if the ranges are not previously defined. For computational purposes, it may be useful to represent the angles using . For example, in the case of proper Euler angles: :\alpha = \operatorname(Z_1 , -Z_2), :\gamma = \operatorname(X_3 , Y_3).


Conversion to other orientation representations

Euler angles are one way to represent orientations. There are others, and it is possible to change to and from other conventions. Three parameters are always required to describe orientations in a
3-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. They can be given in several ways, Euler angles being one of them; see
charts on SO(3) In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot ...
for others. The most used orientation representation are the rotation matrices, the axis-angle and the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, also known as Euler–Rodrigues parameters, which provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description. Expressing rotations in 3D as unit quaternions instead of matrices has some advantages: * Concatenating rotations is computationally faster and numerically more stable. * Extracting the angle and axis of rotation is simpler. * Interpolation is more straightforward. See for example slerp. * Quaternions do not suffer from
gimbal lock Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
as Euler angles do. Regardless, the rotation matrix calculation is the first step for obtaining the other two representations.


Rotation matrix

Any orientation can be achieved by composing three elemental rotations, starting from a known standard orientation. Equivalently, any
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end ...
''R'' can be decomposed as a product of three elemental rotation matrices. For instance: :R = X(\alpha) Y(\beta) Z(\gamma) is a rotation matrix that may be used to represent a composition of extrinsic rotations about axes ''z'', ''y'', ''x'', (in that order), or a composition of
intrinsic rotations The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
about axes ''x''-''y''′-''z''″ (in that order). However, both the definition of the elemental rotation matrices ''X'', ''Y'', ''Z'', and their multiplication order depend on the choices taken by the user about the definition of both rotation matrices and Euler angles (see, for instance, Ambiguities in the definition of rotation matrices). Unfortunately, different sets of conventions are adopted by users in different contexts. The following table was built according to this set of conventions: # Each matrix is meant to operate by pre-multiplying
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
s \begin x \\ y \\ z \end (see Ambiguities in the definition of rotation matrices) # Each matrix is meant to represent an active rotation (the composing and composed matrices are supposed to act on the coordinates of vectors defined in the initial fixed reference frame and give as a result the coordinates of a rotated vector defined in the same reference frame). # Each matrix is meant to represent, primarily, a composition of
intrinsic rotations The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
(around the axes of the rotating reference frame) and, secondarily, the composition of three extrinsic rotations (which corresponds to the constructive evaluation of the R matrix by the multiplication of three truly elemental matrices, in reverse order). #
Right handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more Fine motor skill, dextrous. The other hand, comparatively often the weaker, less dextrous or sim ...
reference frames are adopted, and the
right hand rule In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors. Most of th ...
is used to determine the sign of the angles ''α'', ''β'', ''γ''. For the sake of simplicity, the following table of matrix products uses the following nomenclature: # 1, 2, 3 represent the angles ''α'', ''β'' and ''γ'', i.e. the angles corresponding to the first, second and third elemental rotations respectively. # ''X'', ''Y'', ''Z'' are the matrices representing the elemental rotations about the axes ''x'', ''y'', ''z'' of the fixed frame (e.g., ''X''1 represents a rotation about ''x'' by an angle ''α''). # ''s'' and ''c'' represent sine and cosine (e.g., ''s''1 represents the sine of ''α''). : These tabular results are available in numerous textbooks. For each column the last row constitutes the most commonly used convention. To change the formulas for passive rotations (or find reverse active rotation), transpose the matrices (then each matrix transforms the initial coordinates of a vector remaining fixed to the coordinates of the same vector measured in the rotated reference system; same rotation axis, same angles, but now the coordinate system rotates, rather than the vector). The following table contains formulas for angles ''α'', ''β'' and ''γ'' from elements of a rotation matrix R.


Properties

The Euler angles form a
chart A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
on all of
SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...
, the
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
of rotations in 3D space. The chart is smooth except for a polar coordinate style singularity along . See
charts on SO(3) In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot ...
for a more complete treatment. The space of rotations is called in general "The
Hypersphere of rotations In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there ca ...
", though this is a misnomer: the group
Spin(3) In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...
is isometric to the hypersphere ''S''3, but the rotation space SO(3) is instead isometric to the
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction A ...
RP3 which is a 2-fold quotient space of the hypersphere. This 2-to-1 ambiguity is the mathematical origin of spin in physics. A similar three angle decomposition applies to
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
, the
special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
of rotations in complex 2D space, with the difference that ''β'' ranges from 0 to 2. These are also called Euler angles. The
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...
for SO(3) in Euler angles is given by the Hopf angle parametrisation of SO(3), \textrmV \propto \sin \beta \cdot \textrm\alpha \cdot \textrm\beta \cdot \textrm\gamma, where (\beta, \alpha) parametrise S^, the space of rotation axes. For example, to generate uniformly randomized orientations, let ''α'' and ''γ'' be uniform from 0 to 2, let ''z'' be uniform from −1 to 1, and let .


Geometric algebra

Other properties of Euler angles and rotations in general can be found from the
geometric algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
, a higher level abstraction, in which the quaternions are an even subalgebra. The principal tool in geometric algebra is the rotor \mathbf = \cos(\theta / 2) - I u \sin(\theta / 2) where \theta =
angle of rotation In mathematics, the angle of rotation is a measurement of the amount, of namely angle, that a figure is rotated about a fixed point, often the center of a circle. A clockwise rotation is considered a negative rotation, so that, for instance ...
, \mathbf is the rotation axis (unitary vector) and \mathbf is the pseudoscalar (trivector in \mathbb^3)


Higher dimensions

It is possible to define parameters analogous to the Euler angles in dimensions higher than three. In four dimensions and above, the concept of "rotation about an axis" loses meaning and instead becomes "rotation in a plane." The number of Euler angles needed to represent the group is , equal to the number of planes containing two distinct coordinate axes in ''n''-dimensional Euclidean space. In
SO(4) In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article ''rotation'' means ''rotational dis ...
a rotation matrix is defined by two unit quaternions, and therefore has six degrees of freedom, three from each quaternion.


Applications


Vehicles and moving frames

Their main advantage over other orientation descriptions is that they are directly measurable from a gimbal mounted in a vehicle. As gyroscopes keep their rotation axis constant, angles measured in a gyro frame are equivalent to angles measured in the lab frame. Therefore, gyros are used to know the actual orientation of moving spacecraft, and Euler angles are directly measurable. Intrinsic rotation angle cannot be read from a single gimbal, so there has to be more than one gimbal in a spacecraft. Normally there are at least three for redundancy. There is also a relation to the well-known
gimbal lock Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
problem of
mechanical engineering Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and ...
 . When studying rigid bodies in general, 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
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
,
angular acceleration In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceler ...
,
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
,
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, and
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
are often easiest in body coordinates, because then the moment of inertia tensor does not change in time. If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components. The angular velocity of a rigid body takes a simple form using Euler angles in the moving frame. Also the Euler's rigid body equations are simpler because the inertia tensor is constant in that frame.


Crystallographic texture

In materials science, crystallographic
texture Texture may refer to: Science and technology * Surface texture, the texture means smoothness, roughness, or bumpiness of the surface of an object * Texture (roads), road surface characteristics with waves shorter than road roughness * Texture (c ...
(or preferred orientation) can be described using Euler angles. In texture analysis, the Euler angles provide a mathematical depiction of the orientation of individual crystallites within a polycrystalline material, allowing for the quantitative description of the macroscopic material. The most common definition of the angles is due to Bunge and corresponds to the ''ZXZ'' convention. It is important to note, however, that the application generally involves axis transformations of tensor quantities, i.e. passive rotations. Thus the matrix that corresponds to the Bunge Euler angles is the transpose of that shown in the table above.


Others

Euler angles, normally in the Tait–Bryan convention, are also used in
robotics Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
for speaking about the degrees of freedom of a
wrist In human anatomy, the wrist is variously defined as (1) the Carpal bones, carpus or carpal bones, the complex of eight bones forming the proximal skeletal segment of the hand; "The wrist contains eight bones, roughly aligned in two rows, known ...
. They are also used in
electronic stability control Electronic stability control (ESC), also referred to as electronic stability program (ESP) or dynamic stability control (DSC), is a computerized technology that improves a vehicle's stability by detecting and reducing loss of traction ( skiddi ...
in a similar way. Gun fire control systems require corrections to gun-order angles (bearing and elevation) to compensate for deck tilt (pitch and roll). In traditional systems, a stabilizing gyroscope with a vertical spin axis corrects for deck tilt, and stabilizes the optical sights and radar antenna. However, gun barrels point in a direction different from the line of sight to the target, to anticipate target movement and fall of the projectile due to gravity, among other factors. Gun mounts roll and pitch with the deck plane, but also require stabilization. Gun orders include angles computed from the vertical gyro data, and those computations involve Euler angles. Euler angles are also used extensively in the quantum mechanics of angular momentum. 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. Many mobile computing devices contain
accelerometer An accelerometer is a tool that measures proper acceleration. Proper acceleration is the acceleration (the rate of change of velocity) of a body in its own instantaneous rest frame; this is different from coordinate acceleration, which is accele ...
s which can determine these devices' Euler angles with respect to the earth's gravitational attraction. These are used in applications such as games,
bubble level A spirit level, bubble level, or simply a level, is an instrument designed to indicate whether a surface is horizontal (level) or vertical (plumb). Different types of spirit levels may be used by carpenters, stonemasons, bricklayers, ot ...
simulations, and
kaleidoscope A kaleidoscope () is an optical instrument with two or more reflecting surfaces (or mirrors) tilted to each other at an angle, so that one or more (parts of) objects on one end of these mirrors are shown as a regular symmetrical pattern when v ...
s.


See also

*
3D projection A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object fo ...
* Axis-angle representation *
Conversion between quaternions and Euler angles Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions. This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler ...
* Davenport chained rotations *
Euler's rotation theorem In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed p ...
*
Gimbal lock Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
*
Quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
*
Quaternions and spatial rotation Unit quaternions, known as ''versors'', provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation abou ...
*
Rotation formalisms in three dimensions In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative ...
*
Spherical coordinate system In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...


References


Bibliography

* * * * * *


External links

* * * David Eberly
Euler Angle Formulas
Geometric Tools * An interactive tutorial on Euler angles available at https://www.mecademic.com/en/how-is-orientation-in-space-represented-with-euler-angles
EulerAngles
an iOS app for visualizing in 3D the three rotations associated with Euler angles
Orientation Library
"orilib", a collection of routines for rotation / orientation manipulation, including special tools for crystal orientations * Online tool to convert rotation matrices available a
rotation converter
(numerical conversion) * Online tool to convert symbolic rotation matrices (dead, but still available from the
Wayback Machine The Wayback Machine is a digital archive of the World Wide Web founded by the Internet Archive, a nonprofit based in San Francisco, California. Created in 1996 and launched to the public in 2001, it allows the user to go "back in time" and see ...

symbolic rotation converter

Rotation, Reflection, and Frame Change: Orthogonal tensors in computational engineering mechanics
IOP Publishing
Euler Angles, Quaternions, and Transformation Matrices for Space Shuttle Analysis
NASA {{DEFAULTSORT:Euler Angles Rotation in three dimensions Euclidean symmetries Angle Analytic geometry