The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body.^[1] To describe such an orientation in 3-dimensional Euclidean space three parameters are required. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others.

Euler angles also represent three composed rotations that move a reference frame to a given referred frame. This is equivalent to saying that any orientation can be achieved by composing three elemental rotations (rotations around a single axis), and also equivalent to saying that any rotation matrix can be decomposed as a product of three elemental rotation matrices.

Without considering the possibilities of different signs for the angles or moving the reference frame, there are twelve different conventions divided in two groups. One of them is called "proper" Euler angles and the other Tait–Bryan angles. Sometimes "Euler angles" is used for all of them.

Proper Euler angles[link]

Euler angles – The xyz (fixed) system is shown in blue, the XYZ (rotated) system is shown in red. The line of nodes, labeled N, is shown in green.

Sometimes Tait-Bryan angles are called Euler angles. When this is done, it is necessary to specify when classic Euler angles are referred, using an adjective like "proper" or similar.

Classic definition[link]

Euler angles are a means of representing the spatial orientation of any frame (coordinate system) as a composition of rotations from a frame of reference (coordinate system). In the following, the fixed system is denoted in lower case (x,y,z) and the rotated system is denoted in upper case letters (X,Y,Z).

The geometrical definition (referred sometimes as static) is based on a reference frame and one whose orientation we want to describe, first we define the line of nodes (N) as the intersection of the xy and the XY coordinate planes (in other words, line of nodes is the line perpendicular to both z and Z axis). Then we define its Euler angles as:

• α (or Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

) is the angle between the x-axis and the line of nodes.

• β (or Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

) is the angle between the z-axis and the Z-axis.

• γ (or Failed to parse (Missing texvc executable; please see math/README to configure.): \psi

) is the angle between the line of nodes and the X-axis.

Euler angles between two frames are defined only if both frames have the same handedness. Euler angles are just one of the several ways of specifying the relative orientation of two such coordinate systems. Different authors may use different sets of angles to describe these orientations, or different names for the same angles, leading to different conventions. Therefore any discussion employing Euler angles should always be preceded by their definition.^[2]

Unless otherwise stated, this article will use the convention described in the adjacent drawing, usually named Z-X’-Z’’.

Angle signs and ranges[link]

Normally, angles are defined in such a way that they are positive when they rotate counter-clockwise (the rotation depending on which side of the rotation plane they are observed from, with the positive side being the one containing the positive axis of rotation).

About the ranges:

• α and γ range are defined modulo 2π radians. A valid range could be (−π, π].
• β range covers π radians (but can't be said to be modulo π). For example could be [0, π] or [−π/2, π/2].

The angles α, β and γ are uniquely determined except for the singular case that the xy and the XY planes are identical, the z axis and the Z axis having 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 = π and only (α − γ) is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications.

Angles of a given frame: Geometric derivation.[link]

Projections of Z vector.

Projections of Y vector.

The fastest way to get the Euler Angles of a given frame 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, which is more geometrical. Assuming a frame with unitary vectors (X, Y, Z) as in the main diagram, it can be seen that

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \cos (\beta) = Z_3

And, since

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin^2 x = 1 - \cos^2 x

we have

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \sin (\beta) = \sqrt {1 - Z_3^2}

As Failed to parse (Missing texvc executable; please see math/README to configure.): Z_2

is the double projection of a unitary vector,

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \cos (\alpha) \cdot \sin (\beta) = -Z_2

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \cos (\alpha) = -Z_2 / \sqrt {1 - Z_3^2}

There is a similar construction for Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 90-\beta

and Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(90-\beta) = \sin (\beta)

, this leads to:

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \sin (\beta) \cdot \cos (\gamma) = Y_3

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \cos (\gamma) = Y_3 / \sqrt {1 - Z_3^2}

and finally, using the inverse cosine function,

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \alpha = \arccos (-Z_2 / \sqrt {1 - Z_3^2})

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \beta = \arccos (Z_3)

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \gamma = \arccos (Y_3 / \sqrt {1 - Z_3^2})

It is interesting to note that the inverse cosine function yields 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.^[3]

For computational purposes, it may be useful to represent the angles using atan2(y,x):

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \alpha = \operatorname{atan2} (Z_1 , -Z_2)

Failed to parse (Missing texvc executable; please see math/README to configure.): \qquad \gamma = \operatorname{atan2} (X_3 , Y_3)

Conventions for proper Euler angles[link]

The geometric definition for Euler angles is based in the line of nodes, which must be the intersection of two homologous planes (planes overlapping when the angles are zero). There are six possibilities of choosing the proper Euler angles. Using the static definition they correspond to the three possible homogeneous combinations of planes (XY, XZ and YZ) with the two possible options to measure the angles from (given the line of nodes by the XY planes for example, it can be taken X-N or Y-N as first angle). Hence the six possibilities.

Though is possible to define the angles by geometry, normally the intrinsic rotations equivalence or the extrinsic rotations equivalence are used instead. According with these equivalences, proper Euler angles are equivalent to three combined rotations repeating exactly one axis. The six possible conventions are named after their equivalent rotation composition.

When the intrinsic rotations equivalence is used to name the possible conventions of Euler Angles the names are like Z-X’-Z’’. This means that they are equivalent to three concatenated intrinsic rotations around some moving axes Z, X’ and Z’’ in that order. This composition is non-commutative. It has to be applied in such a way that in the beginning one of the intrinsic axis moves together with the line of nodes. The above diagram convention is usually named this way.

When the extrinsic rotations equivalence is used the given angles are inverted respect the former, meaning that the first angle is the intrinsic rotation and the last one the precession.

If a loose naming convention, like Z-X-Z is used, the name of both conventions would be indistinguishable, even if the angles' order is the opposite. To specify that the given order means intrinsic composition, sometimes a similar notation is used, but stating explicitly which rotation axis are different for each step, as in Z-X’-Z’’. Using this notation, Z-X-Z would mean extrinsic composition. It is also possible to use alternative names like x-y-x or X-Y’-X’’ to clarify.

No geometrical information is lost when using the rotation equivalence because the static parameters can be calculated from the name of the convention. For example, given the convention X-Y’-Z’’, the first rotation is perpendicular to "x" and the third one to "Z". Therefore the planes are the yz and the XY, and the line of nodes is the intersection of these two.

Tait–Bryan angles[link]

Tait–Bryan angles. ZYX (intrinsic composition) convention

The second type of conventions is called Tait–Bryan angles, after Peter Guthrie Tait and George H. Bryan, also known as Nautical or Cardan angles, after Cardan.

Their static difference is the definition for the line of nodes. In the first case two homologous planes were used. In the second one, they are replaced by non-homologous planes (perpendicular when angles are zero).

While proper Euler angles are equivalent to three combined rotations repeating exactly one axis, Tait–Bryan angles are equivalent to three composed rotations in different axes.

Tait–Bryan angles statically defined. Z–X’–Y’’ convention

There are also six Tait–Bryan combinations. They come from the two possible non-homogeneous planes that exist when one is given (given XY, there are two non-homogeneous, XZ and YZ). The three possible planes at the reference frame multiplied by the two options for each one yield the six possible conventions.

There are six possible combinations of this kind, and all of them behave in an identical way. Using the intrinsic rotations equivalence, Tait–Bryan angles correspond with the three rotations with a different axis. Z–X–Y for example. There are also six possibilities of this kind. The enclosed image shows the ZXY convention. The other five proper conventions are obtained by selecting different axes of rotation.

These three angles are normally called "Heading, Elevation and Bank", or "Yaw, Pitch and Roll". The second terms have to be used carefully because they are also the names for the three aircraft principal axes.

Double naming problem[link]

For either Euler or Tait-Bryan angles, it is very simple to convert from an intrinsic (rotating axes) to an extrinsic (static axes) convention, and vice-versa: just swap the order of the operations. An (α, β, γ) rotation using X-Y-Z intrinsic convention is equivalent to a (γ, β, α) rotation using Z-Y-X extrinsic convention; this is true for all Euler or Tait-Bryan axis combinations.

For Tait-Bryan angles, also intrinsic and extrinsic conventions can be used, giving therefore two meanings for every convention name. For example, X-Y-Z, using intrinsic convention, means that a X-rotation is performed, composing intrinsic rotations Y and Z later, but using extrinsic convention means that after the X rotation, extrinsic rotations Y and Z are performed. The meaning is different in both cases. Alternative naming schemas like z-x-y versus Z-X’-Y’’ could be used to prevent confusion.

Relationship with physical motions[link]

See also Givens rotations

Euler angles can be considered as the result of three composed rotations and these conventions are named according to this composition. Since intrinsic rotations produce the same result as extrinsic rotations inverted, there are two possible names for any static convention. For example the intrinsic ZXY and the extrinsic yxz have the same static parameters.

Euler angles as composition of intrinsic rotations[link]

Any target frame can be reached using a specific sequence of intrinsic rotations (mobile frame rotations), whose values are exactly the Euler Angles of the target frame. This example uses the Z-X'-Z" convention.

Starting with an initial set of mobile axes, say XYZ overlapping the reference axes xyz, a composition of three intrinsic rotations (rotations only about the mobile frame axes, assuming active composition) can be used to reach any target frame with an origin coincident with that of XYZ from the reference frame. The value of the rotations are the Euler Angles.

The position of the mobile axes can be reached using three rotations with angles α, β, γ in three ways equivalent to the former definition, as follows:

The XYZ system rotates while the xyz is fixed. Starting with the XYZ system overlapping the reference frame xyz, the same rotations as before can be performed using only rotations around the mobile axes XYZ.

• Rotate the XYZ-system about the Z-axis by α. The X-axis now lies on the line of nodes.
• Rotate the XYZ-system again about the now rotated X-axis by β. The Z-axis is now in its final orientation, and the x-axis remains on the line of nodes.
• Rotate the XYZ-system a third time about the new Z-axis by γ.

Any convention for proper Euler angles is equivalent to three such rotations, with one axis being repeated (ZXZ for example). Tait-Bryan angles are also equivalent to three composed rotations, but in this case, all three rotations are around different axes (ZXY).

Usually conventions are named according with this equivalence.

Euler angles as composition of extrinsic rotations[link]

A rotation represented by Euler angles with (φ,θ,ψ)=(−60°, 30°, 45°) using the 3-1-3 (Z-X-Z) co-moving axes rotations

The same rotation alternatively expressed by (φ,θ,ψ)=(45°, 30°, −60°) using the 3-1-3 (Z-X-Z) fixed axes rotations

Also composition of extrinsic rotations (rotations about the reference frame axes, assuming active composition) can be used to reach any target frame. Let xyz system be fixed while the XYZ system rotates. Start with the rotating XYZ system coinciding with the fixed xyz system.

• Rotate the XYZ-system about the z-axis by γ. The X-axis is now at angle γ with respect to the x-axis.
• Rotate the XYZ-system again about the x-axis by β. The Z-axis is now at angle β with respect to the z-axis.
• Rotate the XYZ-system a third time about the z-axis by α. The first and third axes are identical.

This can be shown to be equivalent to the previous statement:

Let us call (e), (f), (g), (h), the successive frames deduced from the initial (e) reference frame by the successive intrinsic rotations described above. We call u, v, w, t, the successive vectors obtained with that rotation. We write Failed to parse (Missing texvc executable; please see math/README to configure.): (x)_e

for the column matrix representing a vector x in the frame (e). If necessary we add also a lower index to any matrix we wish to operate in a specific frame. We call Failed to parse (Missing texvc executable; please see math/README to configure.): (Z_\alpha)

, Failed to parse (Missing texvc executable; please see math/README to configure.): (X_\beta) , Failed to parse (Missing texvc executable; please see math/README to configure.): (Z_\gamma)

the successive rotations of our example. Thus we can write when describing the intrinsic operations :

Failed to parse (Missing texvc executable; please see math/README to configure.): (1) \qquad (Z_\alpha)_e = (Z_\alpha) \qquad (X_\beta)_f = (X_\beta) \qquad (Z_\gamma)_g = (Z_\gamma)

When describing the intrinsic rotations in the (e) reference frame we must of course transform the matrices used to represent the rotations. Then by the rules of matrix algebra we get :

Failed to parse (Missing texvc executable; please see math/README to configure.): (2) \qquad (t)_e = (Z''_\gamma)_e (X'_\beta)_e (Z_\alpha)_e (u)_e

Failed to parse (Missing texvc executable; please see math/README to configure.): (3) \qquad (X'_\beta)_e = (Z_\alpha)_e (X_\beta)_f (Z_\alpha)_e^t = (Z_\alpha) (X_\beta) (Z_\alpha)^t

Failed to parse (Missing texvc executable; please see math/README to configure.): (4) \qquad (Z''_\gamma)_e = (Z_\alpha)_e (X_\beta)_f (Z_\gamma)_g (X_\beta)_f^t (Z_\alpha)_e^t = (Z_\alpha) (X_\beta) (Z_\gamma) (X_\beta)^t (Z_\alpha)^t

Failed to parse (Missing texvc executable; please see math/README to configure.): (5) \qquad (t)_e = [(Z_\alpha)(X_\beta)(Z_\gamma)(X_\beta)^t (Z_\alpha)^t] [(Z_\alpha)(X_\beta)(Z_\alpha)^t] (Z_\alpha) (u)_e = (Z_\alpha) (X_\beta) (Z_\gamma) (u)_e

Failed to parse (Missing texvc executable; please see math/README to configure.): (6) \qquad (R)_e=(Z''_\gamma)_e (X'_\beta)_e (Z_\alpha)_e = (Z_\alpha) (X_\beta) (Z_\gamma)

The relation (5) can then of course be interpreted in extrinsic manner as a succession of rotations around the (e) axes.

Again, proper Euler angles repeat an axis and Tait–Bryan angles do not. As before, this kind of composition is non-commutative.

Euler rotations[link]

Euler rotations of the Earth. Intrinsic (green), Precession (blue) and Nutation (red)

Euler rotations are defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never 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 and the third one is an intrinsic rotation around an axis fixed in the body that moves.

These rotations are called Precession, Nutation, and intrinsic rotation. While they are rotations when they are applied over individual frames, only precession is valid as a rotation operator, and only precession can be expressed in general as a matrix in the basis of the space.

Gimbal analogy[link]

Left: A three axes z-x-z-gimbal where the external frame and external axis 'x' are not shown and axes 'Y' are perpendicular to each gimbal ring. Right: A simple diagram showing the Euler angles and where the axes 'Y' of intermediate frames are located.

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.

Intermediate frames[link]

The gimbal rings indicate some intermediate frames. They can be defined statically too. Taking some vectors i, j and k over the axes x, y and z, and vectors I, J, K over X, Y and Z, and a vector N over the line of nodes, some intermediate frames can be defined using the vector cross product, as following:

• origin: [i,j,k] (where k = i × j)
• first: [N,k × N,k]
• second: [N,K × N,K]
• final: [I,J,K]

These intermediate frames are equivalent to those of the gimbal. They are such that they differ from the previous one in just a single elemental rotation. This proves that:

• Any target frame can be reached from the reference frame just composing three rotations.
• The values of these three rotations are exactly the Euler angles of the target frame.

Relationship to other representations[link]

Euler angles are one way to represent orientations. There are others, and it is possible to change to and from other conventions.

Matrix orientation[link]

Using the equivalence between Euler angles and rotation composition, it is possible to change to and from matrix convention. First, active rotations and the right handed rule for the positive sign of the angles are assumed. As explained above each global rotation matrix can be interpreted as a composition of intrinsic rotations, that is around the axes of the moving body (the moving reference system), or as a composition of extrinsic rotations around the fixed world axes. We choose to name the different conventions by privileging the latter, for a simple and obvious reason : in a matrix composition like

Failed to parse (Missing texvc executable; please see math/README to configure.): (1) \qquad R= A B C

the composing matrices are elemental, and thus pick out automatically successively different fixed world axes in the order C,B,A. We, of course, do not use the complicated intrinsic expression C"B'A to calculate Failed to parse (Missing texvc executable; please see math/README to configure.): R .

To suppress all existing ambiguities and facilitate the understanding between different professional users we introduce a specific denomination principle more detailed than the existing ones. We introduce following rules :

1/ When naming a global rotation we employ the same names in the same order than the matrix formula used to calculate the global matrix. If Failed to parse (Missing texvc executable; please see math/README to configure.): R=Z_\alpha X_\beta Z_\gamma

we name it Failed to parse (Missing texvc executable; please see math/README to configure.): Z_\alpha X_\beta Z_\gamma

. The consequence of that is that the name illustrates automatically the extrinsic interpretation.

2/ For each matrix we designate explicitly the axis, which we index with the angles (we abbreviate Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_1, ...

with Failed to parse (Missing texvc executable; please see math/README to configure.): 1,2,3)

. That may seem redundant but in fact is not, because there is no valid reason to associate an alphabetic or numeric order to the angles, when the order of rotations is already fixed by rule 1 .

3/ We recall something that should be obvious, that is the composing and composed matrices are supposed to act on the coordinates of vectors defined in the initial fixed world axis and give as a result the coordinates in the same system of a rotated vector, rotation primarily interpreted extrinsicly, but to which we can give globally an equivalent intrinsic interpretation.

Here a table for the right-handed convention (used for example in dynamics) is presented. Right-hand convention refers to the Right hand rule. There exists nothing called "left hand rule", but this expression is used sometimes to refer to the opposite sign convention. Several online tools to perform symbolic rotation composition are available and these kind of tables are not necessary. They are only an example.

Failed to parse (Missing texvc executable; please see math/README to configure.): X_1 Z_2 X_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_2 & - c_3 s_2 & s_2 s_3 \\ c_1 s_2 & c_1 c_2 c_3 - s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 \\ s_1 s_2 & c_1 s_3 + c_2 c_3 s_1 & c_1 c_3 - c_2 s_1 s_3 \end{bmatrix}	Failed to parse (Missing texvc executable; please see math/README to configure.): X_1 Z_2 Y_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_2 c_3 & - s_2 & c_2 s_3 \\ s_1 s_3 + c_1 c_3 s_2 & c_1 c_2 & c_1 s_2 s_3 - c_3 s_1 \\ c_3 s_1 s_2 - c_1 s_3 & c_2 s_1 & c_1 c_3 + s_1 s_2 s_3 \end{bmatrix}
Failed to parse (Missing texvc executable; please see math/README to configure.): X_1 Y_2 X_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_2 & s_2 s_3 & c_3 s_2 \\ s_1 s_2 & c_1 c_3 - c_2 s_1 s_3 & - c_1 s_3 - c_2 c_3 s_1 \\ - c_1 s_2 & c_3 s_1 + c_1 c_2 s_3 & c_1 c_2 c_3 - s_1 s_3 \end{bmatrix}	Failed to parse (Missing texvc executable; please see math/README to configure.): X_1 Y_2 Z_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_2 c_3 & - c_2 s_3 & s_2 \\ c_1 s_3 + c_3 s_1 s_2 & c_1 c_3 - s_1 s_2 s_3 & - c_2 s_1 \\ s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 + c_1 s_2 s_3 & c_1 c_2 \end{bmatrix}
Failed to parse (Missing texvc executable; please see math/README to configure.): Y_1 X_2 Y_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_3 - c_2 s_1 s_3 & s_1 s_2 & c_1 s_3 + c_2 c_3 s_1 \\ s_2 s_3 & c_2 & - c_3 s_2 \\ - c_3 s_1 - c_1 c_2 s_3 & c_1 s_2 & c_1 c_2 c_3 - s_1 s_3 \end{bmatrix}	Failed to parse (Missing texvc executable; please see math/README to configure.): Y_1 X_2 Z_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_3 + s_1 s_2 s_3 & c_3 s_1 s_2 - c_1 s_3 & c_2 s_1 \\ c_2 s_3 & c_2 c_3 & - s_2 \\ c_1 s_2 s_3 - c_3 s_1 & s_1 s_3 + c_1 c_3 s_2 & c_1 c_2 \end{bmatrix}
Failed to parse (Missing texvc executable; please see math/README to configure.): Y_1 Z_2 Y_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_2 c_3 - s_1 s_3 & - c_1 s_2 & c_3 s_1 + c_1 c_2 s_3 \\ c_3 s_2 & c_2 & s_2 s_3 \\ - c_1 s_3 - c_2 c_3 s_1 & s_1 s_2 & c_1 c_3 - c_2 s_1 s_3 \end{bmatrix}	Failed to parse (Missing texvc executable; please see math/README to configure.): Y_1 Z_2 X_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_2 & s_1 s_3 - c_1 c_3 s_2 & c_3 s_1 + c_1 s_2 s_3 \\ s_2 & c_2 c_3 & - c_2 s_3 \\ - c_2 s_1 & c_1 s_3 + c_3 s_1 s_2 & c_1 c_3 - s_1 s_2 s_3 \end{bmatrix}
Failed to parse (Missing texvc executable; please see math/README to configure.): Z_1 Y_2 Z_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_2 c_3 - s_1 s_3 & - c_3 s_1 - c_1 c_2 s_3 & c_1 s_2 \\ c_1 s_3 + c_2 c_3 s_1 & c_1 c_3 - c_2 s_1 s_3 & s_1 s_2 \\ - c_3 s_2 & s_2 s_3 & c_2 \end{bmatrix}	Failed to parse (Missing texvc executable; please see math/README to configure.): Z_1 Y_2 X_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_2 & c_1 s_2 s_3 - c_3 s_1 & s_1 s_3 + c_1 c_3 s_2 \\ c_2 s_1 & c_1 c_3 + s_1 s_2 s_3 & c_3 s_1 s_2 - c_1 s_3 \\ - s_2 & c_2 s_3 & c_2 c_3 \end{bmatrix}
Failed to parse (Missing texvc executable; please see math/README to configure.): Z_1 X_2 Z_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_3 - c_2 s_1 s_3 & - c_1 s_3 - c_2 c_3 s_1 & s_1 s_2 \\ c_3 s_1 + c_1 c_2 s_3 & c_1 c_2 c_3 - s_1 s_3 & - c_1 s_2 \\ s_2 s_3 & c_3 s_2 & c_2 \end{bmatrix}	Failed to parse (Missing texvc executable; please see math/README to configure.): Z_1 X_2 Y_3	Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{bmatrix} c_1 c_3 - s_1 s_2 s_3 & - c_2 s_1 & c_1 s_3 + c_3 s_1 s_2 \\ c_3 s_1 + c_1 s_2 s_3 & c_1 c_2 & s_1 s_3 - c_1 c_3 s_2 \\ - c_2 s_3 & s_2 & c_2 c_3 \end{bmatrix}

For left-hand conventions (positive sign convention against the Right hand rule) it is enough to change the signs of the sinus funcions. When we substitute passive rotations to the active ones we have only to transpose the matrix table. Then each matrix transforms the initial coordinates of a vector remaining fixed to the coordinates of the same vector measured in the globally rotated reference system (same rotation axis, same angles).

When we want to establish a matrix table for left-handed sign convention, we need only change the sign of all the sinusoids in the tables.

Quaternions[link]

Unit quaternions, also known as Euler–Rodrigues parameters, 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.

Geometric algebra[link]

Other representation comes from the Geometric algebra(GA). GA is a higher level abstraction, in which the quaternions are an even subalgebra. The principal tool in GA is the rotor Failed to parse (Missing texvc executable; please see math/README to configure.): \bold\R=[\cos(\theta/2)-I u \sin(\theta/2)]

where Failed to parse (Missing texvc executable; please see math/README to configure.): \bold\theta =

angle of rotation, Failed to parse (Missing texvc executable; please see math/README to configure.): \bold(u) = rotation axis (unitary vector) and Failed to parse (Missing texvc executable; please see math/README to configure.): \bold(I)= pseudoscalar (trivector in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^3)

Properties[link]

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.

The space of rotations is called in general "The Hypersphere of rotations", though this is a misnomer: the group Spin(3) is isometric to the hypersphere S³, but the rotation space SO(3) is instead isometric to the real projective space 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), 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.

The Haar measure for Euler angles has the simple form sin(β).dα.dβ.dγ, 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).

Higher dimensions[link]

It is possible to define parameters analogous to the Euler angles in dimensions higher than three^[4].

The number of degrees of freedom of a rotation matrix is always less than the dimension of the matrix squared. That is, the elements of a rotation matrix are not all completely independent. For example, the rotation matrix in dimension 2 has only one degree of freedom, since all four of its elements depend on a single angle of rotation. A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion.

In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The 4x4 rotation matrices have therefore 6 out of 16 independent components.

Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4.

In general, the number of euler angles in dimension D is quadratic in D; since any one rotation consists of choosing two dimensions to rotate between, the total number of rotations available in dimension D is Failed to parse (Missing texvc executable; please see math/README to configure.): N_{rot}=\binom{D}{2}=D(D-1)/2 , which for D=2,3,4 yields Failed to parse (Missing texvc executable; please see math/README to configure.): N_{rot}=1,3,6 .

Applications[link]

A gyroscope keeps its rotation axis constant. Therefore, angles measured in this frame are equivalent to angles measured in the lab frame

Vehicles and moving frames[link]

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 spacecrafts, 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 problem of Mechanical Engineering ^[5] .

Heading, elevation and bank for an aircraft with axes DIN 9300

The most popular application is to describe aircraft attitudes, normally using a Tait–Bryan convention so that zero degrees elevation represents the horizontal attitude. Tait–Bryan angles represent the orientation of the aircraft respect a reference axis system (world frame) with three angles which in the context of an aircraft are normally called Heading, Elevation and Bank. When dealing with vehicles, different axes conventions are possible.

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, angular acceleration, angular velocity, angular momentum, and kinetic energy 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.

Industrial robot operating in a foundry.

Others[link]

Euler angles, normally in the Tait–Bryan convention, are also used in robotics for speaking about the degrees of freedom of a wrist. They are also used in Electronic stability control 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.

In materials science, crystallographic texture (or preferred orientation) can be described using Euler angles. In texture analysis, the Euler angles provide the necessary mathematical depiction of the orientation of individual crystallites within a polycrystalline material, allowing for the quantitative description of the macroscopic material. ^[6] 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. ^[7]

Many mobile computing devices contain accelerometers 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 simulations, and kaleidoscopes.^{[citation needed]}

See also[link]

• 3D projection
• Axis-angle representation
• Conversion between quaternions and Euler angles
• Euler's rotation theorem
• Quaternion
• Quaternions and spatial rotation
• Rotation formalisms in three dimensions
• Spherical coordinate system

References[link]

Bibliography[link]

• Biedenharn, L. C.; Louck, J. D. (1981), Angular Momentum in Quantum Physics, Reading, MA: Addison–Wesley, ISBN 978-0-201-13507-7 
• Goldstein, Herbert (1980), Classical Mechanics (2nd ed.), Reading, MA: Addison–Wesley, ISBN 978-0-201-02918-5 
• Gray, Andrew (1918), A Treatise on Gyrostatics and Rotational Motion, London: Macmillan (published 2007), ISBN 978-1-4212-5592-7 
• Rose, M. E. (1957), Elementary Theory of Angular Momentum, New York, NY: John Wiley & Sons (published 1995), ISBN 978-0-486-68480-2 
• Symon, Keith (1971), Mechanics, Reading, MA: Addison-Wesley, ISBN 978-0-201-07392-8 
• Landau, L.D.; Lifshitz, E. M. (1996), Mechanics (3rd ed.), Oxford: Butterworth-Heinemann, ISBN 978-0-7506-2896-9 

External links[link]

Wikimedia Commons has media related to: Euler angles

• Weisstein, Eric W., "Euler Angles" from MathWorld.
• Java applet for the simulation of Euler angles available at http://www.parallemic.org/Java/EulerAngles.html.
• http://sourceforge.net/projects/orilib – A collection of routines for rotation / orientation manipulation, including special tools for crystal orientations.
• Online tool to compose rotation matrices available at http://www.aeroengineering.info/eulermatrix.html

Look up yaw in Wiktionary, the free dictionary.

The word yaw can refer to:

• Yaw angle, one of the Tait-Bryan angles, describing the heading of a vehicle or machine, and some other related elements:
  • Yaw system, component responsible for the orientation of a wind turbine towards the wind.
  • Yaw bearing, one of the components to keep optimal heading in a wind turbine.
• Yaw, rotation around the vertical axis in aircraft, an aspect of flight dynamics and some other related elements:
  • Yaw (rotation)
  • an Aerobatic maneuver, rotation around the vertical axis
  • Yaw rate sensor, angular speed of yaw rotation
• Yaw (god), Levantine god
• Yaw drive, wind turbine component
• YAW, Halifax/Shearwater Heliport's IATA code
• Ellen Beach Yaw (1869–1947), concert singer
• Yàw, Thursday, see Akan day names
• Yaw-Yan, Filipino martial art
• Yaws, tropical disease
• Yaws (web server)
• Yaw (ethnic group), a Burmese ethnic group
• YAW (Yoonu Askaan Wi), a Senegalese political party
• Yaw (Shaft angle measurement) The yaw is the substitute term for the angle of the shaft of the Penis. Phrase was coined on South Park.

See also[link]

• Yours (disambiguation), various meanings, most prominently as a pronoun

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

∠, the angle symbol

In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.^[1] Angles are usually presumed to be in a Euclidean plane, but are also defined in non-Euclidean geometry.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc by its radius. In the case of an angle (figure), the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any point and its image by the rotation.

The word angle comes from the Latin word angulus, meaning "a corner". The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Greek ἀγκύλος (ankylοs), meaning "crooked, curved," and the English word "ankle". Both are connected with the Proto-Indo-European root *ank-, meaning "to bend" or "bow".^[2]

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.^[3]

Measuring angles[link]

The size of an angle is characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are called congruent angles.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing the cumulative rotation of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

The measure of angle θ is the quotient of s and r.

In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of the arc s is then divided by the radius of the arc r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = k \frac{s}{r}.

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.

Units[link]

Units used to represent angles are listed below in descending magnitude order. Of these units, the degree and the radian are by far the most commonly used. Angles expressed in radians are dimensionless for the purposes of dimensional analysis.

Most units of angular measurement are defined such that one turn (i.e. one full circle) is equal to n units, for some whole number n. The two exceptions are the radian and the diameter part. For example, in the case of degrees, n = 360. A turn of n units is obtained by setting k = n/(2π) in the formula above. (Proof. The formula above can be rewritten as k = θr/s. One turn, for which θ = n units, corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting n for θ and 2πr for s in the formula, results in k = nr/(2πr) = n/(2π).)

• The turn (or full circle, revolution, rotation, or cycle) is one full circle. A turn can be subdivided in centiturns and milliturns. A turn is abbreviated Failed to parse (Missing texvc executable; please see math/README to configure.): \tau

or rev or rot depending on the application, but just r in rpm (revolutions per minute). 1 turn = 360° = 2π rad  = 400 grad = 4 right angles.

• The quadrant is 1/4 of a turn, i.e. a right angle. It is the unit used in Euclid's Elements. 1 quad. = 90° = π/2 rad = 1/4 turn = 100 grad. In German the symbol ^∟ has been used to denote a quadrant.
• The angle of the equilateral triangle is 1/6 of a turn. It was the unit used by the Babylonians^{[citation needed]}, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.

θ = s/r rad = 1 rad.

• The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One turn is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. When radians are used angles are considered as dimensionless. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system.
• The diameter part (occasionally used in Islamic mathematics) is 1/60 radian. One "diameter part" is approximately 0.95493°.
• The astronomical hour angle is 1/24 of a turn. Since this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. Note that these are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = π/12 rad = 1/6 quad. = 1/24 turn ≈ 16.667 grad.
• The point, used in navigation, is 1/32 of a turn. 1 point = 1/8 of a right angle = 11.25° = 12.5 grad. Each point is subdivided in four quarter-points so that 1 turn equals 128 quarter-points.
• Eratosthenes used a unit of 6° so that a whole turn was divided in 60 units.
• The Babylonians sometimes used the unit pechus of about 2° or 2½°.
• The binary degree, also known as the binary radian (or brad), is 1/256 of a turn.^[4] The binary degree is used in computing so that an angle can be efficiently represented in a single byte (albeit to limited precision). Other measures of angle used in computing may be based on dividing one whole turn into 2ⁿ equal parts for other values of n.^[5]
• The degree, denoted by a small superscript circle (°), is 1/360 of a turn, so one turn is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g. 3.5° for three and a half degrees), but the "minute" and "second" sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics:
• The grad, also called grade, gradian, or gon, is 1/400 of a turn, so a right angle is 100 grads. It is a decimal subunit of the quadrant. A kilometre was historically defined as a centi-grad of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The grad is used mostly in triangulation.
• The mil is approximately equal to a milliradian. There are several definitions ranging from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes), with the milliradian being approximately 0.05729578 degrees (3.43775 minutes). In NATO countries, it is defined as 1/6400th of a circle. Its value is approximately equal to the angle subtended by a width of 1 metre as seen from 1 km away (2π / 6400 = 0.0009817… ≒ 1/1000).
• The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree = 1/21600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
• The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

Positive and negative angles[link]

Although the definition of the measurement of an angle does not support the concept of a negative angle, it is frequently useful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions relative to some reference.

In a two dimensional Cartesian coordinate system, angles are typically defined relative to the positive x-axis with positive angles representing rotations toward the positive y-axis and negative angles representing rotations toward the negative y-axis. When Cartesian coordinates are represented as they commonly are, with the x-axis rightward and the y-axis upward, positive rotations are counterclockwise and negative rotations are clockwise.

In many contexts, an angle of −θ is effectively equivalent to an angle of "one full turn minus θ". For example, an orientation represented as  − 45° is effectively equivalent to an orientation represented as 360° − 45° or 315°. However, a rotation of  − 45° would not be the same as a rotation of 315°.

In three dimensional geometry, "clockwise" and "counterclockwise" have no absolute meaning, so the direction of positive and negative angles must be defined relative to some reference, which is typically a vector passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

In navigation, bearings are measured relative to north. By convention, viewed from above, bearing angle are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

Alternative ways of measuring the size of an angle[link]

There are several alternatives to measuring the size of an angle by the corresponding angle of rotation. The grade of a slope, or gradient is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For very small values (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

In rational geometry the spread between two lines is defined at the square of sine of the angle between the lines. Since the sine of an angle and the sine of its supplementary angle are the same any angle of rotation that maps one of the lines into the other leads to the same value of the spread between the lines.

Astronomical approximations[link]

Astronomers measure angular separation of objects in degrees from their point of observation.

• 1° is approximately the width of a little finger at arm's length.
• 10° is approximately the width of a closed fist at arm's length.
• 20° is approximately the width of a handspan at arm's length.

These measurements clearly depend on the individual subject, and the above should be treated as rough approximations only.

Identifying angles[link]

In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, ...) to serve as variables standing for the size of some angle. (To avoid confusion with its other meaning, the symbol π is typically not used for this purpose.) Lower case roman letters (a, b, c, ...) are also used. See the figures in this article for examples.

In geometric figures, angles may also be identified by the labels attached to the three points that define them. For example, the angle at vertex A enclosed by the rays AB and AC (i.e. the lines from point A to point B and point A to point C) is denoted ∠BAC or BÂC. Sometimes, where there is no risk of confusion, the angle may be referred to simply by its vertex ("angle A").

Potentially, an angle denoted, say, ∠BAC might refer to any of four angles: the clockwise angle from B to C, the anticlockwise angle from B to C, the clockwise angle from C to B, or the anticlockwise angle from C to B, where the direction in which the angle is measured determines its sign (see Positive and negative angles). However, in many geometrical situations it is obvious from context that the positive angle less than or equal to 180 degrees is meant, and no ambiguity arises. Otherwise, a convention may be adopted so that ∠BAC always refers to the anticlockwise (positive) angle from B to C, and ∠CAB to the anticlockwise (positive) angle from C to B.

Types of angles[link]

Right angle.

The complementary angles a and b (b is the complement of a, and a is the complement of b).

Acute (a), obtuse (b), and straight (c) angles. Here, a and b are supplementary angles.

Reflex angle.

• An angle equal to 1/4 turn (90° or π/2 radians) is called a right angle.

  Two lines that form a right angle are said to be perpendicular or orthogonal.

• Angles equal to 1/2 turn (180° or two right angles) are called straight angles.
• Angles equal to 1 turn (360° or four right angles) are called full angles.
• Angles that are not right angles or a multiple of a right angle are called oblique angles.
• Angles smaller than a right angle (less than 90°) are called acute angles ("acute" meaning "sharp").
• Angles larger than a right angle and smaller than a straight angle (between 90° and 180°) are called obtuse angles ("obtuse" meaning "blunt").
• Angles larger than a straight angle but less than 1 turn (between 180° and 360°) are called reflex angles.
• Angles that have the same measure (i.e. the same magnitude) are said to be equal (UK) or congruent (USA). An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g. all right angles are congruent).
• Two angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called vertical angles or opposite angles or vertically opposite angles. These angles are equal in measure.
• Angles that share a common vertex and edge but do not share any interior points are called adjacent angles.
• Two angles that sum to one right angle (90°) are called complementary angles.

  The difference between an angle and a right angle is termed the complement of the angle.

• Two angles that sum to a straight angle (180°) are called supplementary angles.

  The difference between an angle and a straight angle (180°) is termed the supplement of the angle.

• Two angles that sum to one turn (360°) are called explementary angles or conjugate angles.
• An angle that is part of a simple polygon is called an interior angle if it lies on the inside of that simple polygon. A concave simple polygon has at least one interior angle that exceeds 180°.

  In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, or 180°, or 1/2 turn; the measures of the interior angles of a simple quadrilateral add up to 2π radians, or 360°, or 1 turn. In general, the measures of the interior angles of a simple polygon with n sides add up to [(n − 2) × π] radians, or [(n − 2) × 180]°, or (2n − 4) right angles, or (n/2 − 1) turn.

• The angle supplementary to the interior angle is called the exterior angle. It measures the amount of rotation one has to make at this vertex to trace out the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.

  In Euclidean geometry, the sum of the exterior angles of a simple polygon will be one full turn (360°).

• Some authors use the name exterior angle of a simple polygon to simply mean the explementary (not supplementary!) of the interior angle.^[6] This conflicts with the above usage.
• The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. It may be defined as the acute angle between two lines normal to the planes.
• The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.
• Alternate angles, corresponding angle, interior angles and exterior angles are associated with a transversal of a pair of lines by a third.
• A reference angle is the acute version of any angle determined by repeatedly subtracting or adding 180 degrees, and subtracting the result from 180 degrees if necessary, until a value between 0 degrees and 90 degrees is obtained. For example, an angle of 30 degrees has a reference angle of 30 degrees, and an angle of 150 degrees also has a reference angle of 30 degrees (180-150). An angle of 750 degrees has a reference angle of 30 degrees (750-720).^[7]

Angles between curves[link]

The angle between the two curves at P is defined as the angle between the tangents A and B at P

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτός, convex) or cissoidal (Gr. κισσός, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίς, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.^[8]

Dot product and generalisation[link]

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u} \cdot \mathbf{v} = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.

This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their normal vectors and between skew lines from their vector equations.

Inner product[link]

To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\cdot,\cdot\rangle , i.e.

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\mathbf{u},\mathbf{v}\rangle = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{Re}(\langle\mathbf{u},\mathbf{v}\rangle) = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.

or, more commonly, using the absolute value, with

Failed to parse (Missing texvc executable; please see math/README to configure.): |\langle\mathbf{u},\mathbf{v}\rangle| = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|.

The latter definition ignores the direction of the vectors and thus describes the angle between one-dimensional subspaces Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{span}(\mathbf{u})

and  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{span}(\mathbf{v})
spanned by the vectors Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u}
and  Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{v}
correspondingly.

Angles between subspaces[link]

The definition of the angle between one-dimensional subspaces Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{span}(\mathbf{u})

and  Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{span}(\mathbf{v})
given by

Failed to parse (Missing texvc executable; please see math/README to configure.): |\langle\mathbf{u},\mathbf{v}\rangle| = \cos(\theta)\ \|\mathbf{u}\|\ \|\mathbf{v}\|

in a Hilbert space can be extended to subspaces of any finite dimensions. Given two subspaces Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{U},\mathcal{W}

with Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{dim}(\mathcal{U}):=k\leq \operatorname{dim}(\mathcal{W}):=l

, this leads to a definition of Failed to parse (Missing texvc executable; please see math/README to configure.): k

angles called canonical or principal angles between subspaces.

Angles in Riemannian geometry[link]

In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_ij are the components of the metric tensor G,

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}.

Angles in geography and astronomy[link]

In geography, the location of any point on the Earth can be identified using a geographic coordinate system. This system specifies the latitude and longitude of any location in terms of angles subtended at the centre of the Earth, using the equator and (usually) the Greenwich meridian as references.

In astronomy, a given point on the celestial sphere (that is, the apparent position of an astronomical object) can be identified using any of several astronomical coordinate systems, where the references vary according to the particular system. Astronomers measure the angular separation of two stars by imagining two lines through the centre of the Earth, each intersecting one of the stars. The angle between those lines can be measured, and is the angular separation between the two stars.

Astronomers also measure the apparent size of objects as an angular diameter. For example, the full moon has an angular diameter of approximately 0.5°, when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

See also[link]

• Angle bisector
• Argument (complex analysis)
• Astrological aspect
• Central angle
• Clock angle problem
• Complementary angles
• Great circle distance
• Inscribed angle
• Protractor
• Solid angle for a concept of angle in three dimensions.
• Supplementary angles
• Irrational angle
• Angular velocity

Notes[link]

References[link]

• Heiberg, Johan Ludvig (1908). Heath, T. L.. ed. Euclid. The thirteen books of Euclid's Elements. 1. http://books.google.com/books?id=UhgPAAAAIAAJ. 

Attribution

•  This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Angle". Encyclopædia Britannica (11th ed.). Cambridge University Press. 

External links[link]

Wikimedia Commons has media related to: Angles

• Angle Bisectors in a Quadrilateral at cut-the-knot
• Constructing a triangle from its angle bisectors at cut-the-knot
• Angle Estimation – for basic astronomy.
• Angle definition pages with interactive applets.
• Various angle constructions with compass and straightedge
• GonioLab DD – Convert between DecDeg and DegMinSec and vice-versa (requires Java Web Start)