This article derives the main properties of rotations in 3-dimensional space.

- Mathematical formulation
- Rotation around an axis
- The general case
- Quaternions
- Numerical example
- References

The three Euler rotations are one way to bring a rigid body to any desired orientation by sequentially making rotations about axis' fixed relative to the object. However, this can also be achieved with one single rotation (Euler's rotation theorem). Using the concepts of linear algebra it is shown how this single rotation can be performed.

Let (*ê*_{1}, *ê*_{2}, *ê*_{3}) be a coordinate system fixed in the body that through a change in orientation **A** is brought to the new directions

Any vector

rotating with the body is then brought to the new direction

that is to say, this is a linear operator

The matrix of this operator relative to the coordinate system (*ê*_{1}, *ê*_{2}, *ê*_{3}) is

As

or equivalently in matrix notation

the matrix is orthogonal and as a right-handed base vector system is reorientated into another right-handed system the determinant of this matrix has the value 1.

Let (*ê*_{1}, *ê*_{2}, *ê*_{3}) be an orthogonal positively oriented base vector system in **R**^{3}. The linear operator "rotation by angle θ around the axis defined by *ê*_{3}" has the matrix representation

relative to this basevector system. This then means that a vector

is rotated to the vector

by the linear operator. The determinant of this matrix is

and the characteristic polynomial is

The matrix is symmetric if and only if sin *θ* = 0, that is, for *θ* = 0 and *θ* = *π*. The case *θ* = 0 is the trivial case of an identity operator. For the case *θ* = *π* the characteristic polynomial is

so the rotation operator has the eigenvalues

The eigenspace corresponding to *λ* = 1 is all vectors on the rotation axis, namely all vectors

The eigenspace corresponding to *λ* = −1 consists of all vectors orthogonal to the rotation axis, namely all vectors

For all other values of θ the matrix is not symmetric and as sin^{2}*θ* > 0 there is only the eigenvalue *λ* = 1 with the one-dimensional eigenspace of the vectors on the rotation axis:

The rotation matrix by angle θ around a general axis of rotation **k** is given by Rodrigues' rotation formula.

where **I** is the identity matrix and [**k**]_{×} is the dual 2-form of **k** or cross product matrix,

Note that [**k**]_{×} satisfies [**k**]_{×}**v** = **k** × **v** for all vectors **v**.

The operator "rotation by angle θ around a specified axis" discussed above is an orthogonal mapping and its matrix relative to any base vector system is therefore an orthogonal matrix. Furthermore its determinant has the value 1. A non-trivial fact is the opposite, that for any orthogonal linear mapping in **R**^{3} with determinant 1 there exist base vectors *ê*_{1}, *ê*_{2}, *ê*_{3} such that the matrix takes the "canonical form"

for some value of θ. In fact, if a linear operator has the orthogonal matrix

relative to some base vector system (*f̂*_{1}, *f̂*_{2}, *f̂*_{3}) and this matrix is symmetric, the "symmetric operator theorem" valid in **R**^{n} (any dimension) applies saying that it has n orthogonal eigenvectors. This means for the 3-dimensional case that there exists a coordinate system *ê*_{1}, *ê*_{2}, *ê*_{3} such that the matrix takes the form

As it is an orthogonal matrix these diagonal elements B_{ii} are either 1 or −1. As the determinant is 1 these elements are either all 1 or one of the elements is 1 and the other two are −1. In the first case it is the trivial identity operator corresponding to *θ* = 0. In the second case it has the form

if the basevectors are numbered such that the one with eigenvalue 1 has index 3. This matrix is then of the desired form for *θ* = *π*.

If the matrix is asymmetric, the vector

where

is nonzero. This vector is an eigenvector with eigenvalue *λ* = 1. Setting

and selecting any two orthogonal unit vectors *ê*_{1} and *ê*_{2} in the plane orthogonal to *ê*_{3} such that *ê*_{1}, *ê*_{2}, *ê*_{3} form a positively oriented triple, the operator takes the desired form with

The expressions above are in fact valid also for the case of a symmetric rotation operator corresponding to a rotation with *θ* = 0 or *θ* = *π*. But the difference is that for *θ* = *π* the vector

is zero and of no use for finding the eigenspace of eigenvalue 1, and thence the rotation axis.

Defining *E*_{4} as cos *θ* the matrix for the rotation operator is

provided that

that is, except for the cases *θ* = 0 (the identity operator) and *θ* = *π*.

Quaternions are defined similar to *E*_{1}, *E*_{2}, *E*_{3}, *E*_{4} with the difference that the half angle *θ*/2 is used instead of the full angle θ. This means that the first 3 components *q*_{1}, *q*_{2}, *q*_{3} components of a vector defined from

and that the fourth component is the scalar

As the angle θ defined from the canonical form is in the interval

one would normally have that *q*_{4} ≥ 0. But a "dual" representation of a rotation with quaternions is used, that is to say (*q*_{1}, *q*_{2}, *q*_{3}, *q*_{4})}} and (−*q*_{1}, −*q*_{2}, −'*q*_{3}, −*q*_{4}) are two alternative representations of one and the same rotation.

The entities E_{k} are defined from the quaternions by

Using quaternions the matrix of the rotation operator is

Consider the reorientation corresponding to the Euler angles *α* = 10°, *β* = 20°, *γ* = 30° relative to a given base vector system (*f̂*_{1}, *f̂*_{2}, *f̂*_{3}). The corresponding matrix relative to this base vector system is (see Euler angles#Matrix orientation)

and the quaternion is

The canonical form of this operator

with *θ* = 44.537° is obtained with

The quaternion relative to this new system is then

Instead of making the three Euler rotations 10°, 20°, 30° the same orientation can be reached with one single rotation of size 44.537° around *ê*_{3}.

**Kinematics** is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

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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 under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

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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 point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a *rotation group*.

In the theory of three-dimensional rotation, **Rodrigues' rotation formula**, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In other words, the Rodrigues' formula provides an algorithm to compute the exponential map from **so**(3), the Lie algebra of SO(3), to SO(3) without actually computing the full matrix exponential.

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 some seventy years earlier than Hamilton to solve the problem of magic squares. For this reason the dynamics community commonly refers to quaternions in this application as "Euler parameters".

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In mathematics, the **axis–angle representation** of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector **e** indicating the direction of an axis of rotation, and an angle *θ* describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector **e** rooted at the origin because the magnitude of **e** is constrained. For example, the elevation and azimuth angles of **e** suffice to locate it in any particular Cartesian coordinate frame.

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- Shilov, Georgi (1961),
*An Introduction to the Theory of Linear Spaces*, Prentice-Hall, Library of Congress 61-13845.

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