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

The three Euler rotations

Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...

 are one way to bring a rigid object 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

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 point. It also means that the composition of two...

). Using the concepts of linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

 it is shown how this single
rotation can be performed.

Mathematical formulation

Let

be a coordinate system fixed in the body that through a change in orientation is brought to the new directions

Any vector

of the body is then brought to the new direction

i.e. this is a linear operator

The matrix of this operator relative the coordinate system

is

As

or equivalently in matrix notation

the matrix is orthogonal

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

 and as a "right hand" base vector system is re-orientated into another "right hand" system the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 of this matrix has the value 1.

Rotation around an axis

Let


be an orthogonal positively oriented base vector system in .

The linear operator

"Rotation with the angle around the axis defined by "

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

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 of this matrix is


and the characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 is


The matrix is symmetric if and only if , i.e. for
and for .

The case is the trivial case of an identity operator.

For the case the characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 is


i.e. the rotation operator has the eigenvalues

The eigenspace corresponding to is all vectors on the rotation axis, i.e. all vectors


The eigenspace corresponding to consists of all vectors orthogonal to the rotation axis, i.e. all vectors


For all other values of the matrix is un-symmetric and as there is
only the eigenvalue with the one-dimensional eigenspace of the vectors on the rotation axis:

The general case

The operator

"Rotation with the angle around a specified axis"

discussed above is an orthogonal mapping and its matrix relative any base vector system is therefore an
orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

 . Further more its determinant has the value 1.
A non-trivial fact is the opposite, i.e. that for any orthogonal linear mapping in having
determinant = 1 there exist base vectors


such that the matrix takes the "canonical form"


for some value of .

In fact, if a linear operator has the orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

relative some base vector system


and this matrix is symmetric the "Symmetric operator theorem" valid in (any dimension) applies saying

that it has n orthogonal eigenvectors. This means for the 3-dimensional case that there exists a coordinate system

such that the matrix takes the form

As it is an orthogonal matrix these diagonal elements 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 .

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 .

If the matrix is un-symmetric the vector

where


is non-zero. This vector is an eigenvector with eigenvalue


Setting

and selecting any two orthogonal unit vectors in the plane orthogonal to :


such that


form a positively oriented trippel 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
or . But the difference is that for
the vector

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

Defining as the matrix for the
rotation operator is


provided that


i.e. except for the cases (the identity operator) and

Quaternions

Quaternions are defined similar to with
the difference that the half angle is used
instead of the full angle .

This means that the first 3 components are components of a vector defined from

and that the fourth component is the scalar

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

one would normally have that . But a "dual" representation of a rotation with quaternions
is used, i.e.

and

are two alternative representations of one and the same rotation.

The entities are defined from the quaternions by

Using quaternions the matrix of the rotation operator is

Numerical example

Consider the reorientation corresponding to the Euler angles

relative a given base vector system

Corresponding matrix relative to this base vector system is (see Euler angles#Matrix notation)


and the quaternion is

The canonical form of this operator
with is obtained with

The quaternion relative to this new system is then

Instead of making the three Euler rotations


the same orientation can be reached with one single rotation of size around