This article derives the main properties of rotations in 3-dimensional space.
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.
be a coordinate system fixed in the body that through a change in orientation is brought to the new directions
rotating with the body is then brought to the new direction
i.e. this is a linear operator
The matrix of this operator relative to the coordinate system
or equivalently in matrix notation
the matrix is orthogonal and as a "right hand" base vector system is re-orientated into another "right hand" system the determinant of this matrix has the value 1.
Rotation around an axis
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 of this matrix is
and the characteristic polynomial 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 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 rotation matrix by angle around a general axis of rotation
is given by Rodrigues' rotation formula.
where is the identity matrix and is the dual 2-form of or
cross product matrix,
Note that satisfies for all .
The general case
"Rotation with the angle 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, 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
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
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
is non-zero. This vector is an eigenvector with eigenvalue
and selecting any two orthogonal unit vectors in the plane orthogonal to :
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
or . But the difference is that for
is zero and of no use for finding the eigenspace of eigenvalue 1, i.e. the
Defining as the matrix for the
rotation operator is
i.e. except for the cases (the identity operator) and
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.
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
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 orientation)
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