This article derives the main properties of rotations in 3-dimensional space.
Contents
- 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.
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
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
is
As
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
Let
be an orthogonal positively oriented base vector system in
The linear operator
"Rotation with the angle
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
The case
For the case
i.e. the rotation operator has the eigenvalues
The eigenspace corresponding to
The eigenspace corresponding to
For all other values of
The rotation matrix by angle
where
Note that
The general case
The operator
"Rotation with the angle
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
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
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
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 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
is zero and of no use for finding the eigenspace of eigenvalue 1, i.e. the rotation axis.
Defining
provided that
i.e. except for the cases
Quaternions
Quaternions are defined similar to
This means that the first 3 components
and that the fourth component is the scalar
As the angle
one would normally have that
and
are two alternative representations of one and the same rotation.
The entities
Using quaternions the matrix of the rotation operator is
Numerical example
Consider the reorientation corresponding to the Euler angles
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
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