There are two types of rotation transformation that we want to consider:
- Finite rotations, that is a change from one angular orientation to another.
- Continuous and infinitesimal rotations, such as when an object is continuously rotating.
When we first think of these types of rotation we might guess that one of these would be the rate of change of the other and that they would obey similar rules.
However it turns out that continuous and infinitesimal rotations are easily combined using vector addition. However finite rotations are more complicated and require other types of algebra.
Finite rotations
In 2 dimensions rotations can be combined by adding the angles. In 3 dimensions we need to multiply quaternions or matrices.
Can we extend this to more than 3 dimensions? This may not be very practical as our world only has 3 dimensions of space , however I'm hoping that this might give some insights into the nature of rotations.
Here are some approaches, that I can think of, for calculating the rotation of a given point in 3 dimensions:
- Calculate rotation around the axis of rotation.
- Translate the point to the rotation plane, rotate in the plane, then apply the reverse translation from the plane.
- Calculate the rotation as a sequence of two reflections.
In general a rotation occurs in a plane, that is a two dimensional space, which may be embedded in 3D space. It happens that in 3D planes and lines are duals and therefore one can be represented by the other, but this only applies in 3D space, therefore the first of the above methods only applies to 3D.
I will try to investigate whether the second two methods could apply to more than 3 dimensions (if more than 3 dimensions existed).
| Rotations in 'n' dimensions | |
| Rotations as 2 reflections |
Further Reading
This can be used to generate the matrix representation of the axis angle representation of rotation:
