Plane of rotation

It is completely specified by any two non-zero and non-parallel vectors that lie in the plane, that is by any two vectors a and b, such that where ∧ is the exterior product from exterior algebra or geometric algebra (in three dimensions the cross product can be used).

More precisely, the quantity a ∧ b is the bivector associated with the plane specified by a and b, and has magnitude |a| |b| sin φ, where φ is the angle between the vectors; hence the requirement that the vectors be nonzero and nonparallel.

In particular, from the properties of the exterior product it is satisfied by both a and b, and so by any vector of the form with λ and μ real numbers.

For example the negative of the identity matrix in four dimensions (the central inversion), describes a rotation in four dimensions in which every plane through the origin is a plane of rotation through an angle π, so any pair of orthogonal planes generates the rotation.

Any rotation therefore is of the whole plane, i.e. of the space, keeping only the origin fixed.

It is specified completely by the signed angle of rotation, in the range for example −π to π.

One example is shown in the diagram, where the rotation takes place about the z-axis.

The axis of rotation is the line joining the North Pole and South Pole and the plane of rotation is the plane through the equator between the Northern and Southern Hemispheres.

But unlike the axis of rotation the plane generalises into other, in particular higher, dimensions.

[7] A general rotation in four-dimensional space has only one fixed point, the origin.

For example a rotation of α in the xy-plane and β in the zw-plane is given by the matrix A special case of the double rotation is when the angles are equal, that is if α = β ≠ 0.

[8] As already noted the maximum number of planes of rotation in n dimensions is so the complexity quickly increases with more than four dimensions and categorising rotations as above becomes too complex to be practical, but some observations can be made.

If any of the angles are the same then the planes are not unique, as in four dimensions with an isoclinic rotation.

But this is not generally the case: planes are not usually parallel to the axes, and the matrices cannot simply be written down.

But this becomes increasingly difficult to apply in higher dimensions, so it is better to use vectors instead, as follows.

To generate simple rotations only reflections that fix the origin are needed, so the vector does not have a position, just direction.

It does also not matter which way it is facing: it can be replaced with its negative without changing the result.

If x′ is reflected in another, distinct, (n − 1)-dimensional space, described by a unit vector n perpendicular to it, the result is This is a simple rotation in n dimensions, through twice the angle between the subspaces, which is also the angle between the vectors m and n. It can be checked using geometric algebra that this is a rotation, and that it rotates all vectors as expected.

These can be composed to produce more general rotations, using up to n reflections if the dimension n is even, n − 2 if n is odd, by choosing pairs of reflections given by two vectors in each plane of rotation.

This makes them a good fit for describing planes of rotation.

This can generate a rotor through the exponential map, which can be used to rotate an object.

Given a rotor the bivector associated with it can be recovered by taking the logarithm of the rotor, which can then be split into simple bivectors to determine the planes of rotation, although in practice for all but the simplest of cases this may be impractical.

These correspond to the planes of rotation, the eigenplanes of the matrix, which can be calculated using algebraic techniques.

In addition arguments of the complex roots are the magnitudes of the bivectors associated with the planes of rotations.

The form of the characteristic equation is related to the planes, making it possible to relate its algebraic properties like repeated roots to the bivectors, where repeated bivector magnitudes have particular geometric interpretations.