In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.
The main use for planes of rotation is in describing more complex rotations in four-dimensional space and higher dimensions, where they can be used to break down the rotations into simpler parts. This can be done using geometric algebra, with the planes of rotations associated with simple bivectors in the algebra.
Planes of rotation are not used much in two and three dimensions, as in two dimensions there is only one plane (so, identifying the plane of rotation is trivial and rarely done), while in three dimensions the axis of rotation serves the same purpose and is the more established approach.
Mathematically such planes can be described in a number of ways. They can be described in terms of planes and angles of rotation. They can be associated with bivectors from geometric algebra. They are related to the eigenvalues and eigenvectors of a rotation matrix. And in particular dimensions they are related to other algebraic and geometric properties, which can then be generalised to other dimensions.
Definitions
Plane
For this article, all planes are planes through the origin, that is they contain the zero vector. Such a plane in -dimensional space is a two-dimensional linear subspace of the space. It is completely specified by any two non-zero and non-parallel vectors that lie in the plane, that is by any two vectors and , 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 is the bivector associated with the plane specified by and , and has magnitude , where is the angle between the vectors; hence the requirement that the vectors be nonzero and nonparallel.
If the bivector is written , then the condition that a point lies on the plane associated with is simply
This is true in all dimensions, and can be taken as the definition on the plane. In particular, from the properties of the exterior product it is satisfied by both and , and so by any vector of the form
with and real numbers. As and range over all real numbers, ranges over the whole plane, so this can be taken as another definition of the plane.
Plane of rotation
A plane of rotation for a particular rotation is a plane that is mapped to itself by the rotation. The plane is not fixed, but all vectors in the plane are mapped to other vectors in the same plane by the rotation. This transformation of the plane to itself is always a rotation about the origin, through an angle which is the angle of rotation for the plane.
Every rotation except for the identity rotation (with matrix the identity matrix) has at least one plane of rotation, and up to
planes of rotation, where is the dimension. The maximum number of planes up to eight dimensions is shown in this table:
{ | class="wikitable" border="1" ! Dimension | 2 || 3 || 4 || 5 || 6 || 7 || 8 |- ! Number of planes | 1 || 1 || 2 || 2 || 3 || 3 || 4 |}
When a rotation has multiple planes of rotation they are always orthogonal to each other, with only the origin in common. This is a stronger condition than to say the planes are at right angles; it instead means that the planes have no nonzero vectors in common, and that every vector in one plane is orthogonal to every vector in the other plane. This can only happen in four or more dimensions. In two dimensions there is only one plane, while in three dimensions all planes have at least one nonzero vector in common, along their line of intersection.
In more than three dimensions planes of rotation are not always unique. 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 , so any pair of orthogonal planes generates the rotation. But for a general rotation it is at least theoretically possible to identify a unique set of orthogonal planes, in each of which points are rotated through an angle, so the set of planes and angles fully characterise the rotation.
Two dimensions
In two-dimensional space there is only one plane of rotation, the plane of the space itself. In a Cartesian coordinate system it is the Cartesian plane, in complex numbers it is the complex plane. 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 − to . So if the angle is the rotation in the complex plane is given by Euler's formula: