旋转平面

In three-dimensional space（英语：three-dimensional space） there are an infinite number of planes of rotation, only one of which is involved in any given rotation. That is, for a general rotation there is precisely one plane which is associated with it or which the rotation takes place in. The only exception is the trivial rotation, corresponding to the identity matrix, in which no rotation takes place.

In any rotation in three dimensions there is always a fixed axis, the axis of rotation. The rotation can be described by giving this axis, with the angle through which the rotation turns about it; this is the axis angle（英语：axis angle） representation of a rotation. The plane of rotation is the plane orthogonal to this axis, so the axis is a surface normal（英语：surface normal） of the plane. The rotation then rotates this plane through the same angle as it rotates around the axis, that is everything in the plane rotates by the same angle about the origin.

One example is shown in the diagram, where the rotation takes place about the z-axis. The plane of rotation is the xy-plane, so everything in that plane it kept in the plane by the rotation. This could be described by a matrix like the following, with the rotation being through an angle θ (about the axis or in the plane):

${\displaystyle {\begin{pmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{pmatrix}}.}$ 

Another example is the 地球自转. The axis of rotation is the line joining the North Pole and 南极 and the plane of rotation is the plane through the equator（英语：equator） between the Northern（英语：Northern Hemisphere） and Southern（英语：Southern Hemisphere） Hemispheres. Other examples include mechanical devices like a gyroscope（英语：gyroscope） or flywheel（英语：flywheel） which store rotational energy（英语：rotational energy） in mass usually along the plane of rotation.

In any three dimensional rotation the plane of rotation is uniquely defined. Together with the angle of rotation it fully describes the rotation. Or in a continuously rotating object the rotational properties such as the rate of rotation can be described in terms of the plane of rotation. It is perpendicular to, and so is defined by and defines, an axis of rotation, so any description of a rotation in terms of a plane of rotation can be described in terms of an axis of rotation, and vice versa. But unlike the axis of rotation the plane generalises into other, in particular higher, dimensions.^[7]

四维

四维空间中的一般旋转只有一个固定点，即原点。因此，旋转轴不能用于四个维度。但是可以使用旋转平面，并且在四个维度中的每个非平凡旋转都有一个或两个旋转平面。 A general rotation in four-dimensional space（英语：four-dimensional space） has only one fixed point, the origin. Therefore an axis of rotation cannot be used in four dimensions. But planes of rotation can be used, and each non-trivial rotation in four dimensions has one or two planes of rotation.

简单旋转

A rotation with only one plane of rotation is a simple rotation（英语：SO(4)#Simple rotations）. In a simple rotation there is a fixed plane, and rotation can be said to take place about this plane, so points as they rotate do not change their distance from this plane. The plane of rotation is orthogonal to this plane, and the rotation can be said to take place in this plane.

For example the following matrix fixes the xy-plane: points in that plane and only in that plane are unchanged. The plane of rotation is the zw-plane, points in this plane are rotated through an angle θ. A general point rotates only in the zw-plane, that is it rotates around the xy-plane by changing only its z and w coordinates.

${\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&\cos \theta &-\sin \theta \\0&0&\sin \theta &\cos \theta \end{pmatrix}}}$ 

In two and three dimensions all rotations are simple, in that they have only one plane of rotation. Only in four and more dimensions are there rotations that are not simple rotations. In particular in four dimensions there are also double and isoclinic rotations.

双重旋转

In a double rotation（英语：SO(4)#Double rotations） there are two planes of rotation, no fixed planes, and the only fixed point is the origin. The rotation can be said to take place in both planes of rotation, as points in them are rotated within the planes. These planes are orthogonal, that is they have no vectors in common so every vector in one plane is at right angles to every vector in the other plane. The two rotation planes span four-dimensional space, so every point in the space can be specified by two points, one on each of the planes.

A double rotation has two angles of rotation, one for each plane of rotation. The rotation is specified by giving the two planes and two non-zero angles, α and β (if either angle is zero the rotation is simple). Points in the first plane rotate through α, while points in the second plane rotate through β. All other points rotate through an angle between α and β, so in a sense they together determine the amount of rotation. For a general double rotation the planes of rotation and angles are unique, and given a general rotation they can be calculated. For example a rotation of α in the xy-plane and β in the zw-plane is given by the matrix

${\displaystyle {\begin{pmatrix}\cos \alpha &-\sin \alpha &0&0\\\sin \alpha &\cos \alpha &0&0\\0&0&\cos \beta &-\sin \beta \\0&0&\sin \beta &\cos \beta \end{pmatrix}}.}$ 

等斜旋转

A special case of the double rotation is when the angles are equal, that is if α = β ≠ 0. This is called an isoclinic rotation（英语：SO(4)#Isoclinic rotations）, and it differs from a general double rotation in a number of ways. For example in an isoclinic rotation, all non-zero points rotate through the same angle, α. Most importantly the planes of rotation are not uniquely identified. There are instead an infinite number of pairs of orthogonal planes that can be treated as planes of rotation. For example any point can be taken, and the plane it rotates in together with the plane orthogonal to it can be used as two planes of rotation.^[8]

高维

As already noted the maximum number of planes of rotation in n dimensions is

${\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor ,}$ 

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.

Simple rotations can be identified in all dimensions, as rotations with just one plane of rotation. A simple rotation in n dimensions takes place about (that is at a fixed distance from) an (n − 2)-dimensional subspace orthogonal to the plane of rotation.

A general rotation is not simple, and has the maximum number of planes of rotation as given above. In the general case the angles of rotations in these planes are distinct and the planes are uniquely defined. If any of the angles are the same then the planes are not unique, as in four dimensions with an isoclinic rotation.

In even dimensions (n = 2, 4, 6...) there are up to n/2 planes of rotation span the space, so a general rotation rotates all points except the origin which is the only fixed point. In odd dimensions (n = 3, 5, 7, ...) there are n − 1/2 planes and angles of rotation, the same as the even dimension one lower. These do not span the space, but leave a line which does not rotate – like the axis of rotation（英语：Rotation around a fixed axis） in three dimensions, except rotations do not take place about this line but in multiple planes orthogonal to it.^[1]

数学性质

The examples given above were chosen to be clear and simple examples of rotations, with planes generally parallel to the coordinate axes in three and four dimensions. But this is not generally the case: planes are not usually parallel to the axes, and the matrices cannot simply be written down. In all dimensions the rotations are fully described by the planes of rotation and their associated angles, so it is useful to be able to determine them, or at least find ways to describe them mathematically.

反射

Every simple rotation can be generated by two reflections. Reflections can be specified in n dimensions by giving an (n − 1)-dimensional subspace to reflect in, so a two-dimensional reflection is in a line, a three-dimensional reflection is in a plane, and so on. But this becomes increasingly difficult to apply in higher dimensions, so it is better to use vectors instead, as follows.

A reflection in n dimensions is specified by a vector perpendicular to the (n − 1)-dimensional subspace. 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. Similarly unit vector（英语：unit vector）s can be used to simplify the calculations.

So the reflection in a (n − 1)-dimensional space is given by the unit vector perpendicular to it, m, thus:

${\displaystyle \mathbf {x} '=-\mathbf {mxm} \,}$ 

where the product is the geometric product from geometric algebra（英语：geometric algebra）.

If x′ is reflected in another, distinct, (n − 1)-dimensional space, described by a unit vector n perpendicular to it, the result is

${\displaystyle \mathbf {x} ''=-\mathbf {nx} '\mathbf {n} =-\mathbf {n} (-\mathbf {mxm} )\mathbf {n} =\mathbf {nmxmn} }$ 

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.

The quantity mn is a rotor（英语：rotor (mathematics)）, and nm is its inverse as

${\displaystyle (\mathbf {mn} )(\mathbf {nm} )=\mathbf {mnnm} =\mathbf {mm} =1}$ 

So the rotation can be written

${\displaystyle \mathbf {x} ''=R\mathbf {x} R^{-1}}$ 

where R = mn is the rotor.

The plane of rotation is the plane containing m and n, which must be distinct otherwise the reflections are the same and no rotation takes place. As either vector can be replaced by its negative the angle between them can always be acute, or at most π/2. The rotation is through twice the angle between the vectors, up to π or a half-turn. The sense of the rotation is to rotate from m towards n: the geometric product is not commutative（英语：commutative） so the product nm is the inverse rotation, with sense from n to m.

Conversely all simple rotations can be generated this way, with two reflections, by two unit vectors in the plane of rotation separated by half the desired angle of rotation. 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.^[9][10]

二重向量

二重向量s are quantities from geometric algebra（英语：geometric algebra）, clifford algebra（英语：clifford algebra） and the exterior algebra（英语：exterior algebra）, which generalise the idea of vectors into two dimensions. As vectors are to lines, so are bivectors to planes. So every plane (in any dimension) can be associated with a bivector, and every simple bivector（英语：Bivector#Simple bivectors） is associated with a plane. This makes them a good fit for describing planes of rotation.

Every rotation plane in a rotation has a simple bivector associated with it. This is parallel to the plane and has magnitude equal to the angle of rotation in the plane. These bivectors are summed to produce a single, generally non-simple, bivector for the whole rotation. This can generate a rotor（英语：Rotor (mathematics)） through the exponential map（英语：exponential function）, which can be used to rotate an object.

Bivectors are related to rotors through the exponential map (which applied to bivectors generates rotors and rotations using 棣莫弗公式). In particular given any bivector B the rotor associated with it is

${\displaystyle R_{\mathbf {B} }=e^{\frac {\mathbf {B} }{2}}.}$ 

This is a simple rotation if the bivector is simple, a more general rotation otherwise. When squared,

${\displaystyle {R_{\mathbf {B} }}^{2}=e^{\frac {\mathbf {B} }{2}}e^{\frac {\mathbf {B} }{2}}=e^{\mathbf {B} },}$ 

it gives a rotor that rotates through twice the angle. If B is simple then this is the same rotation as is generated by two reflections, as the product mn gives a rotation through twice the angle between the vectors. These can be equated,

${\displaystyle \mathbf {mn} =e^{\mathbf {B} },}$ 

from which it follows that the bivector associated with the plane of rotation containing m and n that rotates m to n is

${\displaystyle \mathbf {B} =\log(\mathbf {mn} ).}$ 

This is a simple bivector, associated with the simple rotation described. More general rotations in four or more dimensions are associated with sums of simple bivectors, one for each plane of rotation, calculated as above.

Examples include the two rotations in four dimensions given above. The simple rotation in the zw-plane by an angle θ has bivector e₃₄θ, a simple bivector. The double rotation by α and β in the xy-plane and zw-planes has bivector e₁₂α + e₃₄β, the sum of two simple bivectors e₁₂α and e₃₄β which are parallel to the two planes of rotation and have magnitudes equal to the angles of rotation.

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. But given the simple bivectors geometric algebra is a useful tool for studying planes of rotation using algebra like the above.^[1][11]

特征值和特征平面

The planes of rotations for a particular rotation using the eigenvalue（英语：eigenvalue）s. Given a general rotation matrix in n dimensions its characteristic equation（英语：secular equation） has either one (in odd dimensions) or zero (in even dimensions) real roots. The other roots are in complex conjugate pairs, exactly

${\displaystyle \left\lfloor {\frac {n}{2}}\right\rfloor ,}$ 

such pairs. These correspond to the planes of rotation, the eigenplane（英语：eigenplane）s of the matrix, which can be calculated using algebraic techniques. In addition argument（英语：argument (complex analysis)）s 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.^[1][12]

另见

• 吉文斯旋转
• Quaternions（英语：Quaternions）
• Rotation group SO(3)（英语：Rotation group SO(3)）
• Rotations in 4-dimensional Euclidean space（英语：Rotations in 4-dimensional Euclidean space）