旋转平面

旋转面、旋转平面（plane of rotation），是一个用于描述空间旋转的抽像概念。

十维以下的旋转平面数量如下表所示：

维数｜零｜一｜二｜三｜四｜五｜六｜七｜八｜九｜十

旋转平面｜0｜0｜1｜1｜2｜2｜3｜3｜4｜4｜5

旋转平面主要用作描述四维空间及以上的旋转，将高维旋转分解为简单的几何代数描述。

数学上，旋转平面可用多种方式描述。可用平面和旋转角度来描述，可用克利福德代数的二重向量来描述。旋转平面又与旋转矩阵的特征值和特征向量有关。

In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions.

旋转平面在二维和三维中使用不多，因为在二维中只有一个平面（因此，识别旋转平面是微不足道的并且很少这样做），而在三维中旋转轴具有相同的目的，并且是更成熟的方法。

Planes of rotation are not used much in and 三维空间s, 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. The main use for them is in describing more complex rotations in 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.

## 定义

### 平面

本文所有平面都通过原点（即包含零向量。）

For this article, all planes are planes through the origin, that is they contain the zero vector. Such a plane in n-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 a and b, such that

$$\mathbf{a} \wedge \mathbf{b} \ne 0,$$

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.

If the bivector a ∧ b is written B, then the condition that a point lies on the plane associated with B is simply

$$\mathbf{x} \wedge \mathbf{B} = 0.$$

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 a and b, and so by any vector of the form

$$\mathbf{c} = \lambda\mathbf{a} + \mu\mathbf{b},$$

with λ and μ real numbers. As λ and μ range over all real numbers, c ranges over the whole plane, so this can be taken as another definition of the plane.

### 旋转平面

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

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

planes of rotation, where n is the dimension.

十维以下的旋转平面数量如下表所示：

维数｜零｜一｜二｜三｜四｜五｜六｜七｜八｜九｜十

旋转平面｜0｜0｜1｜1｜2｜2｜3｜3｜4｜4｜5

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),

$$\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix},$$

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. 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.

## 二维

在二维空间只有一个旋转平面，即空间本身的平面。在笛卡尔坐标系是笛卡尔平面，在复数是复平面。因此，任何旋转都是整个平面空间的旋转，仅原点保持固定。完全由带符号的旋转角度指定，例如在 -π 到 π 的范围内。因此若角度为θ，复平面的旋转则由以下欧拉公式给出：

$$e^{i\theta} = \cos{\theta} + i\sin{\theta},\,$$

笛卡尔平面的旋转则由以下旋转矩阵给出：

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

## 三维

<image>

<image>

三维空间可以有无数个旋转平面，但当旋转平面有了一个，就不能第二个旋转平面。

三维任何旋转都总是只有一个固定的轴，即旋转轴。

这可以用如下矩阵来描述（旋转角度为θ）：$\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}.$

另一例子是地球自转。自转轴是北极至南极的连线，自转平面穿过北半球和南半球之间的赤道平面。

其它例子包括陀螺仪、飞轮等机械装置，通常沿旋转平面储存大量旋转动能。

In 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 representation of a rotation. The plane of rotation is the plane orthogonal to this axis, so the axis is a 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 -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):

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

<image>

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 between the Northern and Southern Hemispheres. Other examples include mechanical devices like a gyroscope or flywheel which store rotational energy in mass usually along the plane of rotation.

在任何三维旋转中，旋转平面都是唯一定义的。它与旋转角度一起完整地描述了旋转。或者，在连续旋转的物体中，旋转特性（例如旋转速率）可以用旋转平面来描述。

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.

## 四维

一般四维旋转只有一个固定点，即原点。因此，四个维度没有旋转轴。但是四维空间可以使用旋转平面，并且在四个维度中的每个非平凡旋转都可以有一至两个旋转平面。

### 简单旋转

仅具有一个旋转平面的旋转是简单旋转。

简单旋转有一个固定的平面，因此点在旋转时不会改变其与该平面的距离。旋转平面与该平面正交，可以说旋转发生在该平面内。

例如，下列矩阵固定 xy 平面：此平面中的点且仅在该平面中的点保持不变。旋转平面是 zw 平面，该平面上的点旋转角度 θ。一般点仅在 zw 平面内旋转，即仅更改其 z 和 w 座标来绕 xy 平面旋转。

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

在二维和三维，所有旋转都是简单旋转，因为只有一个旋转平面。只有在四维以上才存在不是简单旋转的旋转。在四维也存在双重旋转和等斜旋转。

### 双重旋转

双旋转有两个旋转平面，没有固定平面，唯一的固定点是原点。旋转发生在两旋转平面中。这些平面是正交的，也就是说，它们没有共同的向量，因此一个平面中的每个向量都与另一个平面中的每个向量成直角。两个旋转平面跨越四维空间，因此空间中的每个点都可以由两个点指定，每个平面上一个。

双旋转有两个旋转角度，每个旋转平面一个。双重旋转有两个平面和两个非零角度α、β（如果任一角度为零，则是简单旋转）。第一个平面的旋转α点，第二个平面的旋转β点。所有其他点都旋转α和β之间的角度，因此在某种意义上是αβ共同决定了旋转量。对于一般的双旋转，旋转平面和角度是唯一的，并且给定一般的旋转，它们可以被计算。例如，xy 平面中的 α 和 zw 平面中的 β 的旋转由矩阵给出

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

### 等斜旋转

<image>

双旋转的一个特殊情况是角度相等，即 α = β ≠ 0。

例如，在等斜旋转中所有非零点都会旋转相同的角度 α。最重要的是，旋转平面不是唯一标识的。相反，有无数对正交平面可以被视为旋转平面。例如可以任意一点，它旋转所在的平面以及与其正交的平面可以当作两个旋转平面。

## 高维

旋转平面数量公式：$\left\lfloor \frac{n}{2} \right\rfloor$

因此，超过四个维度，复杂性会迅速增加，并且如上所述对旋转进行分类变得过于复杂而不实用，但可以进行一些观察。

简单旋转可以在所有维度上被识别为仅具有一个旋转平面的旋转。n 维的简单旋转围绕会与旋转平面正交的 (n − 2) 维子空间（即距其固定距离）发生。

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 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 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 in three dimensions, except rotations do not take place about this line but in multiple planes orthogonal to it.

## 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

<image>

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 vectors 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:

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

where the product is the geometric product from 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

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

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, and nm is its inverse as

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

So the rotation can be written

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

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. 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 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.

### 二重向量

二重向量s are quantities from geometric algebra, clifford algebra and the 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 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 through the exponential map, 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

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

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

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

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,

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

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

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

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 -plane by an angle θ has bivector e34θ, a simple bivector. The double rotation by α and β in the -plane and -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.

### 特征值和特征平面

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

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

such pairs. 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.