旋轉平面

旋轉平面（），是一個用於描述空間旋轉的抽像概念。

九維以下的旋轉平面數量如下表所示：
{| class="wikitable" border="1"
! 維數
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9
|-
! 旋轉平面
| 0 || 0 || 1 || 1 || 2 || 2 || 3 || 3 || 4 || 4
|}

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

數學上，旋轉平面可用多種方式描述。可用平面和旋轉角度來描述，可用克利福德代數的二重向量來描述。旋轉平面又與旋轉矩陣的特徵值和特徵向量有關。（Mathematically such planes can be described in a number of ways. They can be described in terms of s and . They can be associated with s from . They are related to the  of a .）And in particular s they are related to other algebraic and geometric properties, which can then be generalised to other dimensions.

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. The main use for them is in describing more complex rotations in , where they can be used to break down the rotations into simpler parts. This can be done using , with the planes of rotations associated with  in the algebra.

定義

平面 

本文所有平面都通過原點（即包含零向量。）

For this article, all s are planes through the , that is they contain the . Such a plane in  is a two-dimensional  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  or  (in three dimensions the  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.

旋轉平面 
A plane of rotation for a particular  is a plane that is  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  for the plane.

Every rotation except for the  rotation (with matrix the ) has at least one plane of rotation, and up to

planes of rotation, where  is the dimension. 

九維以下的旋轉平面數量如下表所示：
{| class="wikitable" border="1"
! 維數
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9
|-
! 旋轉平面
| 0 || 0 || 1 || 1 || 2 || 2 || 3 || 3 || 4 || 4
|}

When a rotation has multiple planes of rotation they are always  to each other, with only the origin in common. This is a stronger condition than to say the planes are at s; 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 .

In more than three dimensions planes of rotation are not always unique. For example the negative of the  in four dimensions (the ),

 

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.

二維 
在二維空間只有一個旋轉平面，即空間本身的平面。在笛卡爾坐標系是笛卡爾平面，在複數是複平面。因此，任何旋轉都是整個平面（即整個空間），只保持原點固定。完全由帶符號的旋轉角度指定，例如在 -π 到 π 的範圍內。因此若角度為θ，復平面的旋轉則由以下歐拉公式給出：

 

笛卡爾平面的旋轉則由以下旋轉矩陣給出：

三維 

三維空間可以有無數個旋轉平面，但當旋轉平面有了一個，就不能第二個旋轉平面。

三維任何的旋轉平面，都總是有一個固定的軸，即旋轉軸。

這可以用如下矩陣來描述（旋轉角度為θ）：

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

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  between the  and  Hemispheres. Other examples include mechanical devices like a  or  which store  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.

四維空間中的一般旋轉只有一個固定點，即原點。因此，旋轉軸不能用於四個維度。但是可以使用旋轉平面，並且在四個維度中的每個非平凡旋轉都有一個或兩個旋轉平面。
A general rotation in  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.

簡單旋轉 
A rotation with only one plane of rotation is a . 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 -plane: points in that plane and only in that plane are unchanged. The plane of rotation is the -plane, points in this plane are rotated through an angle . A general point rotates only in the -plane, that is it rotates around the -plane by changing only its  and  coordinates.

 

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.

雙重旋轉 
In a  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,  and  (if either angle is zero the rotation is simple). Points in the first plane rotate through , while points in the second plane rotate through . All other points rotate through an angle between  and , 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  in the -plane and  in the -plane is given by the matrix

等斜旋轉 

A special case of the double rotation is when the angles are equal, that is if . This is called an , 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, . 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.

高維 
As already noted the maximum number of planes of rotation in  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.

Simple rotations can be identified in all dimensions, as rotations with just one plane of rotation. A simple rotation in  dimensions takes place about (that is at a fixed distance from) an -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 () there are up to  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 () there are  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  in three dimensions, except rotations do not take place about this line but in multiple planes orthogonal to it.

數學性質 
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  dimensions by giving an -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  dimensions is specified by a vector perpendicular to the -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 s can be used to simplify the calculations.

So the reflection in a -dimensional space is given by the unit vector perpendicular to it, , thus:

where the product is the geometric product from .

If  is reflected in another, distinct, -dimensional space, described by a unit vector  perpendicular to it, the result is

This is a simple rotation in  dimensions, through twice the angle between the subspaces, which is also the angle between the vectors m and . It can be checked using geometric algebra that this is a rotation, and that it rotates all vectors as expected.

The quantity  is a , and  is its inverse as

So the rotation can be written

where  is the rotor.

The plane of rotation is the plane containing  and , 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 . The rotation is through twice the angle between the vectors, up to  or a half-turn. The sense of the rotation is to rotate from  towards : the geometric product is not  so the product  is the inverse rotation, with sense from  to .

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  reflections if the dimension  is even,  if  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 ,  and the , 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  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  through the , 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  the rotor associated with it is

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

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

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

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  has bivector , a simple bivector. The double rotation by  and  in the -plane and -planes has bivector , the sum of two simple bivectors  and  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.

特徵值和特徵平面 
The planes of rotations for a particular rotation using the s. Given a general rotation matrix in  dimensions its  has either one (in odd dimensions) or zero (in even dimensions) real roots. The other roots are in complex conjugate pairs, exactly

such pairs. These correspond to the planes of rotation, the s of the matrix, which can be calculated using algebraic techniques. In addition 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.

另見 
 吉文斯旋转

另見 
 SO(3)上的卡

註

參考 
 
 
 

几何代数
三维旋转
旋轉對稱
定向 (几何)