旋转平面
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旋转面、旋转平面(英语: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。 [1]
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。
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 |
In three dimensions(英语:three dimensions) it is an alternative to the axis of rotation(英语:Rotation around a fixed axis), but unlike the axis of rotation it can be used in other dimensions, such as two(英语:two dimensions), four(英语:Four-dimensional space) 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 two(英语:two dimension) 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(英语:higher dimensions), where they can be used to break down the rotations into simpler parts. This can be done using geometric algebra(英语:geometric algebra), with the planes of rotations associated with simple bivectors(英语:Bivector#Simple bivectors) in the algebra.[1]
定义
平面
For this article, all planes are planes through the origin, that is they contain the zero vector(英语:zero vector). Such a plane in <span class="ilh-all " data-orig-title="'"`UNIQ--templatestyles-00000015-QINU`"'n-dimensional space" data-lang-code="en" data-lang-name="英语" data-foreign-title="n-dimensional space">[[:n-dimensional space|n-dimensional space]](英语:n-dimensional space) is a two-dimensional linear subspace(英语: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
where ∧ is the exterior product from exterior algebra(英语:exterior algebra) or geometric algebra(英语:geometric algebra) (in three dimensions the cross product(英语: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.[2]
If the bivector a ∧ b is written B, then the condition that a point lies on the plane associated with B is simply[3]
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
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(英语:Linear map) 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(英语:angle of rotation) for the plane.
Every rotation except for the identity(英语:Identity element) rotation (with matrix the identity matrix(英语:identity matrix)) has at least one plane of rotation, and up to
planes of rotation, where n is the dimension.
十维以下的旋转平面数量如下表所示:
When a rotation has multiple planes of rotation they are always orthogonal(英语:orthogonal) to each other, with only the origin in common. This is a stronger condition than to say the planes are at right angle(英语:right angle)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 line of intersection(英语:Plane (geometry)#Line of intersection between two planes).[4]
In more than three dimensions planes of rotation are not always unique. For example the negative of the identity matrix(英语:identity matrix) in four dimensions (the central inversion(英语:Inversion in a point)),
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.[5]