旋轉平面
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旋轉面、旋轉平面(英語:plane of rotation),是一個用於描述空間旋轉的抽像概念。
十維以下的旋轉平面數量如下表所示:
維數 | 零 | 一 | 二 | 三 | 四 | 五 | 六 | 七 | 八 | 九 | 十 |
---|---|---|---|---|---|---|---|---|---|---|---|
旋轉平面 | 0 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
旋轉平面主要用作描述四維空間及以上的旋轉,將高維旋轉分解為簡單的幾何代數描述。 [1]
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。
維數 | 零 | 一 | 二 | 三 | 四 | 五 | 六 | 七 | 八 | 九 | 十 |
---|---|---|---|---|---|---|---|---|---|---|---|
旋轉平面 | 0 | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 |
In three dimensions(英語:three dimensions) it is an alternative to the axis of rotation(英語:Rotation around a fixed axis), but unlike the axis of rotation it can be used in other dimensions, such as two(英語:two dimensions), four(英語:Four-dimensional space) or more dimensions.
旋轉平面在二維和三維中使用不多,因為在二維中只有一個平面(因此,識別旋轉平面是微不足道的並且很少這樣做),而在三維中旋轉軸具有相同的目的,並且是更成熟的方法。
Planes of rotation are not used much in two(英語:two dimension) and 三維空間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(英語:higher dimensions), where they can be used to break down the rotations into simpler parts. This can be done using geometric algebra(英語:geometric algebra), with the planes of rotations associated with simple bivectors(英語:Bivector#Simple bivectors) in the algebra.[1]
定義
平面
For this article, all planes are planes through the origin, that is they contain the zero vector(英語: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="英語" data-foreign-title="n-dimensional space">[[:n-dimensional space|n-dimensional space]](英語:n-dimensional space) is a two-dimensional linear subspace(英語: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(英語:exterior algebra) or geometric algebra(英語:geometric algebra) (in three dimensions the cross product(英語: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.
旋轉平面
A plane of rotation for a particular rotation is a plane that is mapped(英語: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(英語:angle of rotation) for the plane.
Every rotation except for the identity(英語:Identity element) rotation (with matrix the identity matrix(英語:identity matrix)) has at least one plane of rotation, and up to
planes of rotation, where n is the dimension.
十維以下的旋轉平面數量如下表所示:
When a rotation has multiple planes of rotation they are always orthogonal(英語:orthogonal) to each other, with only the origin in common. This is a stronger condition than to say the planes are at right angle(英語: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(英語: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(英語:identity matrix) in four dimensions (the central inversion(英語: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]