旋轉面、旋轉平面，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。 Lounesto 2001 pp 222223 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。 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 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 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 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 algebraLounesto 2001 pp 222223 定義 平面 本文所有平面都透過原點（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 twodimensional of the space It is completely specified by any two nonzero and nonparallel vectors that lie in the plane that is by any two vectors and such that mathbf a wedge mathbf b ne 0 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 nonparallelLounesto 2001 p 38 If the bivector is written then the condition that a point lies on the plane associated with is simplyHestenes 1999 p 48 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 and and so by any vector of the form mathbf c lambdamathbf a mumathbf b 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 leftlfloor frac n2 rightrfloor planes of rotation where is the dimension 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 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 Lounesto 2001 p 222 In more than three dimensions planes of rotation are not always unique For example the negative of the in four dimensions the 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 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 rotationLounesto 2001 p87 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空間的旋轉，僅原點保持固定。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尤拉公式給出： eitheta cos theta isin theta 笛卡爾平面的旋轉則由以下旋轉矩陣給出 Lounesto 2001 pp2728： begin pmatrix cos theta sin theta sin theta cos theta end pmatrix 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。 三維任何旋轉都總是隻有一個固定的軸，即旋轉軸。 這可以用如下矩陣來描述（旋轉角度為 θ）： begin pmatrix cos theta sin theta 0 sin theta cos theta 0 0 0 1 end pmatrix 另一例子是地球自轉。自轉軸是北極至南極的連線，自轉平面穿過北半球和南半球之間的赤道平面。 其它例子包括陀螺儀、飛輪等機械裝置，通常沿旋轉平面儲存大量旋轉動能。 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 begin pmatrix cos theta sin theta 0 sin theta cos theta 0 0 0 1 end pmatrix Another example is the 地球自轉 The axis of rotation is the line joining the North Pole and 南極 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 在任何三維旋轉中，旋轉平面都是唯一定義的。它與旋轉角度一起完整地描述了旋轉。或者，在連續旋轉的物體中，旋轉特性（例如旋轉速率）可以用旋轉平面來描述。 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 dimensionsHestenes 1999 pp 280284 四維 一般只有一個固定點，即原點。因此，四個維度沒有旋轉軸。但是四維空間可以使用旋轉平面，並且在四個維度中的每個非平凡旋轉都可以有一至兩個旋轉平面。 簡單旋轉 僅具有一個旋轉平面的旋轉是。 簡單旋轉有一個固定的平面，因此點在旋轉時不會改變其與該平面的距離。旋轉平面與該平面正交，可以說旋轉發生在該平面內。 例如，下列矩陣固定 xy 平面：此平面中的點且僅在該平面中的點保持不變。旋轉平面是 zw 平面，該平面上的點旋轉角度 θ。一般點僅在 zw 平面內旋轉，即僅更改其 z 和 w 座標來繞 xy 平面旋轉。 begin pmatrix 1 0 0 0 0 1 0 0 0 0 cos theta sin theta 0 0 sin theta cos theta end pmatrix 在二維和三維，所有旋轉都是簡單旋轉，因為只有一個旋轉平面。只有在四維以上才存在不是簡單旋轉的旋轉。在四維也存在雙重旋轉和等斜旋轉。 雙重旋轉 有兩個旋轉平面，沒有固定平面，唯一的固定點是原點。旋轉發生在兩旋轉平面中。這些平面是正交的，也就是說，它們沒有共同的向量，因此一個平面中的每個向量都與另一個平面中的每個向量成直角。兩個旋轉平面跨越四維空間，因此空間中的每個點都可以由兩個點指定，每個平面上一個。 雙旋轉有兩個旋轉角度，每個旋轉平面一個。雙重旋轉有兩個平面和兩個非零角度 α、β（如果任一角度為零，則是簡單旋轉）。第一個平面的旋轉 α 點，第二個平面的旋轉 β 點。所有其他點都旋轉 α 和 β 之間的角度，因此在某種意義上是 αβ 共同決定了旋轉量。對於一般的雙旋轉，旋轉平面和角度是唯一的，並且給定一般的旋轉，它們可以被計算。例如，xy 平面中的 α 和 zw 平面中的 β 的旋轉由矩陣給出 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 等斜旋轉 雙旋轉的一個特殊情況是角度相等，即 α β 0。 例如，在中所有非零點都會旋轉相同的角度 α。最重要的是，旋轉平面不是唯一標識的。相反，有無數對正交平面可以被視為旋轉平面。例如可以任意一點，它旋轉所在的平面以及與其正交的平面可以當作兩個旋轉平面。Lounesto 2001 pp 8389 高維 旋轉平面數量公式：leftlfloor frac n2 rightrfloor 因此，超過四個維度，複雜性會迅速增加，並且如上所述對旋轉進行分類變得過於複雜而不實用，但可以進行一些觀察。 簡單旋轉可以在所有維度上被識別為僅具有一個旋轉平面的旋轉。 n 維的簡單旋轉圍繞會與旋轉平面正交的 n 2 維子空間（即距其固定距離）發生。 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 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 twodimensional reflection is in a line a threedimensional 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 mathbf x mathbf mxm 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 mathbf x mathbf nxmathbf n mathbf nmathbf mxmmathbf n mathbf nmxmn 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 mathbf mnmathbf nm mathbf mnnm mathbf mm 1 So the rotation can be written mathbf x Rmathbf x R1 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 halfturn 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 rotationLounesto 2001 p 5758Hestenes 1999 p 278280 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 nonsimple 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 R_mathbf B efrac mathbf B2 This is a simple rotation if the bivector is simple a more general rotation otherwise When squared R_mathbf B2 efrac mathbf B2 efrac mathbf B2 emathbf B 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 mathbf mn emathbf B from which it follows that the bivector associated with the plane of rotation containing and that rotates to 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 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 aboveDorst Doran Lasenby 2002 pp 7989 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 leftlfloor frac n2 rightrfloor 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 interpretationsDorst Doran Lasenby 2002 pp 145154 另見 吉文斯旋轉 SO 3 上的卡 六自由度 註 參考 Category 幾何代數 Category 三維旋轉 Category 旋轉對稱 Category 定向 幾何