Rotations

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A rotation is a linear transformation (usually in three-dimensional space with a positive definite scalar product) that preserves scalar products. Usually a determinant of +1 is also postulated, else the transformation is called a reflection. If Hepa img196.gif and Hepa img904.gif are (three-dimensional) vectors, R is a rotation and Hepa img905.gif and Hepa img906.gif are the rotated vectors, then

Hepa img907.gif

Let Hepa img908.gif and Hepa img909.gif be orthonormal basis vectors, i.e. Hepa img910.gif . Define matrix elements Hepa img911.gif . Then

Hepa img912.gif

The Hepa img363.gif matrix

Hepa img913.gif

which represents the rotation R, is an orthogonal matrix, since

Hepa img914.gif

If x1, x2, x3 are the components of the vector Hepa img196.gif with respect to the basis Hepa img915.gif ,

Hepa img916.gif

then

Hepa img917.gif

In matrix notation,

Hepa img918.gif

where R is the Hepa img363.gif matrix defined above.

If the rotation R is followed by a second rotation S, the result is a third rotation Q= SR, defined by

Hepa img919.gif

In terms of Hepa img363.gif matrices the composition SR is simply the matrix product, since

Hepa img920.gif

The above formalism treats rotations as active transformations, i.e. the vectors are rotated and the basis vectors are kept fixed. The passive point of view is often adopted, where a vector Hepa img196.gif is not transformed, but its coordinates x1, x2, x3 change because the basis vectors are rotated. If

Hepa img921.gif

then the new coordinates Hepa img922.gif are defined by

Hepa img923.gif

or in matrix notation

Hepa img924.gif

If one passive rotation (coordinate transformation) U is followed by another, V, such that

Hepa img925.gif

then the total result is a third passive rotation P, such that

Hepa img926.gif

Note that the composition of passive rotations, first U and then V, leads to a matrix product, P=UV, in which the order is reversed. The reason for the reversal is that the matrix elements of U and of V are taken with respect to two different bases, Hepa img205.gif and Hepa img927.gif .

A rotation is defined by a rotation axis Hepa img928.gif , and an angle of rotation Hepa img929.gif . With Hepa img930.gif the corresponding rotation matrix is

Hepa img931.gif

In vector notation,

Hepa img932.gif

A general rotation R can also be parameterized by the Euler angles Hepa img41.gif , Hepa img79.gif and Hepa img933.gif , as Hepa img934.gif , where Hepa img935.gif is an active rotation by an angle Hepa img929.gif about the axis Hepa img936.gif . (A different convention is to use Hepa img937.gif instead of Hepa img938.gif , the relation is very simply that Hepa img939.gif . The ranges of the angles are: Hepa img940.gif , Hepa img899.gif , Hepa img941.gif . Explicitly, with Hepa img942.gif , Hepa img943.gif , Hepa img944.gif , Hepa img945.gif , etc., we have

Hepa img946.gif

Example ( Hepa img2.gif Coordinate Systems). A Euclidean coordinate system is determined by an origin Hepa img190.gif and three orthonormal basis vectors Hepa img915.gif . Let Hepa img947.gif be a second Euclidean coordinate system. Let x1,x2,x3 and Hepa img922.gif be the coordinates of a point Hepa img196.gif with respect to the two systems, i.e.

Hepa img948.gif

The coordinate transformations from one system to the other and back are:

Hepa img949.gif

where Hepa img950.gif , i.e. R is the rotation defined by Hepa img951.gif .

Suppose one has measured three reference points Hepa img952.gif , Hepa img953.gif , and Hepa img954.gif in the two systems in order to determine the coordinate transformation. The three distances Hepa img955.gif , Hepa img956.gif and Hepa img957.gif should be independent of the coordinate system; this gives three constraints

Hepa img958.gif

One should make a least squares fit in order to get the constraints exactly satisfied (the Hepa img111.gif of the fit gives a consistency check of the measurements). Define Hepa img959.gif . Then

Hepa img960.gif

if one defines Hepa img961.gif . Similarly, Hepa img962.gif . It follows that Hepa img963.gif , and the Hepa img363.gif matrix R can be found from the linear equation

Hepa img964.gif

The solution for R is unique whenever the vectors Hepa img965.gif and Hepa img966.gif are linearly independent. Finally,

Hepa img967.gif