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
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and
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are (three-dimensional) vectors, R is a rotation and
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and
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are the rotated vectors, then
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Let
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and
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be orthonormal basis vectors, i.e.
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. Define matrix elements
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. Then
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The
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matrix
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which represents the rotation R, is an orthogonal matrix, since
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If x1, x2, x3 are the components of the vector
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with respect to the basis
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,
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then
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In matrix notation,
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where R is the
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matrix defined above.
If the rotation R is followed by a second rotation S, the result is a third rotation Q= SR, defined by
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In terms of
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matrices the composition SR is simply the matrix product, since
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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
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is not transformed, but its coordinates x1, x2, x3 change because the basis vectors are rotated. If
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then the new coordinates
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are defined by
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or in matrix notation
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If one passive rotation (coordinate transformation) U is followed by another, V, such that
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then the total result is a third passive rotation P, such that
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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,
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and
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.
A rotation is defined by a rotation axis
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, and an angle of rotation
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. With
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the corresponding rotation matrix is
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In vector notation,
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A general rotation R can also be parameterized by the Euler angles
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,
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and
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, as
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, where
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is an active rotation by an angle
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about the axis
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. (A different convention is to use
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instead of
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, the relation is very simply that
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. The ranges of the angles are:
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,
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,
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. Explicitly, with
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,
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,
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,
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, etc., we have
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Example (
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Coordinate Systems). A Euclidean coordinate system is determined by an origin
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and three orthonormal basis vectors
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. Let
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be a second Euclidean coordinate system. Let x1,x2,x3 and
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be the coordinates of a point
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with respect to the two systems, i.e.
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The coordinate transformations from one system to the other and back are:
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where
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, i.e. R is the rotation defined by
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.
Suppose one has measured three reference points
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,
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, and
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in the two systems in order to determine the coordinate transformation. The three distances
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,
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and
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should be independent of the coordinate system; this gives three constraints
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One should make a least squares fit in order to get the constraints exactly satisfied (the
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of the fit gives a consistency check of the measurements). Define
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. Then
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if one defines
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. Similarly,
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. It follows that
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, and the
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matrix R can be found from the linear equation
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The solution for R is unique whenever the vectors
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and
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are linearly independent. Finally,
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