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The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system (y counterclockwise from x) by pre-multiplication (the rotation matrix R applied on the left of the column vector v to be rotated).
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
These parameters can be written as the elements of a 3 × 3 matrix A, called a rotation matrix. Typically, the coordinates of each of these vectors are arranged along a column of the matrix (however, beware that an alternative definition of rotation matrix exists and is widely used, where the vectors' coordinates defined above are arranged by ...
A rotation of the vector through an angle θ in counterclockwise direction is given by the rotation matrix: = ( ), which can be viewed either as an active transformation or a passive transformation (where the above matrix will be inverted), as described below.
Each matrix is meant to operate by pre-multiplying column vectors [] (see Ambiguities in the definition of rotation matrices) Each matrix is meant to represent an active rotation (the composing and composed matrices are supposed to act on the coordinates of vectors defined in the initial fixed reference frame and give as a result the ...
where is the k-th 3-vector measurement in the reference frame, is the corresponding k-th 3-vector measurement in the body frame and is a 3 by 3 rotation matrix between the coordinate frames. [ 1 ] a k {\displaystyle a_{k}} is an optional set of weights for each observation.
Let P and Q be two sets, each containing N points in .We want to find the transformation from Q to P.For simplicity, we will consider the three-dimensional case (=).The sets P and Q can each be represented by N × 3 matrices with the first row containing the coordinates of the first point, the second row containing the coordinates of the second point, and so on, as shown in this matrix:
An xy-Cartesian coordinate system rotated through an angle to an x′y′-Cartesian coordinate system In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and ...