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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. [2] [3] A rotation of axes is a linear map [4] [5] and a rigid transformation.
The trace of a rotation matrix is equal to the sum of its eigenvalues. For n = 2, a rotation by angle θ has trace 2 cos θ. For n = 3, a rotation around any axis by angle θ has trace 1 + 2 cos θ. For n = 4, and the trace is 2(cos θ + cos φ), which becomes 4 cos θ for an isoclinic rotation.
The rotation group is a Lie group of rotations about a fixed point. This (common) fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions. For a particular rotation: The axis of rotation is a line of its ...
A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L 1. Then reflect P′ to its image P′′ on the other side of line L 2. If lines L 1 and L 2 make an angle θ with one another, then points P and P′′ will make an angle 2θ around point O, the ...
The composition of two rotations is itself a rotation. Let (a 1, b 1, c 1, d 1) and (a 2, b 2, c 2, d 2) be the Euler parameters of two rotations. The parameters for the compound rotation (rotation 2 after rotation 1) are as follows:
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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 (n = 2, 4, 6...) there are up to n / 2 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 (n = 3, 5, 7, ...
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