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For example the following matrix fixes the xy-plane: points in that plane and only in that plane are unchanged. The plane of rotation is the zw-plane, points in this plane are rotated through an angle θ. A general point rotates only in the zw-plane, that is it rotates around the xy-plane by changing only its z and w coordinates.
into two skew-symmetric matrices A 1 and A 2 satisfying the properties A 1 A 2 = 0, A 1 3 = −A 1 and A 2 3 = −A 2, where ∓θ 1 i and ∓θ 2 i are the eigenvalues of A. Then, the 4D rotation matrices can be obtained from the skew-symmetric matrices A 1 and A 2 by Rodrigues' rotation formula and the Cayley formula. [9]
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 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 has two angles of rotation, one for each plane of rotation, through which points in the planes rotate. If these are ω 1 and ω 2 then all points not in the planes rotate through an angle between ω 1 and ω 2. Rotations in four dimensions about a fixed point have six degrees of freedom. A four-dimensional direct motion in general ...
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.
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There are thus 10 two-dimensional crystallographic point groups: C 1, C 2, C 3, C 4, C 6, D 1, D 2, D 3, D 4, D 6; The groups may be constructed as follows: C n. Generated by an element also called C n, which corresponds to a rotation by angle 2π/n.