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  2. Rotation of axes in two dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_of_axes_in_two...

    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.

  3. Euler–Rodrigues formula - Wikipedia

    en.wikipedia.org/wiki/Euler–Rodrigues_formula

    A rotation about the origin is ... The parameters for the compound rotation (rotation 2 after rotation 1) are as follows: ... (±1, 0, 0, 0). Rotations of 180 degrees ...

  4. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    This has the convenient implication for 2 × 2 and 3 × 3 rotation matrices that the trace reveals the angle of rotation, θ, in the two-dimensional space (or subspace). For a 2 × 2 matrix the trace is 2 cos θ, and for a 3 × 3 matrix it is 1 + 2 cos θ. In the three-dimensional case, the subspace consists of all vectors perpendicular to the ...

  5. Rotation (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Rotation_(mathematics)

    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.

  6. Rotations and reflections in two dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotations_and_reflections...

    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 ...

  7. Quaternions and spatial rotation - Wikipedia

    en.wikipedia.org/wiki/Quaternions_and_spatial...

    3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]

  8. Bride 'Upset' After Groom's Friend Makes Joke When Wedding ...

    www.aol.com/bride-upset-grooms-friend-makes...

    The groom disagreed with his wife, countering that his friend was "just joking." "But I don’t find anything funny about that," the bride insisted.

  9. Plane of rotation - Wikipedia

    en.wikipedia.org/wiki/Plane_of_rotation

    describes a rotation in four dimensions in which every plane through the origin is a plane of rotation through an angle π, so any pair of orthogonal planes generates the rotation. But for a general rotation it is at least theoretically possible to identify a unique set of orthogonal planes, in each of which points are rotated through an angle ...