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  2. Rotation (mathematics) - Wikipedia

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

    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 ...

  3. 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.

  4. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    Reverse rotate the axis-point pair which was done in step 1 (undoing step 1) ... This is a matrix form of Rodrigues' rotation formula, (or the equivalent, ...

  5. Rodrigues' rotation formula - Wikipedia

    en.wikipedia.org/wiki/Rodrigues'_rotation_formula

    In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...

  6. Rotation formalisms in three dimensions - Wikipedia

    en.wikipedia.org/wiki/Rotation_formalisms_in...

    Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.

  7. Euler's rotation theorem - Wikipedia

    en.wikipedia.org/wiki/Euler's_rotation_theorem

    A rotation represented by an Euler axis and angle. In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two ...

  8. 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 ...

  9. Rotations in 4-dimensional Euclidean space - Wikipedia

    en.wikipedia.org/wiki/Rotations_in_4-dimensional...

    In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement.