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for 90°, 180°, and 270° counter-clockwise rotations. A 180° rotation (middle) followed by a positive 90° rotation (left) is equivalent to a single negative 90° (positive 270°) rotation (right).
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 y axes counterclockwise through an angle .
Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations , which have no fixed points, and (hyperplane) reflections , each of them having an entire ( n − 1) -dimensional flat of ...
When viewed at a position along the positive z-axis, the ¼ turn from the positive x-to the positive y-axis is counter-clockwise. For left-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn is clockwise. Interchanging the labels of any two axes reverses the handedness.
Two-dimensional rotation can occur in two possible directions or senses of rotation.Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to the left, and back up to the top.
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.
In 3D, rotations have three degrees of freedom, a degree for each linearly independent plane (bivector) the rotation can take place in. It has been known that pairs of quaternions can be used to generate rotations in 4D, yielding six degrees of freedom, and the geometric algebra approach verifies this result: in 4D, there are six linearly ...
Therefore the set of rotations has a group structure, known as a rotation group. The theorem is named after Leonhard Euler, who proved it in 1775 by means of spherical geometry. The axis of rotation is known as an Euler axis, typically represented by a unit vector ê. Its product by the rotation angle is known as an axis-angle vector.