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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.
A plane rotation around a point followed by another rotation around a different point results in a total motion which is either a rotation (as in this picture), or a translation. A motion of a Euclidean space is the same as its isometry: it leaves the distance between any two points unchanged after the transformation.
In contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, a glide reflection does both, and an improper rotation combines a change in handedness with a normal rotation. If a fixed point is taken as the origin of a Cartesian coordinate system, then every point can be given coordinates as a displacement ...
The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is ...
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
To define a spherical coordinate system, one must designate an origin point in space, O, and two orthogonal directions: the zenith reference direction and the azimuth reference direction. These choices determine a reference plane that is typically defined as containing the point of origin and the x– and y–axes , either of which may be ...
The identity rotation corresponds to the point at the center of the ball. Rotations through an angle 𝜃 between 0 and π (not including either) are on the same axis at the same distance. Rotation through angles between 0 and − π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin ...
If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse of a rotation is also a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than ...