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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 .
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 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 ...
To plot any dot from its spherical coordinates (r, θ, φ), where θ is inclination, the user would: move r units from the origin in the zenith reference direction (z-axis); then rotate by the amount of the azimuth angle (φ) about the origin from the designated azimuth reference direction, (i.e., either the x– or y–axis, see Definition ...
An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dφ/dt. Next, the terms in the acceleration in the inertial frame are ...
Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. 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 ...
Improper rotation, also called rotation-reflection: rotation about an axis by an angle θ, combined with reflection in the plane through the origin perpendicular to the axis. Rotation-reflection by θ = 360°/n for any positive integer n is denoted S n (from the Schoenflies notation for the group S n that it generates if n is even).
The two rotation planes span four-dimensional space, so every point in the space can be specified by two points, one on each of the planes. A double rotation has two angles of rotation, one for each plane of rotation. The rotation is specified by giving the two planes and two non-zero angles, α and β (if either angle is zero the rotation is ...