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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 .
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.
Rotation of an object in two dimensions around a point O. Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point.
Namely, they have positive values when they represent a rotation that appears clockwise when looking in the positive direction of the axis, and negative values when the rotation appears counter-clockwise. The opposite convention (left hand rule) is less frequently adopted. About the ranges (using interval notation):
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.
A rotation through angle θ with non-standard axes. If a standard right-handed Cartesian coordinate system is used, with the x-axis to the right and the y-axis up, the rotation R(θ) is counterclockwise. If a left-handed Cartesian coordinate system is used, with x directed to the right but y directed down, R(θ) is clockwise.
For example, the daily rotation of the Earth is clockwise when viewed from above the South Pole, and counterclockwise when viewed from above the North Pole (considering "above a point" to be defined as "farther away from the center of earth and on the same ray"). The shadow of a horizontal sundial in the Northern Hemisphere rotates clockwise
An xy-Cartesian coordinate system rotated through an angle to an x′y′-Cartesian coordinate system 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 ...