Rotating

Rotating refers to the motion of an object or reference frame around an axis or central point in which the distances to the axis remain unchanged. Rotation is distinguished from translation, where every point moves the same distance in a parallel direction.

In two dimensions, a rotation is a transformation about a fixed point by a certain angle, preserving the size and shape of figures. The angle is measured in degrees or radians, with positive values indicating counterclockwise rotation and negative values indicating clockwise rotation. The set of all 2D rotations forms the group SO(2).

In three dimensions, a rotation changes orientation without changing position, and it can be described by an axis and an angle, or represented by a rotation matrix, Euler angles, unit quaternions, or an axis–angle pair. Unlike 2D rotations, 3D rotations generally do not commute; the order of successive rotations affects the final orientation. The mathematical structure for 3D rotations is the special orthogonal group SO(3).

Rotations can be described in active or passive terms. An active rotation moves the object itself, whereas a passive rotation changes the observer’s reference frame. These viewpoints are related by a change of coordinates and can yield different descriptions of the same physical situation.

Common quantities associated with rotation include angular velocity (how fast the orientation changes) and angular momentum (an object’s rotational motion tendency). Units include radians per second for velocity and radians for angle, with radians being a dimensionless measure.

Rotating underpins many phenomena and technologies, from the spinning of celestial bodies and gyroscopes to computer graphics, robotics, and navigation systems. Its properties of symmetry and invariance make it central to physics, engineering, and mathematics.