Rotated

Rotated is the past tense and past participle of rotate, and it is also used as an adjective to describe something that has been turned around an axis or center by a specified angle. The term appears across disciplines such as geometry, computer graphics, engineering, and everyday language. Its etymology traces to Latin rotatus, from rota, meaning wheel.

In geometry, rotation is a rigid motion that preserves distances and angles while changing an object's orientation. In two dimensions, rotating a point (x, y) by angle θ about the origin yields (x cos θ − y sin θ, x sin θ + y cos θ). The corresponding linear transform is the 2D rotation matrix. In three dimensions, rotations are represented by matrices, quaternions, or Euler angles, and they form the rotation groups SO(2) in the plane and SO(3) in space. Rotated figures remain congruent to their originals, with 360 degrees restoring the initial pose.

In computing and imaging, rotating an image or object reorients it within a plane or space. Operations may use interpolation methods such as nearest neighbor, bilinear, or bicubic to estimate new pixel values, balancing sharpness and artifacts. In computer graphics and CAD, rotation is often combined with translations and scalings within homogeneous coordinates for complex transformations.

Other contexts include astronomy and geodesy, where rotation describes the turning of the Earth or the orientation of coordinate frames relative to inertial space. The term is also encountered in everyday language to describe objects that have been turned, for example, rotated signs or rotated components.