GegenUhrzeigersinn

GegenUhrzeigersinn is the German term for counterclockwise rotation, defined as the direction opposite to the motion of clock hands when viewed from above a fixed reference frame. The standard German phrasing in prose is gegen den Uhrzeigersinn, but the combined form GegenUhrzeigersinn or hyphenated Gegen-Uhrzeigersinn also appears in technical usage. The concept is widely used in mathematics, physics and engineering to describe positive angular orientation.

In two-dimensional geometry, rotation by an angle theta around the origin in the GegenUhrzeigersinn direction is represented by the rotation matrix [[cos theta, -sin theta], [sin theta, cos theta]]. A point (x, y) is transformed to (x cos theta - y sin theta, x sin theta + y cos theta). In polar coordinates, the angle increases in the GegenUhrzeigersinn direction, and angular velocity is considered positive for counterclockwise rotation, in accordance with the right-hand rule.

Context and caveats: While GegenUhrzeigersinn denotes counterclockwise rotation in the standard mathematical sense, some coordinate systems—such as computer graphics frameworks with a vertical axis that increases downward—can invert the apparent rotation direction. Therefore the applied convention depends on the reference frame and software, and it is common to specify the coordinate system when giving rotation directions.

Examples: Rotating (1, 0) by 90 degrees GegenUhrzeigersinn yields (0, 1); a rotation by -90 degrees yields (0, -1). The term is also used in discussions of angular momentum, torque directions, and orientation in design and animation workflows.