Rotationsvinkel

Rotationsvinkel, or rotation angle, is the measure of how much a rigid body is rotated about a fixed point or axis. In Swedish, rotationsvinkel is the term used for this concept. In two dimensions the rotation is typically described with respect to the origin, while in three dimensions it is described with respect to a chosen axis.

In 2D, rotation by angle θ maps a point (x, y) to (x', y') = (x cos θ − y sin θ, x sin θ + y cos θ). This can be written as a matrix multiplication p' = R(θ) p, with R(θ) = [[cos θ, −sin θ], [sin θ, cos θ]].

Angle units: θ is commonly expressed in radians or degrees; the relation is θ_deg = θ × 180/π. By convention, positive angles denote counterclockwise rotation when the perpendicular viewing direction is standard.

In 3D, a rotation is specified by an axis and an angle, often represented by an axis-angle pair, a rotation matrix, or a quaternion. Rodrigues' rotation formula gives the rotated vector v' = v cos θ + (u × v) sin θ + u (u · v) (1 − cos θ) for a unit axis u.

Applications include computer graphics, robotics, and geometry, where the rotation angle is used to describe orientation changes, interpolate orientations, and compose transformations. The concept is central to linear algebra and rigid-body kinematics.