In geometry, a rotation is understood to be a self-image of Euclidean space with at least one fixed point, which leaves all distances invariant and maintains the orientation . If the orientation is reversed, there is a mirroring (geometry) or rotating mirroring .
Since rotations leave lengths (and hence angles) invariant, every rotation is a congruence map . In two- and three-dimensional space, every rotation has a certain angle of rotation. Rotations whose angles of rotation differ by 360 ° or a multiple thereof are identical.
In the plane , any true rotation (i.e. not rotation about zero angle) leaves only one point fixed, the center of rotation. If a point and its image is different , then the angle does not depend on and defines the angle of rotation. A rotation by 180 ° causes the same mapping of the plane as a point reflection at the center of rotation .
In three-dimensional space, every real rotation leaves exactly one straight line, the axis of rotation. Each plane perpendicular to the axis of rotation is rotated by the same angle of rotation, whereby its point of intersection with the axis is the fixed point.
In analytic geometry , rotations are special equidistant affine maps. If a Cartesian coordinate system is selected , the origin of which is on the axis of rotation, the translational component becomes zero. The rotation is then described by a rotation matrix . A rotation with a translation component can also be described as a matrix in homogeneous coordinates .