Improper rotation

A rotating mirror is a congruence of the three-dimensional Euclidean space into itself. It is formed by successive application of a rotation about any axis followed by reflection in a plane, which is intersected at right angles from the axis. Due to the plane reflection is an orientation-reversing motion of the Euclidean space.

In the special case of a rotation of 180 ° ( angle of rotation) creates a point reflection at the intersection of the rotation axis and mirror plane. Here, however, the axis of rotation is not uniquely determined: each can line through the center of the point reflection serve as an axis.

Alternatively, a rotation-reflection described as a composition of a rotation and a reflection point of the room at a point of the axis of rotation. However, in this case the angle of rotation by 180 °, or in comparison to the original definition is modified.

Occasionally rotary reflections are defined as the linear maps in it. These arise as a sequential execution of a rotation about an axis through the origin, followed by the mirroring of the original plane normal to the rotational axis.

Since each rotation-reflection has a fixed point, each affine rotation-reflection can be a suitable choice of the coordinate origin constitute only by an orthogonal matrix with determinant -1:

By suitable rotation of the Cartesian coordinate system can be used in the form of

With brought (cf. orthogonal group ).