Rotating mirroring

If the point P is first rotated around the black axis of rotation and then mirrored on the blue plane (or vice versa), the projection takes place on the point Q. If, on the other hand, it is mirrored after the rotation at the inversion center (red point in the blue plane) (or vice versa), the projection takes place on the point Q '.

A rotation mirroring is a congruence mapping of the three-dimensional Euclidean space in itself. It is composed of a rotation and a reflection on a plane that is intersected at right angles by the axis of rotation. A related mapping is rotational inversion , which consists of a rotation and a reflection at a point on the axis of rotation. In both cases, the sequence of the sub-operations rotation and mirroring does not matter during execution. Rotational mirroring and rotational inversion provide the same result if (i) the center of inversion is the intersection of the mirror plane with the axis of rotation and (ii) the two angles of rotation differ by. Both images are movements in Euclidean space that reverse the orientation because of the reflections . ${\ displaystyle \ pi = 180 ^ {\ circ}}$

The rotation angles 0 ° and 180 ° provide particularly simple results:

A rotational mirroring by 0 ° (= rotational inversion by 180 °) is a simple plane mirroring: The point P in the adjacent figure is projected vertically downwards along the blue line.
A rotational mirroring by 180 ° (= rotational inversion by 0 °) is a point mirroring at the intersection of the mirror plane with the axis of rotation (in the figure the red point in the blue plane): The point P is therefore projected diagonally backwards and downwards along the red line .

Since it is actually a point reflection, the result in this case does not depend on the position of the axis as long as it goes through the inversion center.

If the origin of a Cartesian coordinate system is placed in the center of inversion, a rotation mirror is represented by an orthogonal matrix with determinant -1. Also, if the axis is chosen as the axis of rotation, the shape takes ${\ displaystyle A}$ ${\ displaystyle z}$ ${\ displaystyle A}$

{\ displaystyle A = {\ begin {pmatrix} \ cos \ varphi & - \ sin \ varphi & 0 \\\ sin \ varphi & \ cos \ varphi & 0 \\ 0 & 0 & -1 \ end {pmatrix}}}

on. In a rotary inversion, the matrix has the same shape, so it is only by replacing. ${\ displaystyle \ varphi}$ ${\ displaystyle \ varphi + \ pi}$

Repeatedly applying a rotation mirror with the angle provides rotation mirroring and ordinary rotations alternately. The corresponding angles are , , , ... Is so a rotation is a multiple of , so that a total of only finitely many different pictures appear in this process. These form a group which to describe crystal structures and molecular symmetries used rotating mirror group . ${\ displaystyle \ varphi}$ ${\ displaystyle \ varphi}$ ${\ displaystyle 2 \ varphi}$ ${\ displaystyle 3 \ varphi}$ ${\ displaystyle \ varphi = {\ tfrac {2 \ pi} {n}}}$ ${\ displaystyle 2 \ pi}$

literature

Martin Nitschke: Geometry. Application-related basics and examples . Carl-Hanser-Verlag, 2005, ISBN 3-446-22676-1 , p. 98 ff .

Web links

Commons : Improper rotation - collection of images, videos and audio files