Inversion in a point

In Euclidean geometry, the inversion of a point "X" in respect to a point "P" is a point "X"* such that "P" is the midpoint of the line segment with endpoints "X" and "X"*. In other words, the vector from "X" to "P" is the same as the vector from "P" to "X"*.

The formula for the inversion in "P" is

:x*=2a−x

where a, x and x* are the position vectors of "P", "X" and "X"* respectively.

This mapping is an isometric involutive affine transformation which has exactly one fixed point, which is "P".

In odd-dimensional Euclidean space it does "not" preserve orientation, it is an indirect isometry.

Geometrically in 3D it amounts to rotation about an axis through "P" by an angle of 180°, combined with reflection in the plane through "P" which is perpendicular to the axis; the result does not depend on the orientation (in the other sense) of the axis. Notations for the type of operation, or the type of group it generates, are $overline\{1\}$, "C_i", "S₂", and 1×. The group type is one of the three symmetry group types in 3D without any pure rotational symmetry, see cyclic symmetries with "n"=1.

The following point groups in three dimensions contain inversion:
*"C"_"n"h and "D"_"n"h for even "n"
*"S"_2"n" and "D"_"n"d for odd "n"
*"T"_h, "O"_h, and "I"_h

Closely related to inverse in a point is reflection in respect to a plane, which can be thought of as a "inversion in a plane".

Inversion with respect to the origin

Inversion with respect to the origin corresponds to additive inversion of the position vector, and also to scalar multiplication by −1. The operation commutes with every other linear transformation, but not with translation: it is in the central of the general linear group. "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion, also called parity transformation.

ee also

* Affine involution
* Circle inversion
* Parity (physics)