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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 
, Ci, 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:
- Cnh and Dnh for even n
- S2n and Dnd for odd n
- Th, Oh, and Ih
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.
See also
