Inversion symmetry

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 even-dimensional Euclidean space, say 2N-dimensional space, the inversion in a point P is equivalent to N rotations over angles π in each plane of an arbitrary set of N mutually orthogonal planes intersecting at P. These rotations are mutually commutative. Therefore inversion in a point in even-dimensional space is an orientation-preserving isometry or direct isometry.

In odd-dimensional Euclidean space, say (2N + 1)-dimensional space, it is equivalent to N rotations over π in each plane of an arbitrary set of N mutually orthogonal planes intersecting at P, combined with the reflection in the 2N-dimensional subspace spanned by these rotation planes. Therefore it reverses rather than preserves 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_nh and D_nh for even n
S_2n and D_nd 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.

Inversion symmetry

Inversion with respect to the origin

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