Rotation-reflection axes
In geometry, an improper rotation, also called rotoreflection,rotary reflection, or rotoinversion is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to that axis.
In 3D, equivalently it is the combination of a rotation and an inversion in a point on the axis. Therefore it is also called a rotoinversion or rotary inversion. A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation.
In both cases the operations commute. Rotoreflection and rotoinversion are the same if they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection.
An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis. This is called an n-fold improper rotation if the angle of rotation is 360°/n. The notation Sn (German, Spiegel, for mirror) denotes the symmetry group generated by an n-fold improper rotation (not to be confused with the same notation for symmetric groups). The notation n is used for n-fold rotoinversion; i.e., rotation by an angle of rotation of 360°/n with inversion. The Coxeter notation for S2n is [2n+,2+], and orbifold notation is n×, order 2n.
The direct subgroup, index 2, is Cn, [n]+, (nn), order n, as the rotoreflection generator applied twice.
S2n for odd n contain inversion, with S2 = Ci is the group generated by inversion. S2n contain indirect isometries but not inversion for even n. In general, if odd p is a divisor of n, then S2n/p is a subgroup of S2n. For example S4 is a subgroup of S12.
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