Improper rotation

In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation. It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry.

There is a distinction between rotary reflection and rotary inversion symmetry operations and their associated symmetry elements. Rotary reflections are generally used to describe the symmetry of individual molecules and are defined as a 360°/n rotation about an n-fold rotation axis followed by a reflection over a mirror plane perpendicular to the n-fold rotation axis. Rotoinversions are generally used to describe the symmetry of crystals and are defined as a 360°/n rotation about an n-fold rotation axis followed by an inversion through the origin. Although rotary reflection operations have a rotoinversion analogue and vice versa, rotoreflections and rotoinversions of the same order need not be identical. For example, a 6-fold rotoinversion axis and its associated with symmetry operations are distinct from those resulting from a 6-fold reflection axis.