rotoreflection

Rotoreflection is a geometric transformation that combines rotation and reflection in a single operation. It is a type of isometry, meaning it preserves distances between points in space, and is commonly studied in the context of symmetry operations in mathematics and physics. The concept is particularly relevant in the analysis of molecular symmetry, crystallography, and group theory.

In two-dimensional space, rotoreflection can be visualized as a rotation about a fixed point followed by a reflection across a line passing through that point. The combined effect depends on the angle of rotation and the orientation of the reflection line. For example, a rotoreflection by an angle of 180 degrees (π radians) is equivalent to a simple reflection across a line through the center of rotation. In three dimensions, rotoreflection involves rotating an object around an axis followed by reflecting it across a plane perpendicular to that axis.

Rotoreflections are classified under the broader category of improper rotations, which also include inversion (a 180-degree rotation combined with reflection through the origin). In group theory, rotoreflections are represented by elements of dihedral groups and other symmetry groups, where they play a key role in defining the symmetry properties of geometric shapes and physical systems. For instance, the dihedral group *Dn* describes the symmetries of a regular *n*-sided polygon, including rotoreflections that map the polygon onto itself.

The mathematical representation of a rotoreflection can be expressed using matrices or complex numbers. In two dimensions, a rotoreflection by an angle *θ* about the origin can be represented as a composition of a rotation matrix and a reflection matrix. The resulting transformation matrix combines both operations into a single linear transformation. Rotoreflections are also fundamental in the study of point groups and space groups, where they help classify the symmetry elements of crystals and molecules.