rotoinversion

Rotoinversion is a geometric operation in three-dimensional space that combines a rotation with a simultaneous inversion through a central point. It is a type of improper rotation, meaning it is not a pure rotation. The operation can be visualized as rotating an object by a certain angle around an axis and then reflecting all points of the object through the origin, also known as a point reflection. Mathematically, rotoinversion is represented by a matrix that is the product of a rotation matrix and an inversion matrix. The inversion matrix has negative values on the diagonal and zeros elsewhere. The order of operations, whether rotation followed by inversion or vice versa, does not affect the final result. Rotoinversion is a symmetry operation found in crystallography and molecular symmetry. For example, a molecule or crystal lattice can possess rotoinversion symmetry if it looks identical after undergoing this combined transformation. The angle of rotation and the presence of inversion define the specific rotoinversion operation, often denoted as S_n, where n relates to the rotation angle. For instance, S2 is equivalent to simple inversion.