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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'''. 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''''' (''S'' for ''Spiegel'', German for Mirror ) denotes the symmetry group generated by an ''n''-fold improper rotation (not to be confused with the same notation for Symmetric Group s). The notation is used for '''''n''-fold rotoinversion''', i.e. rotation by an angle of rotation of 360°/''n'' with inversion. In the wider sense, an improper rotation is an '''indirect . An indirect isometry is an Affine Transformation with an Orthogonal Matrix that has a determinant of −1. A proper rotation is an ordinary rotation. In the wider sense, a proper rotation is a '''direct isometry''', i.e., an element of ''E''+(3): it can also be the identity, a rotation with a translation along the axis, or a pure translation. A direct isometry is an affine transformation with an orthogonal matrix that has a determinant of 1. In the wider senses, the composition of two improper rotations is a proper rotation, and the product of an improper and a proper rotation is an improper rotation. When studying the symmetry of a physical system under an improper rotation (e.g., if a system has a mirror symmetry plane), it is important to distinguish between vectors and Pseudovectors (as well as Scalars and Pseudoscalar s, and in general; between Tensor s and Pseudotensor s), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion). SEE ALSO |
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