Window function: Difference between revisions

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In [[geometry]], an '''improper rotation''', also called '''rotoreflection''' or '''rotary reflection''' is, depending on context, a [[linear transformation]] or [[affine transformation]] which is the combination of a [[Rotation (geometry)|rotation]] about an axis and a reflection in a plane.


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'''.
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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 '''''S<sub>n</sub>''''' (''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 <math>\bar{n}</math> is used for '''''n''-fold rotoinversion''', i.e. rotation by an angle of rotation of 360°/''n'' with inversion.
In a wider sense, an "improper rotation" may be defined as any '''[[Euclidean group#Direct and indirect isometries|indirect isometry]]''', i.e., an element of ''E''(3)\''E''<sup>+</sup>(3) (see [[Euclidean group]]): thus it can also be a pure reflection in a plane, or have a [[glide reflection|glide plane]]. 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 defined as a '''direct isometry''', i.e., an element of ''E''<sup>+</sup>(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 either the narrower or the wider senses, the composition of two improper rotations is a proper rotation, and the composition 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 [[vector (geometry)|vector]]s and [[pseudovector]]s (as well as [[scalar (mathematics)|scalars]] and [[pseudoscalar (mathematics)|pseudoscalar]]s, and in general; between [[tensor]]s and [[pseudotensor]]s), since the latter transform differently under proper and improper rotations (in 3 dimensions, pseudovectors are invariant under inversion).
 
==See also==
*[[Isometry]]
*[[Orthogonal group]]
 
[[Category:Euclidean symmetries]]
[[Category:Lie groups]]

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