Improper rotation

Improper rotation

In 3D 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 about an axis and an inversion about the origin.

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 groups). The notation ar{n} 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 isometry, i.e., an element of "E"(3)"E"+(3) (see Euclidean group): it can also be a pure reflection in a plane, or have a 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 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 pseudoscalars, and in general; between tensors and pseudotensors), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).

ee also

*Isometry
*Orthogonal group


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Rotation (mathematics) — Rotation of an object in two dimensions around a point O. In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a… …   Wikipedia

  • Rotation group — This article is about rotations in three dimensional Euclidean space. For rotations in four dimensional Euclidean space, see SO(4). For rotations in higher dimensions, see orthogonal group. In mechanics and geometry, the rotation group is the… …   Wikipedia

  • Improper — ;In mathematics *Improper rotation *Improper integral *Improper fraction *Improper prior *Improper distribution *Improper point *Improper limits;Other *Improper English *Improper motion *Improper nounee also* Proper …   Wikipedia

  • Rotation matrix — In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian… …   Wikipedia

  • Rotation (disambiguation) — The term rotation can be used in several ways and includes various topics. Generally, it is used to denote (in 3D) the rotation of movement of a rigid body in such a way that any given point of that body remains at a constant distance from some… …   Wikipedia

  • Rotation in office — Rotation in office, or term limits, dates back to the American Revolution, and prior to that to the democracies and republics of antiquity. The council of 500 in ancient Athens rotated its entire membership annually, as did the ephorate in… …   Wikipedia

  • Improper motion — The improper motion of a star refers to the change of its coordinates on the sky not originating from the motion of the star itself, as opposed to proper motion. Obvious sources of improper motion include diurnal motion (caused by the rotation of …   Wikipedia

  • Wikipedia:Reference desk/Mathematics — The Wikipedia Reference Desk covering the topic of mathematics. Mathematics #eee #f5f5f5 #eee #aaa #aaa #aaa #00f #36b #000 #00f mathematics Wikipedia:Refere …   Wikipedia

  • Point groups in three dimensions — In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries… …   Wikipedia

  • Pseudovector — In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation (a transformation that can be expressed as an inversion …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”