Improper rotation
Encyclopedia
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 a reflection in a plane perpendicular to the axis.
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 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 pseudovector
s (as well as scalars
and pseudoscalars, and in general; between tensor
s and pseudotensor
s), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
or affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
which is the combination of a rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
about an axis and a reflection in a plane perpendicular to the axis.
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
Angle of rotation
In mathematics, the angle of rotation is a measurement of the amount, the angle, that a figure is rotated about a fixed point, often the center of a circle....
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
Mirror image
A mirror image is a reflected duplication of an object that appears identical but reversed. As an optical effect it results from reflection off of substances such as a mirror or water. It is also a concept in geometry and can be used as a conceptualization process for 3-D structures...
. 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
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...
) denotes the symmetry group generated by an n-fold improper rotation (not to be confused with the same notation for symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
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 isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
, i.e., an element of E(3)\E+(3) (see Euclidean group
Euclidean group
In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
): it can also be a pure reflection in a plane, or have a glide plane
Glide reflection
In geometry, a glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line. Reversing the order of combining gives the same result...
. An indirect isometry is an affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
with an orthogonal matrix
Orthogonal matrix
In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
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 pseudovector
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...
s (as well as scalars
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
and pseudoscalars, and in general; between tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s and pseudotensor
Pseudotensor
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation preserving coordinate transformation , but gains an additional sign flip under an orientation reversing coordinate transformation In physics and mathematics, a pseudotensor is usually...
s), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).