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Improper rotation
 
 
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In 3D geometry
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 is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication....
 or affine transformation
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
 which is the combination of a rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
 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
Inversion in a point

In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*....
 on the axis. Therefore it is also called a rotoinversion or rotary inversion.

In both cases the operations commute.






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In 3D geometry
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 is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication....
 or affine transformation
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
 which is the combination of a rotation
Rotation

A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A Three-dimensional space object rotates around a line called an axis....
 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
Inversion in a point

In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*....
 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
Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
. 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 with one surface polished, which leads to reflection and another opaque. The most familiar type of mirror is the plane mirror, which has a flat surface....
) 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 on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
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, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
, 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. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
): 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....
. An indirect isometry is an affine transformation
Affine transformation

In geometry, an affine transformation or affine map or an affinity between two vector spaces consists of a linear transformation followed by a translation :...
 with an orthogonal matrix
Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...
 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 , i.e. a transformation that rotates vectors and pseudovectors by an arbitrary angle about an arbitrary axis, but gains an additional sign flip under an improper rotation: a transformation that can be expressed as a proper rotation...
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

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
s and pseudotensor
Pseudotensor

In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under a proper rotation, but gains an additional sign flip under an improper rotation ....
s), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).

See also

  • Isometry
    Isometry

    In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
  • Orthogonal group
    Orthogonal group

    In mathematics, the orthogonal group of degree n over a field F is the group of n-by-n orthogonal matrix with entries from F, with the group operation that of matrix multiplication....