Improper rotation - Citizendia

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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

Equivalently it is the combination of a rotation and an inversion in a point on the axis. 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 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" 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 S_n (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). A mirror is an object with a surface that has good Specular reflection; that is it is smooth enough to form an Image. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying The notation $\bar{n}$ 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. For the Mechanical engineering and Architecture usage see Isometric projection. 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. In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional 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 An indirect isometry is an affine transformation with an orthogonal matrix that has a determinant of −1. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T

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). In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Physics, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as Improper rotations History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Physics and Mathematics, a pseudotensor is usually a quantity that transforms like a Tensor under a Proper rotation, but gains an additional

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