Editing Improper rotation (section)

==As an indirect isometry ==

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 [[Euclidean group|E]](3)\E<sup>+</sup>(3): 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.