This definition is not equivalent to that requiring a sign flip under improper rotations, but it is general to all vector spaces.
12.
Improper rotations correspond to orthogonal matrices with determinant, and they do not form a group because the product of two improper rotations is a proper rotation.
13.
Improper rotations correspond to orthogonal matrices with determinant, and they do not form a group because the product of two improper rotations is a proper rotation.
14.
Coxeter's bracket notation is also included, based on reflectional Coxeter groups, and modified with plus superscripts accounting for rotations, improper rotations and translations.
15.
When describing the improper rotation symmetry element ( in 3 dimensional Euclidean space ) it is always specified that the reflection plane be perpendicular to the rotation axis.
16.
Orthogonal transformations in two-or three-dimensional Euclidean space are stiff reflections, or combinations of a rotation and a reflection ( also known as improper rotations ).
17.
The label'pseudo'can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor.
18.
The rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number.
19.
In some literature, the term " rotation " is generalized to include improper rotations, characterized by orthogonal matrices with determinant " 1 ( instead of + 1 ).
20.
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.
