In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space...

term rotation is generalized to include improper rotations, characterized by orthogonal matrices with a determinant of −1 (instead of +1). An improper rotation...

The improper rotation is represented by the symbol Sn where n is the order. Since the improper rotation is the combination of a proper rotation and a...

pseudoscalars and pseudotensors, both of which gain an extra sign-flip under improper rotations compared to a true scalar or tensor. Physical examples of pseudovectors...

the transformation. But a (proper) rotation also has to preserve the orientation structure. The "improper rotation" term refers to isometries that reverse...

do not form a group because the product of two improper rotations is a proper rotation. The rotation group is a group under function composition (or...

centers of inversion, C for proper rotation axes, σ for mirror planes, and S for improper rotation axes (rotation-reflection axes). C and S are usually...

particular, a symmetry element can be a mirror plane, an axis of rotation (either proper and improper), or a center of inversion. For an object such as a molecule...

groups section. The improper rotation groups are denoted by Sn. These groups are characterized by an n-fold improper rotation axis Sn, where n is necessarily...

theorem). In dimension at most three, any improper rigid transformation can be decomposed into an improper rotation followed by a translation, or into a sequence...

see how our definition of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the...

has the symmetry of a tetrahedron. Td includes improper rotation operations, T excludes improper rotation operations, and Th is T with the addition of an...

\operatorname {cl} S} . In crystallography, an overline indicates an improper rotation or a negative number: 3 ¯ {\displaystyle {\overline {3}}} is the Hermann–Mauguin...

g. a proper rotation) but additionally changes sign under an orientation-reversing coordinate transformation (e.g., an improper rotation), which is a...

the sign of the third vector if improper rotations are allowed). * Quaternions can be implicitly converted to a rotation-like matrix (12 multiplications...

rotation, of a computer monitor or display Earth's rotation Improper rotation or rotoreflection, a rotation and reflection in one Internal rotation,...

this way, it is unwise to plan crops years in advance. Improper implementation of a crop rotation plan may lead to imbalances in the soil nutrient composition...

Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations...

(E), rotation operation or proper rotation (Cn), reflection operation (σ), inversion (i) and rotation reflection operation or improper rotation (Sn)....

same as a half-turn rotation (180° or π radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances...

are mirror images one of the other (a glide plane or a four-fold improper rotation takes one into the other). There are four sodium ions around each...

classified as either proper or improper rotations. For linear molecules, either clockwise or counterclockwise rotation about the molecular axis by any...

(often called "pure") rotation symmetry. No inversion, reflection, roto-inversion or roto-reflection (i.e., improper rotation) symmetry exists in such...

a single reflection, a rotation (2 reflections), or an improper rotation (3 reflections). In four dimensions, double rotations are added that represent...

inversion point Improper rotation, which consists of a rotation about an axis followed by an inversion. Rotation axes (proper and improper), reflection planes...

(mathematics) Curl (mathematics) Fleming's left-hand rule for motors Improper rotation ISO 2 Oersted's law Poynting vector Pseudovector Reflection (mathematics)...

centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane...

orthogonal group. The Euclidean group E(n) comprises all translations, rotations, and reflections of E n {\displaystyle \mathbb {E} ^{n}} ; and arbitrary...

element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1). The geometric...

Reflection in a plane through the origin, denoted σ. Improper rotation, also called rotation-reflection: rotation about an axis by an angle θ, combined with reflection...