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...

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

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...

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...

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...

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...

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

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...

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

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...

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

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

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...

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

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

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

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

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...

restricted rotation about a single bond (barrier to rotation ≥ ca. 23 kcal/mol) are said to exhibit atropisomerism. A chiral compound can contain no improper axis...

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

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...

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...

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

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

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