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Rotation composed with a reflection
In geometry, an improper rotation (also called rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space
Improper_rotation
Geometric transformation which produces an identical image
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
Symmetry_operation
Matrix representing a Euclidean rotation
term rotation is generalized to include improper rotations, characterized by orthogonal matrices with a determinant of −1 (instead of +1). An improper rotation
Rotation_matrix
Group of rotations in 3 dimensions
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
3D_rotation_group
Physical quantity that changes sign with improper rotation
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
Pseudovector
Motion of a certain space that preserves at least one point
the transformation. But a (proper) rotation also has to preserve the orientation structure. The "improper rotation" term refers to isometries that reverse
Rotation_(mathematics)
Movement with a fixed point is rotation
see how our definition of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the
Euler's_rotation_theorem
Horizontal line immediately above a portion of writing
\operatorname {cl} S} . In crystallography, an overline indicates an improper rotation or a negative number: 3 ¯ {\displaystyle {\overline {3}}} is the Hermann–Mauguin
Overline
Type of physical quantity
g. a proper rotation) but additionally changes sign under an orientation-reversing coordinate transformation (e.g., an improper rotation), which is a
Pseudotensor
Point, line, or plane about which a molecule or crystal is symmetric
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
Symmetry_element
Notation to represent symmetry in point groups
also inversion center and improper rotation operations. I (the chiral icosahedral group) indicates that the group has the rotation axes of an icosahedron
Schoenflies_notation
Classification system for crystals
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
Crystallographic_point_group
Mathematical transformation that preserves distances
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
Rigid_transformation
Symmetry of molecules of chemical compounds
perpendicular to it, leaves the molecule unchanged. Also called an n-fold improper rotation axis, it is abbreviated Sn. Examples are present in tetrahedral silicon
Molecular_symmetry
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
List of character tables for chemically important 3D point groups
List_of_character_tables_for_chemically_important_3D_point_groups
Geometric symmetry operation
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
Point_reflection
Groups of point isometries in 3 dimensions
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
Point groups in three dimensions
Point_groups_in_three_dimensions
Branch of mathematics that studies the properties of groups
(E), rotation operation or proper rotation (Cn), reflection operation (σ), inversion (i) and rotation reflection operation or improper rotation (Sn).
Group_theory
Correspondence between quaternions and 3D rotations
the sign of the third vector if improper rotations are allowed). * Quaternions can be implicitly converted to a rotation-like matrix (12 multiplications
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Geometric property of some molecules and ions
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
Chirality_(chemistry)
Agricultural practice of changing crops
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
Crop_rotation
Mnemonic for 3D vectors orientations and rotations
(mathematics) Curl (mathematics) Fleming's left-hand rule for motors Improper rotation ISO 2 Oersted's law Poynting vector Pseudovector Reflection (mathematics)
Right-hand_rule
Symmetry group of a configuration in space
centering), the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane
Space_group
Linear algebra operation
Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations
Orthogonal_transformation
Isometry group of Euclidean space
orthogonal group. The Euclidean group E(n) comprises all translations, rotations, and reflections of E n {\displaystyle \mathbb {E} ^{n}} ; and arbitrary
Euclidean_group
Blue tectosilicate mineral
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
Sodalite
Group of geometric symmetries with at least one fixed point
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
Point_group
Type of symmetry
(often called "pure") rotation symmetry. No inversion, reflection, roto-inversion or roto-reflection (i.e., improper rotation) symmetry exists in such
Centrosymmetry
a single reflection, a rotation (2 reflections), or an improper rotation (3 reflections). In four dimensions, double rotations are added that represent
Cartan–Dieudonné_theorem
Property of an object that is not congruent to its mirror image
plane of symmetry S1, an inversion center of symmetry S2, or a higher improper rotation (rotoreflection) Sn axis of symmetry is achiral. (A plane of symmetry
Chirality_(mathematics)
Shape with seven sides
elements are: a 7-fold proper rotation axis C7, a 7-fold improper rotation axis, S7, 7 vertical mirror planes, σv, 7 2-fold rotation axes, C2, in the plane of
Heptagon
Ordered arrangement of atoms, ions, or molecules in a crystalline material
inversion point Improper rotation, which consists of a rotation about an axis followed by an inversion. Rotation axes (proper and improper), reflection planes
Crystal_structure
2024 aviation accident in Nepal
were analyzed by the AAIC to determine the performance of flight crew. Rotation rates of above 3° per second were common, with 18 recorded instances of
2024 Saurya Airlines Bombardier CRJ200 crash
2024_Saurya_Airlines_Bombardier_CRJ200_crash
Notation to represent symmetry in point groups, plane groups and space groups
symmetry axes. Rotation axes are denoted by a number n – 1, 2, 3, 4, 5, 6, 7, 8, ... (angle of rotation φ = 360°/n). For improper rotations, Hermann–Mauguin
Hermann–Mauguin_notation
Movement of an object which leaves at least one point unchanged
orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as a proper rotation (from which an axis of rotation can be found as described
Rotation
Vector satisfying some of the criteria of an eigenvector
Pseudo-Euclidean space Null vector Indefinite orthogonal group Orientation Improper rotation Symplectic structure Multilinear algebra Multilinear algebra Tensor
Generalized_eigenvector
Transformation of a geometric space preserving structure
known as proper or rigid) and indirect (or improper) motions. Direct motions include translations and rotations, which preserve the orientation of a chiral
Motion_(geometry)
Group of transformations under which the object is invariant
orientation-reversing isometries (reflections, glide reflections and improper rotations), as long as those isometries map this particular X to itself. The
Symmetry_group
Designation of positions within a crystallographic space group
(such as inversion points, improper rotation points, and intersections of rotation axes with mirror planes or other rotation axes), but other designations
Wyckoff_positions
Ligands that lack mirror symmetry but have two-fold rotational symmetry
asymmetric induction only requires that the ligand be chiral (i.e. have no improper rotation axis). Asymmetry (i.e. absence of any symmetry elements) is not required
C2-Symmetric_ligands
Geometrical property
cyclic group Z. In 3D, a rotary reflection, rotoreflection or improper rotation is a rotation about an axis combined with reflection in a plane perpendicular
Symmetry_(geometry)
Choice of reference for distinguishing an object and its mirror image
quantity that changes sign with improper rotation Rotation formalisms in three dimensions – Ways to represent 3D rotationsPages displaying short descriptions
Orientation_(vector_space)
Mathematical descriptions of a rotation group
reverses orientation is called an improper rotation. Every improper rotation of three-dimensional Euclidean space is a rotation followed by a reflection in
Charts_on_SO(3)
Mapping from a Euclidean space to itself
rotation. Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is
Reflection_(mathematics)
any composition of these basic transformations, including rotations and improper rotations and most generally similarity transformations. However, shear
Conformal linear transformation
Conformal_linear_transformation
Topics referred to by the same term
rotation, of a computer monitor or display Earth's rotation Improper rotation or rotoreflection, a rotation and reflection in one Internal rotation,
Rotation_(disambiguation)
Problem in physics and celestial mechanics
"The second orthogonality conditions in the theory of proper and improper rotations. II. The intrinsic vector". Journal of Research of the National Bureau
N-body_problem
Family of linear transformations
makes them improper transformations. If Λ is a proper orthochronous Lorentz transformation, then TΛ is improper antichronous, PΛ is improper orthochronous
Lorentz_transformation
Conclusion of the human pregnancy
Mechanical fetal injury may be caused by improper rotation of the fetus.
Childbirth
Feature of a system that is preserved under some transformation
Spatial rotation: These spatial symmetries are classified as proper rotations and improper rotations. The former are just the 'ordinary' rotations; mathematically
Symmetry_(physics)
space Null vector Indefinite orthogonal group Orientation (geometry) Improper rotation Symplectic structure Multilinear algebra Tensor Classical treatment
Outline_of_linear_algebra
Classification of a two-dimensional repetitive pattern
groups, and modified with plus superscripts accounting for rotations, improper rotations and translations. An orbifold can be viewed as a polygon with
Wallpaper_group
Classification system for symmetry groups in geometry
ρ1ρ2 is a rotation of angle π/q, and σ0,2 = ρ0ρ2 represents a rotation of angle π/2. [p,q]+, , is an index 2 subgroup represented by two rotation generators
Coxeter_notation
Class of molecular symmetry
Cartesian coordinate system. n-fold rotation–reflection, or improper rotation, Sn about an n-fold axis of improper rotation is composed of two successive geometry
Symmetry of diatomic molecules
Symmetry_of_diatomic_molecules
Space formed by the ''n''-tuples of real numbers
it preserves orientation, while for odd n it is reversed (see also improper rotation). Rn understood as an affine space is the same space, where Rn as
Real_coordinate_space
called 2n-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle
Cyclic symmetry in three dimensions
Cyclic_symmetry_in_three_dimensions
Chemical compound
threefold improper rotational symmetry and a mirror plane. The threefold rotational axis (which is normal to the layers) is termed improper because the
Lithium_cobalt_oxide
Line or vector perpendicular to a curve or a surface
mathematics Pseudovector – Physical quantity that changes sign with improper rotation Tangential and normal components – Mathematical vector components
Normal_(geometry)
Crystallographic law
illustrated in the figure. Rotoinversion is also known as improper rotation, rotoreflection, or rotation-reflection. René Just Haüy showed in 1784 that the law
Law of symmetry (crystallography)
Law_of_symmetry_(crystallography)
Operator in quantum field theory
related to spin Pseudovector – Physical quantity that changes sign with improper rotation Pseudotensor – Type of physical quantity Induced representation –
Pauli–Lubanski_pseudovector
Polyhedral compound
antiprisms). It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide. This compound
Compound_of_twenty_octahedra
Tensor operator generalizes the notion of operators which are scalars and vectors
pseudovector operators are invariant up to a sign: +1 for proper rotations and −1 for improper rotations. Since operators can be shown to form a vector operator
Tensor_operator
Method to avoid stock loss
is improperly rotated, newer stock will be sold first, leaving out of date stock sitting on the shelves which will have to be thrown away. Rotation also
Stock_rotation
Polyhedral compound
axis of rotational symmetry. It arises from superimposing two copies of the corresponding prismatic compound of antiprisms (without rotational freedom)
Prismatic compound of antiprisms with rotational freedom
Prismatic_compound_of_antiprisms_with_rotational_freedom
Group of symmetries of an n-dimensional hypercube
sends each vertex to the opposite vertex, and various improper rotations (a combination of a rotation about an axis and a reflection in a plane perpendicular
Hyperoctahedral_group
Distribution of an uncertain quantity
prior is called an improper prior. However, the posterior distribution need not be a proper distribution if the prior is improper. This is clear from
Prior_probability
Space in mathematics and theoretical physics
is assumed. What is the "rotations group" depends on exact definition of a rotation. "O" groups contain improper rotations. Transforms that preserve
Pseudo-Euclidean_space
Polyhedral compound
The compound of twenty octahedra with rotational freedom is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra, considered
Compound of twenty octahedra with rotational freedom
Compound_of_twenty_octahedra_with_rotational_freedom
Polyhedral compound
tetrahedra with rotational freedom will have a convex hull of a nonuniform truncated cuboctahedron Compounds of twelve tetrahedra with rotational freedom Stellated
Compound of twelve tetrahedra with rotational freedom
Compound_of_twelve_tetrahedra_with_rotational_freedom
Polyhedral compound
The compound of six tetrahedra with rotational freedom is a uniform polyhedron compound made of a symmetric arrangement of 6 tetrahedra, considered as
Compound of six tetrahedra with rotational freedom
Compound_of_six_tetrahedra_with_rotational_freedom
1979 aviation accident in Chicago, Illinois
Safety Board (NTSB) found that as the aircraft was beginning its takeoff rotation, engine number one (the left engine) separated from the left wing, flipping
American_Airlines_Flight_191
Inability of a person to correctly determine their body position in space
strong visual cues. In aviation, spatial disorientation can result in improper perception of the attitude of the aircraft, referring to the orientation
Spatial_disorientation
Polyhedral compound
(without rotational freedom). This compound shares its vertices with the compound of twelve pentagrammic crossed antiprisms with rotational freedom. Cartesian
Compound of twelve pentagonal antiprisms with rotational freedom
Compound_of_twelve_pentagonal_antiprisms_with_rotational_freedom
American astronomer (1928–2016)
galaxy rotation rates. She uncovered the discrepancy between the predicted and observed angular motion of galaxies by studying galactic rotation curves
Vera_Rubin
Matrix realization of the Clifford algebra
space is invariant under proper rotations, and so the two spaces are spinorially equivalent. However, if improper rotations are also taken into consideration
Weyl–Brauer_matrices
2013 aviation accident in Afghanistan
all seven people on board. The subsequent investigation concluded that improperly secured cargo broke free during the take-off and rolled to the back of
National_Airlines_Flight_102
Circulation density in a vector field
alternative notation rot F is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the
Curl_(mathematics)
Measure of observed changes in the apparent locations of stars
proper motion reveal larger structures like stellar streams, the general rotation of the Milky Way disk, and the random motions of stars in the Galactic
Proper_motion
Integration method to calculate volume
the distance between the function and the axis of rotation. This works only if the axis of rotation is horizontal (example: y = 3 or some other constant)
Disc_integration
Representation of a tensor in Euclidean space
corresponds to two types of orthogonal transformation: (+1) for rotations and (−1) for improper rotations (including reflections). There are considerable algebraic
Cartesian_tensor
2007 aviation accident in South Africa
International Airport to O. R. Tambo International Airport. During the rotation from runway 01, engine No.2 detached from the right wing. The pilots then
Nationwide Airlines Flight 723
Nationwide_Airlines_Flight_723
Disease of the feet of hooved animals
into the hoof. Endocrinopathy Endocrinopathy is usually the result of improper insulin regulation, and is most commonly seen with pituitary pars intermedia
Laminitis
Permanent indentation of a hard surface
very close proximity. The very small indentations can quickly lead to improper operation, such as chattering or excess vibration, which in turn can accelerate
Brinelling
Fastener consisting of a U-shaped bracket through which a pin is placed
a fastener, a clevis provides a method of allowing rotation in some axes while restricting rotation in others. There are two main types of clevis pins:
Clevis_fastener
Kind of motor used in drilling
compressed air, or other gas, can be used for mud motor input power. Normal rotation of the bit while using a mud motor can be from 60 rpm to over 100 rpm.
Mud_motor
Difference in shape from a mirror image
symmetry. In terms of point groups, all chiral figures lack an improper axis of rotation (Sn). This means that they cannot contain a center of inversion
Chirality
Deformity of the hip
bone), post-Perthes deformity, osteomyelitis, and post traumatic (due to improper healing of a fracture between the greater and lesser trochanter). Shepherd's
Coxa_vara
Dwarf planet with a ring and two moons
that is now unclear. Haumea's elongated shape, together with its rapid rotation, rings, and high albedo (from a surface of crystalline water ice), is thought
Haumea
Type of fibrous joint
with a cast or splint, contractile activity by these muscles can cause improper alignment of the broken bones during healing. Betts, J. Gordon (2013).
Syndesmosis
Feeding livestock on forage
seasonal, or rotational within a grazing period. Longer rotations are found in ley farming, alternating arable and fodder crops; in rest rotation, deferred
Grazing
1995 aviation accident in Missouri
continue takeoff when the aircraft was below the calculated rotation airspeed. ATI had provided improper training, which resulted in them failing to understand
Air Transport International Flight 782
Air_Transport_International_Flight_782
while a skater is in motion. Regular wheel rotation helps distribute wear evenly, similar to tire rotation in car. This practice prevents any single wheel
Inline_skate_tuning
American baseball pitcher (1977–2017)
resurrecting his career. By mid-season, he was back in the Blue Jays' rotation. He posted a 5–3 win–loss record with a 3.19 ERA for the Blue Jays in 16
Roy_Halladay
Maintenance of motor vehicle tires
and fast cornering, constant heavy cargo transportation and rough roads, improper pressure (under- or overinflation) as the tire with low pressure cause
Tire_maintenance
Form of train incident
vertical bounce, track shift under a train, excessive speed) improper operation of points, or improper observance of signals protecting them (signal errors)
Derailment
Angle between the vertical axis and the steering axis of a steered wheel, in side view
Caster displacement moves the steering axis ahead of the axis of wheel rotation, as with the front wheels of a shopping cart. Caster angle moves the steering
Caster_angle
International motorcycle racing
riders, one for manufacturers and, as of 2024, a Teams World Championship.[improper synthesis? (See discussion.)] The motorcycles that race in the championship
Superbike_World_Championship
initially scheduled to be held before 12 June 2027; however, the planned rotation agreed to by GERB and PP–DB failed to materialise in March 2024, and no
June 2024 Bulgarian parliamentary election
June_2024_Bulgarian_parliamentary_election
2000 aviation accident in California
the three-member crew was not injured. Post-accident investigation found improper maintenance to the left main landing gear was at fault. Emery Worldwide
Emery Worldwide Airlines Flight 17
Emery_Worldwide_Airlines_Flight_17
Human postural problem
shoulder. It is commonly believed that digitalisation combined with the improper use of digital devices have resulted in the prevalence of sedentary lifestyles
Rounded_shoulder_posture
IMPROPER ROTATION
IMPROPER ROTATION
Boy/Male
Muslim
Proper Name.
Boy/Male
Muslim
Proper
Boy/Male
Hindu, Indian
Proper
Girl/Female
African, Arabic, French, Indian, Muslim, Sindhi
Right and Proper; Suitable; Proper
Boy/Male
Tamil
Yetharth | யேதாரà¯à®¤Â
Proper, Possibility
Yetharth | யேதாரà¯à®¤Â
Boy/Male
Hindu
Proper, Possibility
Girl/Female
Muslim
Proper Name.
Boy/Male
Muslim
Proper Name.
Girl/Female
Indian
Improper, Fear-causing
Girl/Female
Muslim
Proper Name.
Girl/Female
Indian
Right, Proper
Boy/Male
Tamil
Yatharth | யதாரà¯à®¤Â
Proper, Possibility
Yatharth | யதாரà¯à®¤Â
Boy/Male
Hindu
Proper, Possibility
Boy/Male
Indian
Proper
Boy/Male
Muslim
Proper Name.
Girl/Female
Tamil
Improper, Fear-causing
Girl/Female
Muslim
Proper name
Girl/Female
Muslim
Right, Proper
Girl/Female
Muslim
Proper Name.
Boy/Male
Tamil
Selven | ஸேலà¯à®µà¯‡à®¨Â
Proper
IMPROPER ROTATION
IMPROPER ROTATION
Boy/Male
Tamil
Ayobaahu | அயோபாஹà¯à®‚
One of the kauravas
Female
Hebrew
 Variant spelling of Hebrew Alma, ALMAH means "maiden." Compare with another form of Almah.
Girl/Female
Tamil
Girl/Female
Bengali, Hindu, Indian
A Desert Plant
Girl/Female
American, British, Chinese, Christian, Danish, Dutch, English, French, German, Greek, Jamaican, Latin, Norse, Scandinavian, Swedish
Follower of Christ; Stone Church; The Anointed One; Variant Form of Christine; Christian Woman
Boy/Male
Indian
Servant of God (Allah)
Boy/Male
Arabic
Table Companion; Associate
Surname or Lastname
English (Kent)
English (Kent) : apparently a habitational name from a lost or unidentified place, but possibly a variant of Moorfoot (see Morfitt).
Boy/Male
Arabic, Muslim
Blessing of Allah
Boy/Male
American, Anglo, Australian, British, Christian, Danish, Dutch, English, Finnish, French, German, Swedish, Swiss
Red Wolf; Wolf Counsel; Wise Wolf
IMPROPER ROTATION
IMPROPER ROTATION
IMPROPER ROTATION
IMPROPER ROTATION
IMPROPER ROTATION
n.
Wrong or improper pronunciation.
a.
Not peculiar or appropriate to individuals; general; common.
v. i.
To grow better; to advance or make progress in what is desirable; to make or show improvement; as, to improve in health.
v. t.
To use or employ to good purpose; to make productive; to turn to profitable account; to utilize; as, to improve one's time; to improve his means.
a.
Rightly so called; strictly considered; as, Greece proper; the garden proper.
a.
Belonging to the natural or essential constitution; peculiar; not common; particular; as, every animal has his proper instincts and appetites.
v. t.
To make better; to increase the value or good qualities of; to ameliorate by care or cultivation; as, to improve land.
a.
Not according to facts; inaccurate; erroneous.
a.
Befitting one's nature, qualities, etc.; suitable in all respect; appropriate; right; fit; decent; as, water is the proper element for fish; a proper dress.
n.
One who, or that which, improves.
v. t.
To disapprove; to find fault with; to reprove; to censure; as, to improve negligence.
a.
Pertaining to one of a species, but not common to the whole; not appellative; -- opposed to common; as, a proper name; Dublin is the proper name of a city.
a.
Not proper; not suitable; not fitted to the circumstances, design, or end; unfit; not becoming; incongruous; inappropriate; indecent; as, an improper medicine; improper thought, behavior, language, dress.
imp. & p. p.
of Improve
a.
Not proper or peculiar; improper.
adv.
In an improper manner; not properly; unsuitably; unbecomingly.
n.
Improper.
v. i.
To increase; to be enhanced; to rise in value; as, the price of cotton improves.
adv.
Properly; hence, to a great degree; very; as, proper good.
v. t.
To appropriate; to limit.