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Topic: Improper rotations

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Rotation group

Quaternions and spatial rotation

Rotation

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	Rotation group - Wikipedia, the free encyclopedia
		Such a rotation is a linear transformation that preserves the length of vectors, and also preserves the orientation, or handedness, of space.
		The composition of two rotations is a rotation, and every rotation has a unique inverse which is again a rotation.
		The group of all proper and improper rotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).
	en.wikipedia.org /wiki/SO(3) (1538 words)

	Improper rotation - Wikipedia, the free encyclopedia
		In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a perpendicular plane.
		Equivalently it is the combination of a rotation and an inversion in a point on the axis.
		In the wider sense, an improper rotation is an indirect isometry, i.e., an element of E(3)\E
	en.wikipedia.org /wiki/Improper_rotation (315 words)

	KU Symmetry in Crystallography Notes (Site not responding. Last check: 2007-10-07)
		An improper rotation may be thought of as occurring in two steps, first a proper rotation is performed, followed by an inversion through a particular point on the rotation axis.
		Note that it is not necessary for either the rotation operation or the inversion center to exist as an operation of the group for the improper rotation axis to exist, e.g.
		Rotation axes not in the equatorial plane are drawn with the symbol representing the type of axis at the projection point on the equatorial plane.
	xrayweb.msg.ku.edu /notes/symmetry.html (4414 words)

	Vector (spatial) - Wikipedia, the free encyclopedia
		Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field.
		rotation of a vector field results in a correspondingly rotated scalar field for the divergence and a correspondingly rotated vector field for the curl
		This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations.
	en.wikipedia.org /wiki/Vector_(spatial) (2834 words)

	I
		Improper rotations are composite symmetry operations, that is they consist of two symmetry operations performed in succession.
		rotation about the given improper rotation axis, followed by reflection through a plane perpendicular to the axis.
		Like improper rotation axes, glide planes are composite symmetry elements formed by combining the reflection operation and a translation.
	www.chemistry.ohio-state.edu /~woodward/ch754/sym_2d.htm (1524 words)

	English Papers, Term papers, Pg.8, 051022
		"Improper rotation is a combination of rotation followed by reflection in the plane perpendicular to the axis of rotation.
		The symmetry element associated with improper rotation is an axis, which is the same axis as that used for the rotation part of the improper rotation.
		Improper rotations are designated with the letter S. Reflection across the xy plane, which is perpendicular to the z axis doesn't move any of the atoms for this planar molecule."
	www.englishpapers.net /lib/essay/0_8.html (1431 words)

	NIST: Methane Symmetry Operations - Cont. 3D Rotation-Reflec.
		Rotations corresponding to the sense-reversing operations, as defined in (eq.
		This is just opposite to the situation to be encountered in Section 15, where rigid-body rotations of the methane molecule in free space are considered.
		Because a given symmetry species may occur more than once in a given rotational manifold, we follow a convention introduced by Jahn [6] and number ground state levels of identical J value and symmetry species with right superscripts (1), (2), etc., beginning with the member of each set at lowest energy.
	physics.nist.gov /Pubs/Methane/chap08.html (623 words)

	NIST: Methane Symmetry Operations - Improper rotations
		Improper rotations thus correspond to permutation-inversion operations, with the permuted indices related by (eq.
		Figure 4 illustrates: (a) an arbitrary instantaneous configuration of the methane molecule; (b) the transformation of vibrational displacement vectors required for the point group operation S
		The transformation of center-of-mass coordinates is not illustrated.
	physics.nist.gov /Pubs/Methane/chap043.html (416 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		In two dimensions, a rotation group of order $n$ (that is, consisting of rotations of $2\pi k/n$ radians for integers $k$) is denoted as the cyclic group $C_n$.
		If we include inversion (improper rotations) then we generate the dihedral group $D_n$, the group of symmetries of a regular $n$-gon.
		Suppose a wallpaper group has rotations of order $q$ and let $a$ be the shortest nonzero vector in L, the lattice group.
	splorg.org:8080 /~tobin/projects/symmetry/space.tex~ (621 words)

	Specification of Parameters in POLAR. (Site not responding. Last check: 2007-10-07)
		This set corresponds to reflection about the "x-y" plane, or rotation about the "z" axis by 180 degrees followed by inversion, that is, the set corresponds to an improper rotation.
		For non-chiral systems, such improper rotations are not important, for chiral systems the effect is to reverse the chirality.
		Fortunately, the operation to convert from an improper to a proper rotation is simple: all that is necessary is to reverse the sign of all the coefficients in an eigenvector.
	www.cachesoftware.com /mopac/Mopac2002manual/node270.html (666 words)

	ROT=n (C) (Site not responding. Last check: 2007-10-07)
		In the calculation of the rotational contributions to the thermodynamic quantities the symmetry number of the molecule must be supplied.
		The symmetry number of a point group is the number of equivalent positions attainable by pure rotations.
		This number cannot be assumed by default, and may be affected by subtle modifications to the molecule, such as isotopic substitution.
	www.cup.uni-muenchen.de /cicum/software/mopac7/node124.html (120 words)

	Vector - Wikipedia (Site not responding. Last check: 2007-10-07)
		Examples are "moving north at 90 m.p.h" or "pulling towards the center of Earth with a force of 70 Newtons".
		In differential geometry, physics, and engineering, the term vector usually refers quantities that are closely related to the coordinates from Euclidean space (or to tangent spaces of a differentiable manifold)—the notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations.
		This is a quantity that transforms like a three-vector under proper rotations, but gains an additional sign flip under improper rotations.
	wikipedia.findthelinks.com /ve/Vector.html (1624 words)

	III.C. CRYSTALS, SYMMETRY, AND DIFFRACTION
		The repeat distance between points in a particular row of the reciprocal lattice is inversely proportional to the interplanar spacing between the nets of the crystal lattice that are normal to this row of points.
		A four-fold rotation axis, parallel to c and through the origin of a tetragonal unit cell (a=b), moves a point at x, y, z to a point at (y, -x, z) by a rotation of 90 about the axis.
		This is the result of, first, a two-fold rotation about an axis through the origin and parallel to b (x, y, z to -x, y, -z) and then an inversion about the origin (-x, y, -z to x, -y, z).
	em-outreach.ucsd.edu /web-course/Sec-III.C.1-C.5/Sec-III.C.1-C.5.html (3671 words)

	Molecular Symmetry Examples
		The rotation axes and mirror planes are then represented according to predefined conventions that can be changed and tuned for each particular case.
		Inversion and reflection are particular cases of improper rotations.
		Crystallographers and solid state scientists in general prefer the use of the Herman-Mauguin or international notation, in which the improper rotations are rotation-inversion rather than rotation-reflection operations.
	www.uniovi.es /qcg/d-MolSym (1440 words)

	scalars (Site not responding. Last check: 2007-10-07)
		Scalar quantities have magnitude, but not a direction and should thus be distinguished from vectors.
		More formally, a scalar is a quantity that is invariant under coordinate rotations (or Lorentz transformations, for relativity).
		A related concept is a pseudoscalar, which is invariant under proper rotations but (like a pseudovector) flips sign under improper rotations.
	www.yourencyclopedia.net /Scalars (312 words)

	Concise Space-Group Symbols
		A is a superscript symbol denoting the axis of rotation
		The matrices for improper rotations (-1, -2, -3, -4 and -6) are identical except that the signs are reversed.
		The symbols for face-diagonal 2-fold rotations are 2' and 2".
	www.chem.gla.ac.uk /~louis/software/wingx/hlp/gen110.htm (993 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		For example, we may permute the verticies of a triangle, rotate a sphere through any arbitrary angle, or rotate a cube by a multiple of ninety degrees or invert it through its center; these are all symmetry operations.
		In general the symmetry group of a regular $n$-sided polygon (an ``$n$-gon'') is the dihedral group $D_n$, and the ``equivalent regions'' of that polygon are found by dividing the polygon into sectors, where every vertex and the midpoint of every edge lie on a sector boundary.
		If we designate the identity transformation as $e$, a quarter-turn rotation by $r$, mirroring along the two diagonals as $d_1, d_2$, and mirroring along horizontal and vertical bisectors as $b_1, b_2$, then we have $D_{4}=\{e, r, r^2, r^3, d_1, d_2, b_1, b_2\}$.
	splorg.org:8080 /~tobin/projects/symmetry/symmetry.tex,v (2749 words)

	Mineralogy Notes 2
		We will discuss symmetry groups made up of rotation and inversion operations only which are called the point groups, each of which is one of the 32 crystal classes.
		Following the rules of groups, there is a limited number of ways in which the 10 proper and im proper rotations can be combined to form groups, that is, there are 32 possible combinations to form groups.
		Each of the 10 allowed proper and improper rotations is, by itself, one of the 32 point groups, and we have seen stereographic projections of each of these.
	ruby.colorado.edu /~smyth/G30102.html (3477 words)

	Untitled (Site not responding. Last check: 2007-10-07)
		In this case because of the rather unusual nature of the rotation operators we shall derive the results for both operators just to be absolutely sure.
		An improper rotation can usually be expressed as the product of a proper rotation plus the inversion operator I.
		can be obtained by first calculating the character for the corresponding proper rotation and then multiply the result by i.
	pauline.berkeley.edu /textbook/prob2-10.html (427 words)

	ANR IMPACT - Successful Crop Rotations -- Key to Profitable Farming (Site not responding. Last check: 2007-10-07)
		The decision as to which crop to use for rotation depends on many factors, including soil type, microclimate and, most important, the presence or absence of plant diseases.
		If a farmer chooses the wrong rotation, the result is likely to be substantial economic loss or even crop failure.
		UCCE farm advisors have studied various crop rotations in the Imperial Valley, keeping a record of potentially damaging conditions to be avoided.
	ucanr.org /delivers/impactview.cfm?impactnum=428&mainunitnum=1086 (382 words)

	1. Introduction to Molecular Symmetry (Site not responding. Last check: 2007-10-07)
		Note that the ‘v’ and ‘h’ labelling refers to the relationship between the planes and the principal rotation axis not to the plane of the molecule.
		The principal rotation axis of the benzene molecule is a C
		Improper axes are often the most difficult symmetry elements to spot.
	www.hull.ac.uk /php/chsajb/symmetry&spectroscopy/ho_1.html (1962 words)

	Untitled Normal Page (Site not responding. Last check: 2007-10-07)
		Improper crop rotations As a result of population growth, land shortage and economic pressures, farmers in some areas have adopted cereal-based, intensive crop rotations, based particularly on rice and wheat, in place of the more balanced cereal-legume rotations that were formerly found.
		This is a contributory cause of soil fertility decline.
		Because of such pressure in the short term, labour, land and capital resources cannot be spared to care for the land, for example green manuring or soil conservation structures.
	www.fao.org /docrep/V4360E/V4360E08.htm (3390 words)

	Anorganische Chemie Arbeitsgruppe Deiseroth (Uni-GH-Siegen)
		These can include rotation axes, mirror planes and inversion centers.
		Rotation through an angle of 360/n degrees, where n is an integer.
		Improper rotations are regular rotations followed by a reflection in the plane perpendicular to the axis of rotation.
	www.uni-siegen.de /~anchem/lehre/talksws0304/montemayor_Symmetry.htm (439 words)

	Improper rotation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		Improper rotation -- Facts, Info, and Encyclopedia article
		An improper rotation of an object thus produces a rotation of its (A likeness in which left and right are reversed) mirror image.
		Improper rotations can be (Click link for more info and facts about represented) represented by 3×3 (Click link for more info and facts about orthogonal matrices) orthogonal matrices with (A determining or causal element or factor) determinants of −1.
	www.absoluteastronomy.com /encyclopedia/i/im/improper_rotation.htm (177 words)

	Kalée Gregory: Symmetry Module II
		Recall that you do an improper rotation Sn by rotating a molecule 360/n degrees about a Cn axis and then reflecting it through a plane perpendicular to that axis.
		The most common kind of improper rotation axis is an S1 axis, which is the same thing as a mirror plane.
		By coincidence, the crystals themselves were handed too, and Pasteur was able physically to separate the ones that rotated light clockwise from the ones that rotated it counterclockwise.
	kalee.tock.com /portfolio/sym2.html (1740 words)

	Symmetry and Group Theory Lecture Notes
		rotation by 360°/n about a particular axis defined as the n-fold rotation axis.
		If plane is parallel to the principle rotation axis, but bisects angle between 2 C
		The corresponding operation is rotation of the molecule by 180° about an axis.
	www.newark.rutgers.edu /~cheminfo/undergrad/chem207/SymmetryGroupTheory.html (600 words)

	Description of a Magnetic Structure
		Each of the elements of the magnetic group acts on the magnetic moment with the rotation and translation appropriate to the corresponding element in the crystallographic group.
		and are the rotation and translation operators associated with one of the elements in the magnetic group and
		One must remember that magnetic moment is an axial vector so that improper rotations introduce an additional inversion.
	www.ill.fr /dif/ccsl/mk4man/c6node1.html (279 words)

	Finding a homeomorphism from circle to square with rotation
		No suppose you have a set of vectors on the unit circle (a.i + b.j with a^2+b^2=1), and the rotations of these vectors around the circle, described by 2*2 unitary matrices.
		This case is easy, I think, because then the rotation by p on the circle would also be a rotation by p around the center of the square (Intuition, also see answer given by Willem de Boer above).
		However I am also wondering about the case of both normal rotations and improper rotations, that is rotations combined with a flip around one of the axis (e.g a change in signe of one of the coordinates).
	www.groupsrv.com /science/post-680193.html (1006 words)

	Group Theory
		n indicates a rotation of 360/n where n = 1,....
		Each of these Symmetry Operations is associated with a Symmetry Element which is a point, a line, or a plane about which the operation is performed such that the molecule's orientation and position before and after the operation are indistinguishable.
		This implies n-fold rotational symmetry about the axis.
	www.science.siu.edu /chemistry/tyrrell/group_theory/sym1.html (487 words)

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