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Topic: Euclidean group

	Euclidean group
		In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry.
		In the terms of the Erlangen programme, Euclidean geometry is therefore a specialisation of affine geometry.
		Another use of a Euclidean group is for the kinematics of a rigid body, in classical mechanics.
	publicliterature.org /en/wikipedia/e/eu/euclidean_group.html (461 words)

	Erlangen programme
		Then, by abstracting the underlying groups of symmetries from the geometries, the relationships between them can be re-established at the group level.
		Since the group of affine geometry is a subgroup of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry; but not the other way round.
		For example the group of projective geometry in n dimensions is the symmetry group of n-dimensional projective space (the matrix group of size n+1, quotiented by scalar matrices).
	www.ebroadcast.com.au /lookup/encyclopedia/er/Erlangen_programme.html (586 words)

	Euclidean group - Wikipedia, the free encyclopedia
		The Euclidean group is a subgroup of the group of affine transformations.
		Examples of such groups are, in 1D, the group generated by a translation of 1 and one of √2, and, in 2D, the group generated by a rotation about the origin by 1 radian.
		for any point group: the group of all isometries which are a combination of an isometry in the point group and a translation; for example, in the case of the group generated by inversion in the origin: the group of all translations and inversion in all points; this is the generalized dihedral group of R
	en.wikipedia.org /wiki/Euclidean_group (1463 words)

	Search Results for geometry
		Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development.
		Euclidean geometry] but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it excepting the determination of a constant, which cannot be fixed a priori.
		In particular he wrote Lectures on differential geometry (1894), Lectures on the theory of groups of substitutions (1900), Lectures on the theory of continuous groups (1918), Lectures on the theory of functions of a complex variable (1901) and Lectures on the theory of algebraic numbers (1923).
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?BIOGS=1&TOPICS=1&CURVES=1&REFS=1&BIBLI=1&SOCIETIES=1"=1&CHRON=1&WORD=geometry&CONTEXT=1 (17345 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		An opertion of a given group that caries a given figure (collection of points), into coincidence with itself (with permutation of the points) is called a superpostion of the figure, the superpostions of a given figure form the subgroups of the group of operations.
		Euclidean geometry is based on the concept of a Euclidean vector space Vn over the field R of real numbers in which a positive definite symmetric bilenear form f is distinguished with the result that to every vector pain (x,n), there corresponds a real number f(x,n)....
		Group Bn of orientation preserving isometries, also called the group of motions, is a normal subgroup of Bn with index 2.
	www.rhfweb.com /ngeom.html (3070 words)

	PlanetMath: Coxeter group
		is a finitely generated group, which carries a presentation of the form
		Cross-references: Euclidean space, generated by, isomorphic, effective, subgroup, transformation, clear, hyperplane, fixed, image, vector, linear transformation, reflection, orthogonal transformations, group, Euclidean vector space, real, finite, satisfies, integers, presentation, finitely generated group
		This is version 5 of Coxeter group, born on 2006-01-21, modified 2006-01-27.
	planetmath.org /encyclopedia/CoxeterGroup.html (154 words)

	PlanetMath: geometry
		Familiar examples from Euclidean geometry are the length of line segments, areas of triangles, and angles.
		Most obviously, the Euclidean plane itself is a manifold since its points can be described by pairs of real numbers according to various coordiante systems (Cartesian coordinates, polar coordinates, etc.) Thus, the concept of manifold fulfills the desire of its inventors that Euclidean space and surfaces be of the same ontological status.
		The study of Lie groups forms an important branch of group theory and is of relevance to other branches of mathematics.
	planetmath.org /encyclopedia/Geometry.html (4225 words)

	Euclidean plane isometry - Wikipedia, the free encyclopedia
		The set of Euclidean plane isometries forms a group under composition, which is the two-dimensional case of the Euclidean group.
		Neither the full group nor the even subgroup are abelian; for example, reversing the order of composition of two parallel mirrors reverses the direction of the translation they produce.
		Since the even subgroup is normal, it is the kernel of a homomorphism to a quotient group, where the quotient is isomorphic to a group consisting of a reflection and the identity.
	en.wikipedia.org /wiki/Euclidean_plane_isometry (2426 words)

	Examples of Groups
		Another important group is called Euclidean group It consists of all the transformations of the plane which do not alter distances.
		Given a geometric figure in the plane the symmetry group of the figure consists of all isometries that transform points on the figure to points on the figure.
		The group of permutations on n objects is called the symmetric group on n objects and is symbolized by Sn.
	members.tripod.com /~dogschool/examples.html (1438 words)

	Philosophy of Real Mathematics: Klein 2-geometry
		Or, it might be nice to let G be the whole Euclidean group and A some abelian group on which it acts: this would decategorify to the Euclidean group.
		The relation of geometry and group theory began with some realization sort of like this: the group G of Eulidean transformations acts on everything in the plane, in a way that preserves all geometrical structure.
		Groups of the first sort stabilize a point in the projective plane; groups of the second sort stabilize a line.
	www.dcorfield.pwp.blueyonder.co.uk /2006/05/klein-2-geometry.html (7515 words)

	Crystallographic Topology - Orbifold 2
		There are 17 plane groups (wallpaper groups) defining the symmetry in all patterns that repeat by 2-dimensional lattice translations in Euclidean 2-space.
		The notation under the crystallographic drawing is the standard plane group name and that under the orbifold drawing is our notation for the Euclidean 2-orbifold with S, D, and RP denoting sphere, disk, and real projective plane, respectively.
		The annulus and Möbius band in row one are derived from plane groups pm and cm by first cutting out an asymmetric unit bounded by those portions of the mirrors denoted by double stripes and matching the ends together as indicated in Fig.
	www.ornl.gov /ortep/topology/Aorbfld2.html (1385 words)

	The stratification of 3D geometry
		This concept of stratification is closely related to the groups of transformations acting on geometric entities and leaving invariant some properties of configurations of these elements.
		Attached to the projective stratum is the group of projective transformations, attached to the affine stratum is the group of affine transformations, attached to the metric stratum is the group of similarities and attached to the Euclidean stratum is the group of Euclidean transformations.
		the metric group is a subgroup of the affine group and both are subgroups of the projective group.
	www.cs.unc.edu /~marc/tutorial/node25.html (322 words)

	Group actions
		In algebra, geometry and topology we often exploit the fact that important structures arise from families of morphisms that are indexed by a group.
		For example, rotations in the plane about the origin are indexed by the unimodular group of complex numbers; we say that this group acts on the plane and the orbit of a point at distance r from the origin is the circle of radius r.
		G is a Lie group, so G has a differentiable structure with respect to which its binary operation and the taking of inverses is smooth, and X is a smooth manifold.
	www.maths.man.ac.uk /~kd/curves/node4.html (797 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		In 1939, Wigner published his fundamental paper on little groups, where they are defined to be the maximal subgroup of the Poincar{\'e} group whose transformations leave the four-momentum of a relativistic particle invariant.
		He observed that little groups are locally isomorphic to the rotation and Euclidean groups for massive and massless particles, respectively.
		The rotation group has been long-known, however the applications of the groups that are locally isomorphic to the Euclidean group have been relatively of a more recent interest.
	www.physics.umd.edu /robot/contrac/baskal.html (407 words)

	Space and Time
		In so-called projective geometry, for example, which originated with the technique of perspective in drawing, the group is larger than the Euclidean, and two rectangles which are not equal in the Euclidean sense may be equal in the projective sense.
		The larger the group, the more we abstract from reality: projective geometry is more abstract than the Euclidean, but all geometry is abstract, that is, it takes away infinitely much from our perception of reality.
		But this is a very small group (only two elements); a pyramid is more symmetric, the group of motions leaving it unchanged is larger; a cube is even more symmetric, but the sphere is supremely symmetric--here the group is infinite.
	www.albany.edu /~rn774/fall96/spacetime.html (3651 words)

	Group Theory application in Robotics, Computer Vision and Computer Graphics (Site not responding. Last check: 2007-11-06)
		Group theory, the ultimate theory for symmetry, is a powerful tool that has a direct impact on research in robotics, computer vision, computer graphics and medical image analysis.
		Instead, it relies heavily on intuitions in (1) 3D Euclidean space, images and patterns; (2) a geometric computational model; and (3) concrete, real world applications in robotics, computer vision, computer graphics and medical image analysis drawing from the instructor’s many years of research experience and from an emerging, vibrant, interdisciplinary international research community.
		One cause of this shortage is the discrepancy between the ideal algebraic formulation of symmetry, namely group theory, and the instantiation of symmetry in the noisy physical world.
	www.cs.cmu.edu /afs/cs.cmu.edu/misc/mosaic/common/omega/Web/People/yanxi/www/newtest.htm (1054 words)

	3. Discrete Subgroups of Euclidean Group
		The considered groups are discrete, so in every such group we can choose a translation whose vector is minimal.
		The second class contains the groups with a translation subgroup generated by one single translation - the symmetry group of friezes.
		The function sending one group of isometries into another preserves the type of transformations if it sends translations in translations, rotations in rotations, reflections in reflections and glide reflections in glide reflections.
	members.tripod.com /vismath1/ana/ana3.htm (533 words)

	Symmetry Group (Site not responding. Last check: 2007-11-06)
		Example 3.1 Let G be the group of motions of the plane generated by translations, rotations and reflexions (in a line); we call this the Euclidean group in 2 dimensions and may write it E
		For example, the property of being a polygon is a Euclidean property; the number of vertices and sides of a polygon is a Euclidean invariant.
		It is easy to see why the symmetry groups are the same; for the centers of the faces of a cube are the vertices of a regular octahedron, and the centers of the faces of a regular octahedron are the vertices of a cube.
	www.mi.sanu.ac.yu /vismath/hil/ped3.htm (1390 words)

	93-82 A Geometric Approach for Denoting and Intersecting TR Subgroups of the Euclidean Group (Site not responding. Last check: 2007-11-06)
		Although there are many ways to denote a group and compute group intersections, it is not obvious that there exists a uniform computational denotation for the symmetry groups of surfaces, given the diversity of these groups --- the finite, infinite, discrete and continuous subgroups of the Euclidean group.
		In this article a geometric denotation in terms of {\em characteristic invariants} is proposed for an important family of subgroups of the Euclidean group, namely the \TR subgroups, each of which is a semidirect product of a translation subgroup {\bf T} and a rotation subgroup {\bf R} of the Euclidean group.
		Therefore, the group theoretical formalization of surface contacts is feasible in applications involving transformations between solids in Euclidean space, such as robotics, computer graphics, computer vision and mechanical design.
	dimacs.rutgers.edu /TechnicalReports/abstracts/1993.old/93-82.html (270 words)

	Mailgate: sci.math.research: Re: Invariant Rings of Finite Euclidean Reflection Groups (Site not responding. Last check: 2007-11-06)
		Some of these >groups have familiar geometric interpretations: W(H_3) is the symmetry >group of the icosahedron and its dual, W(H_4) is the symmetry group of the >120-cell and its dual, and W(F_4) is the symmetry group of the 24-cell; >these are the sporadic regular polytopes.
		One more reasonable approach is to regard the invariance of a homogeneous polynomial of given degree d under the group -- equivalently, under a set of generators of the group -- as a set of simultaneous linear conditions on the polynomial's coefficients.
		You already know that D_4 is the group of signed permutation matrices the product of whose signs is +1, and that its invariant ring R(D_4) is polynomial with generators x1x2x3*x4 and the first three symmetric functions in x1^2, x2^2, x3^2, x4^2.
	mailgate.supereva.it /sci/sci.math.research/msg04174.html (933 words)

	Henri Poincare: Science and Hypothesis: Chapter 5: Experiment and Geometry (Site not responding. Last check: 2007-11-06)
		Now, this is a property which in either Euclidean or non-Euclidean space belongs to the straight line, and belongs to it alone.
		These new verifications are therefore impossible if the bodies move according to the Euclidean group; but they become possible if we suppose the bodies to move according to the Lobatschewskian group.
		Suppose, for instance, that we have a large sphere of radius R, and that its temperature decreases from the centre to the surface of the sphere according to the law of which I spoke when I was describing the nonEuclidean world.
	spartan.ac.brocku.ca /~lward/Poincare/Poincare_1905_06.html (3605 words)

	Weyl's Generalization of Klein's Erlangen Program (Site not responding. Last check: 2007-11-06)
		According to Klein, geometry is characterized by its group of symmetries (by symmetry is meant a transformation of space which leaves all objective properties of space invariant) and investigates the mathematical objects which are invariant (simply called "invariants") under this given group.
		For instance, projective geometry is more general than Euclidean geometry: this is translated in the fact that the Euclidean group is a subgroup of the projective group.
		So the group of symmetries should be an "abstract" group rather than a concrete group (by concrete I mean consisting of transformations).
	www.math.mcgill.ca /~malkoun/Erlangen_Program/Erlangen_Program.html (436 words)

	Everything about Brillouin Zone (Site not responding. Last check: 2007-11-06)
		A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete symmetry group.
		A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector).
		In geometry, the fundamental domain of a symmetry group of an object or pattern is a part of the pattern, as small as possible, which, based on the symmetry, determines the whole object or pattern.
	cdd770d9c100aa9a889bdd63e2f2de00.es.wikimiki.org (8963 words)

	Poincare (Site not responding. Last check: 2007-11-06)
		In the same paper, Inonu and Wigner give a detailed exposition of the contraction of the O(3) rotation group into the E(2) group, namely the Euclidean group in two-dimensional space consisting of rotations around the origin and translations in two orthogonal directions.
		Can the E(2)-like little group for massless particles be obtained from the O(3)-like little group by a group contraction procedure.
		The little group takes the form of transformations on a cylindrical surface consisting of rotations (helicity) and up-down translations (gauge transformations).
	www.physics.umd.edu /robot/einstein/eicontr.html (575 words)

	Symmetry - Wikipedia, the free encyclopedia
		For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space.
		G is the group of affine transformations with a matrix A with determinant 1 or -1, i.e.
		In another meaning of the word, the rotation group of an object is the symmetry group within E+(n), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries.
	en.wikipedia.org /wiki/Symmetry (3461 words)

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