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Topic: Affine space

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	Affine geometry - Wikipedia, the free encyclopedia
		Affine geometry can be explained as the geometry of vectors, not involving any notions of length or angle.
		Affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0.
		The term affine space is used in projective geometry as the complement of the points (hyperplane) at infinity (see also projective space).
	en.wikipedia.org /wiki/Affine_geometry (756 words)

	Affine space - Wikipedia, the free encyclopedia
		In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space.
		An affine space is a set with a faithful transitive vector space action, a principal homogeneous space with a vector space action.
		An affine subspace of a vector space V is a subset closed under affine combinations of vectors in the space.
	en.wikipedia.org /wiki/Affine_space (1025 words)

	Affine - Wikipedia, the free encyclopedia
		Affine space, an abstract structure that generalises the affine-geometric properties of Euclidean space
		Affine transformation, a linear transformation followed by a translation between two vector spaces, or equivalently, a transformation that preserves all affine combinations
		Affine representation, a continuous group homomorphism whose values are automorphisms of an affine space
	en.wikipedia.org /wiki/Affine (194 words)

	Affine space: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-07)
		In mathematics, euclidean space is a generalization of the 2- and 3-dimensional spaces studied by euclid....
		An affine space is a space in which you can subtract two points to form a vector pointing from one point to the other.
		In mathematics, the affine group of any affine space over a field k is the group (mathematics)group of all invertible affine transformations from...
	www.absoluteastronomy.com /encyclopedia/a/af/affine_space.htm (1146 words)

	Affine and Projective Spaces
		Projective n-space is an extension of affine n-space by the inclusion of an additional coordinate, say [w], such that not all coordinates are zero at once.
		Homogenization recasts the equation in projective space and all of the terms are of the same degree.
		For example, the circle x^2 + y^2 - 1 = 0 in affine 2-space is the cross-section at w = 1 of the unit cone x^2 + y^2 - w^2 = 0 in projective space.
	www.science.gmu.edu /~jsteidel/806-prj/spaces.html (713 words)

	Affine geometry - TheBestLinks.com - Analytic geometry, Euclidean geometry, Geometry, Linear algebra, ... (Site not responding. Last check: 2007-11-07)
		Affine geometry - TheBestLinks.com - Analytic geometry, Euclidean geometry, Geometry, Linear algebra,...
		Affine geometry, Analytic geometry, Euclidean geometry, Geometry, Linear...
		The ratio of the area of the envelope to the area of the triangle is affine invariant, and so only needs to be calculated from a simple case such as a unit isosceles right angled triangle to give ${3 \over 4} \log_e(2)-{1 \over 2}, i.e.$
	www.thebestlinks.com /Affine_geometry.html (733 words)

	COMP 290-001: Lecture Notes: Design Example for an Affine-Geometry Kernel
		The vector space is connected with the point space as follows: For each pair of points in the point space the difference between the two points is a vector in the vector space.
		A vector is an element of the vector space of an affine space.
		Assuming the reference frame for the new space is given in the reference frame of the points, the affine transformation is easily specified as follows; the base point of the reference frame maps to the origin and each vector in the basis of the reference frame maps to the unit vectors of the standard basis.
	www.mpi-sb.mpg.de /~kettner/courses/lib_design_03/notes/gkernel.html (1098 words)

	Affine space (Site not responding. Last check: 2007-11-07)
		Then A is an affine space and K is called the coefficient field.
		In an affine space, it is possible to fix a point and coordinate axis such that every point...
		Dimension:: The dimension of an affine space is the same as that of.
	www.serebella.com /encyclopedia/article-Affine_space.html (1075 words)

	Affine geometry (Site not responding. Last check: 2007-11-07)
		Affine geometry can be explained as the of vectors not involving any notions of length angle.
		The term affine space is used in projective geometry as complement of the points (hyperplane) at infinity also projective space).
		An space A for a vector space V is just such a principal homogeneous one then has to recover scalar multiplication A as a well-defined concept.
	www.freeglossary.com /Affine_geometry (710 words)

	PlanetMath: geometrization of $\mathbb{R}^n$ (Site not responding. Last check: 2007-11-07)
		This function is also known as a metric, but one needs to be careful with the term "metric" because it is sometimes also used to refer to an inner product.
		has such a special name “affine space” is because it is acted upon by the set of affine transformations, which are transforms of the form
		) It is worth mentioning that the name “affine space” is used primarily in geometry and in commutative algebra.
	www.planetmath.org /encyclopedia/AffineSpace.html (389 words)

	Encyclopedia article on Space [EncycloZine]
		The memory space of a computer is often thought as a one dimensional array of memory cells.
		The Space character (ASCII value 32) is used as the blank separating words (same as Orthographic space), and corresponds to the spacebar on the keyboard.
		In board games, space is often the term used to describe the divisions to which the result of the dice roll (or similar alternative) denotes the movement.
	encyclozine.com /Space (681 words)

	ipedia.com: Affine geometry Article (Site not responding. Last check: 2007-11-07)
		It is the geometry of affine space, of a given dimension n over a field K. The case where...
		Affine geometry can be explained as the geometry of vectorss, not involving any notions of length or angle.
		The ratio of the area of the envelope to the area of the triangle is affine invariant, and so only needs to be calculated from a simple case such as a unit isosceles right angled triangle to give, i.e.
	www.ipedia.com /affine_geometry.html (739 words)

	Affine representation: Encyclopedia topic (Site not responding. Last check: 2007-11-07)
		An affine representation of a topological (topological: in mathematics, a topological group g is a group that is also a topological space...
		Since the affine group (affine group: in mathematics, the affine group of any affine space over a field k is the group...
		We may ask whether a given affine representation has a fixed point (fixed point: in mathematics, a fixed point of a function is a point that is mapped to itself by the...
	www.absoluteastronomy.com /reference/affine_representation (293 words)

	PlanetMath: affine space (Site not responding. Last check: 2007-11-07)
		In Algebraic Geometry, we consider affine space as a topological space, with the usual Zariski topology (see also algebraic set, affine variety).
		“Glueing” several copies of affine space one obtains a projective space.
		This is version 4 of affine space, born on 2005-05-05, modified 2006-02-21.
	planetmath.org /encyclopedia/AffineSpace3.html (113 words)

	Affine geometry - Term Explanation on IndexSuche.Com
		Affine space is distinguished from a vector_space of the same dimension by 'forgetting' the origin ''0''.
		The affine group is generated by the general linear group and the translations and is the GL(n,K).
		An affine space ''A'' for a vector space ''V'' is just such a principal_homogeneous_space; one then has to recover scalar multiplication on ''A'' as a well-defined concept.
	www.indexsuche.com /Affine_geometry.html (610 words)

	-= Selected Publications =-
		An arbitrary nonlinear automorphism of the affine space A
		On algebraical maximality of the affine subgroup in the group of biregular automorphisms of the symplectic space of the four dimension.
		Affine group as a subgroup of biregular transformation group of an affine space, Naukovi zapysky University "Kiev Mohyla Academy" ser.
	www.ukma.kiev.ua /~yubod/publications.htm (361 words)

	Introduction and First Examples (Site not responding. Last check: 2007-11-07)
		The syntax for defining an affine space over a ring is similar to that employed when defining a polynomial ring over another ring.
		That is, there is a 1 - 1 correspondence between subschemes of an affine ambient space and the ideals of the coordinate ring of that ambient.
		For example, given a linear system L and a point p lying in the projective space on which L is defined, a subsystem of L consisting of those hypersurfaces of L which contain p may be defined implicitly.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1151.htm (2913 words)

	Affine space - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		In mathematics, an affine space may be defined somewhat abstractly as a set on which a vector space acts transitively.
		Imagine that Smith knows that a certain point is the origin, and Jones believes that another point -- call it p -- is the origin.
		Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the "affine structure" -- i.e., the values of affine combinations, defined as linear combinations in which the sum of the coefficients is 1.
	www.encyclopedia-online.info /Affine_space (247 words)

	Re: Einstein could have predicted expanding universe in 1905
		It is then made into an affine space by choosing an absolute hyperplane and calling it the 'hyperplane at infinity'.
		The affine space is given a concept of distance and turned into Euclidean (Minkowski) space by making the hyperplane at infinity into an elliptic (hyperbolic) space by choosing an absolute elliptic (hyperbolic) polarity on this hyperplane.
		An affine space admits the dilatations as congruences.
	www.lns.cornell.edu /spr/2003-11/msg0056053.html (437 words)

	position vector - Page 2
		An "affine space" is a set of points such that any line through one of the points is contained in the space.
		A space of positions in a plane (or the space of times on a line) is an affine space.
		Then the sum of two position vectors may not lie in the area of the space and thus the square fails to be a vector space.
	www.physicsforums.com /showthread.php?p=827838 (2939 words)

	Ambients
		For the purposes of this chapter, any scheme is contained in some ambient space, either an affine space or one of a small number of standard projective spaces: these are projective space itself, possibly weighted, and rational scrolls.
		A procedure to change the print names of the coordinate functions of the ambient space A. It leaves A unchanged except that the visible names of the first #N coordinate functions are replaced by the strings of N and the rest return to their default.
		For projective spaces, one talks about the homogeneous coordinate ring and restricts attention to homogeneous polynomials, that is, polynomials whose terms all have the same weight with respect to a single grading, but nonetheless one is working inside a polynomial ring.
	wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text1129.htm (2287 words)

	Affine space (Site not responding. Last check: 2007-11-07)
		In mathematics, an affine space may be defined somewhatabstractly as a set on which a vector space acts transitively.
		Albeit somewhat jocular, the following characterization may be easier to understand: an affine space is what is left of avector space after you've forgotten which point is the origin.
		Here is the punch line: Smith knows the "linear structure", but both Smith and Jones know the"affine structure" -- i.e., the values of affine combinations,defined as linear combinations in which the sum of the coefficients is 1.
	www.therfcc.org /affine-space-52231.html (225 words)

	MATTER and SPACE with TORSION
		Abstract: Equations are obtained describing the curvature and torsion of general metric-affine space G4 or, in accordance with the unified field theory, the distribution and motion of matter.
		Equations (3)-(4) do not possess such a dualism, for they operate only with the geometrical parameters of G4 space; that is, they describe distribution and motion of the matter, which itself is simply curvature and torsion of space-time.
		As a result, the final solution of equations (7) in G4 space for a spherically symmetric stationary field of massless fluid with spin is given by formulae (5), (6), (11), (27).
	www.acadjournal.com /2003/v9/part4/p1 (4231 words)

	Affine stratum
		Affine geometry differs from projective geometry by identifying a special plane, called the plane at infinity.
		Strictly speaking, this plane is not part of the affine space, the points contained in it can't be expressed through the usual non-homogeneous 3-vector coordinate notation used for affine, metric and Euclidean 3D space.
		For the (more restrictive) affine group parallelism is added as a new invariant property.
	www.cs.unc.edu /~marc/tutorial/node27.html (249 words)

	Glossary
		An affine or projective algebraic set is called a variety if it is irreducible in its Zariski topology i.e.
		The functor X → A(X) is an antiequivalence of the full subcategory of affine varieties and the category of affine domains over k (finitely generated k-algebras which are integral domains).
		A homogenous space is a variety X such that there is an algebraic group G and a transitive action on X for which GxX → X is a morphism.
	www.math.purdue.edu /~dvb/algeom2.html (786 words)

	Schemes
		If X is an affine space for which no projective closure has been computed, the projective closure will be a projective space with this space as its first standard patch.
		The ith affine patch of the scheme X. The number of affine patches is dependent on the type of projective ambient space in which X lies, but for instance, the standard projective space of dimension n has n + 1 affine patches.
		This intrinsic returns a sequence of maps from affine spaces to the projective space P whose images are these affine pieces of a decomposition.
	www.umich.edu /~gpcc/scs/magma/text1019.htm (7582 words)

	[No title]
		The description we have just given of projective space using coordinates is external; next, we want to investigate an internal description.
		In the affine case this was easy: we simply took lists of polynomials (that clearly gave maps from one affine space to another), restricted them to an algebraic subset of the domain, and looked at conditions that ensured that the image landed in another algebraic set.
		It is almost always the case that there is no single formula (defining a map on all of projective space or even on all of the projective set) that works everywhere to define morphisms.
	odin.mdacc.tmc.edu /~krc/agathos/proj.html (999 words)

	algebraic variety in TutorGig Encyclopedia
		An 'affine algebraic variety' is essentially the set of common zeroes of a set of polynomials, and is one of the central objects of study in classical (and to some extent, modern) algebraic geometry.
		A scheme is a locally ringed space such that every point has a neighbourhood, which, as a locally ringed space, is isomorphic to a spectrum of a ring.
		Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety simply mean that the affine charts have trivial nilradical.
	www.tutorgig.com /ed/algebraic_variety (991 words)

	Affine group: Encyclopedia topic (Site not responding. Last check: 2007-11-07)
		In mathematics (mathematics: A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement), the affine group of any affine space (affine space: in mathematics, an affine space is an abstract structure that generalises the affine...
		There is more than one convenient way to describe the structure of affine groups.
		[follow hyperlink for more...]) : this is given on the affine space (affine space: in mathematics, an affine space is an abstract structure that generalises the affine...
	www.absoluteastronomy.com /reference/affine_group (214 words)

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