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	Affine transformation - Wikipedia, the free encyclopedia
		An affine transformation is invertible iff A is invertible.
		The invertible affine transformation form the affine group, which has the general linear group of degree n as subgroup and itself is a subgroup of the general linear group of degree n+1.
		An affine subspace of a vector space (sometimes called a linear manifold) is a coset of a linear subspace; i.e., it is the result of adding a constant vector to every element of the linear subspace.
	en.wikipedia.org /wiki/Affine_transformation (1053 words)

	Affine geometry - Wikipedia, the free encyclopedia
		In geometry, affine geometry is geometry not involving any notions of origin, length or angle, but with the notion of subtraction of points giving a vector.
		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.
	en.wikipedia.org /wiki/Affine_geometry (746 words)

	PlanetMath: affine transformation (Site not responding. Last check: 2007-10-08)
		Compositions of affine maps is again an affine map.
		In affine geometry, the set of affine transformations on the affine space becomes a group under compositions of maps.
		This is version 10 of affine transformation, born on 2004-10-24, modified 2005-02-15.
	planetmath.org /encyclopedia/AffineTransformation.html (117 words)

	Various Geometries
		Affine Geometry is not concerned with the notions of circle, angle and distance.
		Affine transformations preserve collinearity of points: if three points belong to the same straight line, their images under affine transformations also belong to the same line and, in addition, the middle point remains between the other two points.
		Analytically, affine transformations are represented in the matrix form f(x) = Ax + b, where the determinant det(A) of a square matrix A is not 0.
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	Affine transformation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		An affine subspace of a vector space is a (Click link for more info and facts about coset) coset of a (Click link for more info and facts about linear subspace) linear subspace; i.e., it is the result of adding a constant vector to every element of the linear subspace.
		The set of all invertible affine transformations forms a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group under the operation of composition of functions.
		To visualise the general affine transformation of the (Click link for more info and facts about Euclidean plane) Euclidean plane, take labelled (A quadrilateral whose opposite sides are both parallel and equal in length) parallelograms ABCD and A′B′C′D′.
	www.absoluteastronomy.com /encyclopedia/a/af/affine_transformation.htm (734 words)

	Geometric Operations - Affine Transformation
		In many imaging systems, detected images are subject to geometric distortion introduced by perspective irregularities wherein the position of the camera(s) with respect to the scene alters the apparent dimensions of the scene geometry.
		Applying an affine transformation to a uniformly distorted image can correct for a range of perspective distortions by transforming the measurements from the ideal coordinates to those actually used.
		Affine transformations are most commonly applied in the case where we have a detected image which has undergone some type of distortion.
	homepages.inf.ed.ac.uk /rbf/HIPR2/affine.htm (1396 words)

	Augmented Reality Home Page - Minimal Calibration
		Affine reprojection (Koenderink and van Doorn 1991; Mundy and Zisserman 1992) is used by the U of R system to create the projection of any other point defined in the affine coordinate frame.
		The affine coordinates of any point can be computed from Equation 3 if its projection in at least two views is known and the projections of the affine basis points are also known in those views.
		The affine coordinates of the white point of a nail attached to our metal frame were computed from two views of the scene.
	www.se.rit.edu /~jrv/research/ar/thesis.html (3794 words)

	ASCII version of Phys. Rev. A 44, 2730
		Growth processes resulting in self affine interfaces have attracted particular interest during the last few years because of their relevance to a number of phenomena of practical importance, including thin film growth by vapor deposition, two phase viscous flow in porous media, formation of biological patterns, and sedimentation of granular materials (see, e.g., Refs.
		A single valued standard self affine function h (x) satisfies the relation h (x) apeq lambda^{-H}h (lambda x), where lambda is a parameter and H is the Houmllder or roughness exponent.
		These self affine functions are obtained by plotting the displacement h (x) of a particle randomly walking in one dimension as a function of time (denoted here by x).
	www.nd.edu /~alb/paper/PRA44_2730 (2199 words)

	Affine Arithmetic Project (Site not responding. Last check: 2007-10-08)
		Affine arithmetic is somewhat similar to Hansen's generalized interval arithmetic [10], but differs in several important details.
		Affine arithmetic may also be compared to the ellipsoid calculus of Chernousko, Kurzhanski, and Ovseevich [8,9], in that both attempt to record linear correlations between quantities.
		In general, the internal approximation errors of affine arithmetic operations depend quadratically on the width of the input intervals.
	www.dcc.unicamp.br /~stolfi/EXPORT/projects/affine-arith/Welcome.html (806 words)

	Copyright (c) 1996, 1997, 1998 Thomas E. Burge. All rights reserved.
		The Affine Toolkit is copyright (c) 1995-1998 Thomas Burge.
		The Affine Toolkit is based on only published materials regarding the RenderMan Standard and the general subject of computer graphics.
		Affine is a trademark of Thomas E. Burge
	www.affine.org (1735 words)

	Operations on Affine Algebras
		Given an ideal I of an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the ideal J. PreimageIdeal(I) : RngMPolRes -> RngMPol
		Given an ideal I of an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the ideal I' of P such that the image of I' under the natural epimorphism P -> Q is I. PreimageRing(I) : RngMPolRes -> RngMPol
		Given an affine algebra Q which is the quotient ring P/J, where P is a polynomial ring and J an ideal of P, return the polynomial ring P. OriginalRing(Q) : RngMPolRes -> Rng
	www.math.lsu.edu /magma/text1136.htm (571 words)

	Finite Dimensional Affine Algebras (Site not responding. Last check: 2007-10-08)
		If an affine algebra is defined over a field and has finite dimension considered as a vector space over its coefficient field, extra special operations are available on its elements.
		Given an element f of a finite dimensional affine algebra Q defined over a field, return the representation matrix of f, which is a d by d matrix over the coefficient field of Q (where d is the dimension of Q) which represents f.
		Given an element f of a finite dimensional affine algebra Q defined over a field, return whether f is a unit.
	www.math.lsu.edu /magma/text1138.htm (358 words)

	On Radon's Theorem For Affine Kähler Immersions (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		On Radon's Theorem For Affine Kähler Immersions (ResearchIndex)
		On Radon's Theorem For Affine Kähler Immersions (1993)
		1 the Cartan-Norden theorem for affine Kaehler immersions (context) - Nomizu, Podesta - 1991
	citeseer.ist.psu.edu /ivanov93radon.html (327 words)

	Larry D. Bradley: Iterated Function Systems
		Fractals reproducing realistic shapes, such as mountains, clouds, or plants, can be generated by the iteration of one or more affine transformations.
		The form of the attractor is given through the choice of the coefficients a, b, c, d, e, and f, which uniquely determine the affine transformation.
		What is truly amazing is that only 28 numbers are necessary to generate this infinitely complex image: four 2 x 2 transformation matrices, four 2 x 1 translational vectors, and four weighted probabilities for the transformations (each attractor).
	www.pha.jhu.edu /~ldb/seminar/ifs.html (173 words)

	Affine space (Site not responding. Last check: 2007-10-08)
		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)

	Affine (Site not responding. Last check: 2007-10-08)
		But the only quadrilaterals affine to a square are the parallelograms.) A function between vector spaces is affine if it preserves affine combinations, that is, linear combinations in which the coefficients add up to 1.
		The reason that we usually don't learn the phrase "affine transformation" as undergraduates is that any affine transformation is a linear transformation followed by a translation (i.e.
		The notion of affine space has been described as what's left when one removes the origin from a linear space.
	www.cis.upenn.edu /~bcpierce/types/archives/1997-98/msg00120.html (340 words)

	The Affine Connection (Site not responding. Last check: 2007-10-08)
		There is generally no connection between vectors in tangent spaces at different points except a topological construct called a fiber-bundle that is about vectors at nearby points on the manifold being nearly equal.
		On a manifold an affine connection is a mathematical structure that allows parallel transport of vectors along a path.
		If the affine connection is symmetric then this sphere may rotate as it traverses a loop, but it will not expand or shrink.
	www.cap-lore.com /MathPhys/AffineConnec.html (269 words)

	Affine Structure (Site not responding. Last check: 2007-10-08)
		Informally, the affine structure of the given spacetime specifies which of the curves in the spacetime are the "straight lines", not presupposing the metric of the spacetime yet.
		That is, we can separate the notion of "straight line" or "geodesic" and the notion of metric, and the affine structure is concerned only with the former.
		A space (or spacetime) which has an affine structure is an affine space, as Sklar said.
	www.bun.kyoto-u.ac.jp /%7Esuchii/affine.str.html (101 words)

	Affine: User Interface (Site not responding. Last check: 2007-10-08)
		Math for a discussion of the affine transformations it's calculating.
		The affine transformation to accomplish the source to target mapping is show in the bottom table.
		If they are, an error message is given and the affine output is blanked.
	www.cbc.yale.edu /courseware/affine/readers/interface.html (202 words)

	IMA Preprints: November 1995 Abstract (Site not responding. Last check: 2007-10-08)
		Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets.
		In this case, affine invariant edged are obtained as a weighted difference of images at different scales.
		These edge detectors are the basis both to extend the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration.
	www.ima.umn.edu /preprints/November1995/1360abs.html (224 words)

	Affine Algebras (Quotient rings)
		If an affine algebra has finite dimension considered as a vector space over the coefficient field, extra special operations are available on its elements.
		If an affine algebra has finite dimension considered as a vector space over its coefficient field, extra special operations are available on its elements.
		Given an element f of a finite dimensional affine algebra Q, return the representation matrix of f, which is a d by d matrix over the coefficient field of Q (where d is the dimension of Q) which represents f.
	www.math.colostate.edu /manuals/magma/htmlhelp/text407.html (1050 words)

	Affine curvature evolution of 2D curves and 3D surfaces for smoothing applications (Site not responding. Last check: 2007-10-08)
		Affine curvature evolution of 2D curves and 3D surfaces for smoothing applications
		The cat image smoothed with the affine evolution flow for 2,8 and 128 time units.
		This is because the Gaussian curvature estimate vanishes everywhere on the surface except for some small numerical preturbation at the extremeties.
	www.cim.mcgill.ca /~mdesco/affine.html (237 words)

	Affine 2D Analogue
		This flow is what all the the literature on affine evolutions claim to be the affine analogue of the curve shortening flow in Euclidean space.
		Now, in affine geometry, the claimed analogue is the functional is affine length and the fastest way to shrink it is the evolution in the affine normal direction which corresponds to an Euclidean curvature deformation in the Euclidean normal direction.
		We have argued that the affine curve evolution is not necessarily the fastest way to shrink the affine perimeter.
	www.cim.mcgill.ca /~mdesco/shape_report/report/node4.html (794 words)

	Introduction and ToC: Affine and projective planes (Site not responding. Last check: 2007-10-08)
		In all known cases will be the characteristic of the field is the prime involved in the order of the affine plane; in general it is any prime dividing the order n of the plane.
		Toward this end we introduce a notion of ``linear equivalence'': two affine planes are linearly equivalent if their hulls are isomorphic.
		Section 4 is a detailed discussion of the affine case, Section 5 introduces linear equivalence and tameness, Section 6 is largely concerned with translation planes, and Section 7 discusses the Hamada-Sachar Conjecture and presents our partial solution.
	www.lehigh.edu /efa0/public/www-data/aandptoc.html (1162 words)

	Affine and Euclidean Geometric Heat Equation for Anisotropic Smoothing
		In this work, we study the geometric smoothing of curves and surfaces via the geometric heat equation known as the deformation by local curvature.
		We emphasize the paper on the affine analogue of this flow and its formulation in terms of Euclidean curvature and normal.
		We investigate the key mathematical properties of both the Euclidean and the affine geometric heat equation that make them sensible smoothing methods.
	www.cim.mcgill.ca /~mdesco/shape_report/report/index.html (236 words)

	Affine Transformations
		The transformations that move lines into lines, while preserving their intersection properties, are special and interesting, because they will move all polylines into polylines and all polygons into polygons.
		Every affine transformation can be expressed as a transformation that fixes some special point (the "origin") followed by a simple translation of the entire plane.
		Evidently coefficients A, B, D, and E determine a linear transformation and coefficients C and F determine a parallel translation: that is, such three-by-three matrices describe affine two-dimensional transformations.
	www.quantdec.com /GIS/affine.htm (3073 words)

	Representations of rank two affine Hecke algebras (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		I have made special effort to describe how the classification here relates to the classifications by Langlands parameters (coming from p-adic group theory) and by indexing triples (coming from a q-analogue of the...
		Calibrated representations of affine Hecke algebras - Ram (1998)
		Affine Hecke algebras, cyclotomic Hecke algebras and Clifford..
	citeseer.ist.psu.edu /236822.html (491 words)

	Affine Geometry (Site not responding. Last check: 2007-10-08)
		Affine transformations map affine subspaces to affine subspaces.
		Two parallel lines are lines in an affine plane which do not meet.
		Since affine transformations preserve planes and incidence, their images lie in an affine plane and do not meet.
	www-groups.dcs.st-and.ac.uk /~john/geometry/Lectures/L13.html (229 words)

	3D Affine Evolution
		The 3D affine differential geometry and setting up the problem is more complicated than the the 2D case.
		It turns out that the simplest possible affine surface evolution is moving every point of the surface according to the Gaussian curvature raised to some exponent.
		Analogous to the inflection points issue in 2D, it resolves the ``non-existence'' problem of 3D affine differential geometry at parabolic points, where one of the two principal curvature vanishes.
	www.cim.mcgill.ca /~mdesco/shape_report/report/node5.html (230 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).
		The product of two affine varieties is affine, and its coordinate ring is the the tensor product of the coordinate rings of the factors.
	www.math.purdue.edu /~dvb/algeom2.html (786 words)

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