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# Topic: Affine transformation

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 Affine transformation - Wikipedia, the free encyclopedia The invertible affine transformations form the affine group, which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n+1. A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. 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   (1084 words)

 PlanetMath: affine transformation An affine property is a geometry property that is preserved by an affine transformation. One can more generally define an affine transformation to be an order-preserving bijection between two affine geometries. This is version 23 of affine transformation, born on 2004-10-24, modified 2006-06-16. planetmath.org /encyclopedia/AffineTransformation.html   (387 words)

 Affine - Wikipedia, the free encyclopedia 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 Affine connection, a connection on the tangent bundle of a differentiable manifold en.wikipedia.org /wiki/Affine   (190 words)

 [No title] Note that transformations lower in the table inherit the invariants of those above, but because they possess their own groups of definitive axioms as well, the converse is not true.} The transformation of the part face shown in the example image above is approximated by a planar \em{affine} transformation. Affine transformations are most commonly applied in the case where we have a detected image which has undergone some type of distortion. Notice that we have defined a transformation which works well for objects at the depth of the door frame, but nearby objects have been distorted because the affine transform cannot account for distortions at widely varying depths. www.cee.hw.ac.uk /hipr/src/affine.hpr   (1335 words)

 [No title] From these it is possible to compute the Affine transformation T that transforms the first configuration to the second one. The Affine transformation is given by six parameters (a1, a2, a3, a4, tx, ty). Affine transformations are stored as points R (a1,a2,a3,a4,tx,ty) in parameter space. www.geocities.com /ResearchTriangle/Campus/7971/affine.htm   (272 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. Recall that, for a linear transformation, the first column of the matrix contains the coefficients of the point where the first basis vector (1, 0) is sent and the second column contains the coefficients of the point where the second basis vector (0, 1) is sent. www.quantdec.com /GIS/affine.htm   (3073 words)

 AffineTransform (Java 2 Platform SE v1.4.2) This constant indicates that the transform defined by this object is an identity transform. Transforms an array of relative distance vectors by this transform. An identity transform is one in which the output coordinates are always the same as the input coordinates. java.sun.com /j2se/1.4.2/docs/api/java/awt/geom/AffineTransform.html   (5071 words)

 Transformation of coordinates (Projective; Affine; Metric) The metric transformations are a subset of the affine transformations. The affine transformations are a subset of the projective transformations. The affine properties are a subset of the metric properties. www.ping.be /~ping1339/coortf.htm   (1901 words)

 Maths - Affine Transforms - Martin Baker An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). In general, an affine transformation is a composition of rotations, translations, dilations, and shears. Therefore affine transformations are good for represent the movement of solid objects whose position can be represented by a combination of its centre coordinates and its orientation. www.euclideanspace.com /maths/geometry/affine   (537 words)

 COMP 290-001: Lecture Notes: Design Example for an Affine-Geometry Kernel An affine transformation is a linear mapping from an affine space to an affine space that preserves affine combinations. This implies that the affine transformation of the difference of two points (a vector) is the same as the difference of two affine transformed points. 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. photon.poly.edu /~hbr/cs903-F00/lib_design/notes/gkernel.html   (1098 words)

 Geometric Operations - Affine Transformation Note that transformations lower in the table inherit the invariants of those above, but because they possess their own groups of definitive axioms as well, the converse is not true. The transformation of the part face shown in the example image above is approximated by a planar affine transformation. Notice that we have defined a transformation which works well for objects at the depth of the door frame, but nearby objects have been distorted because the affine plane transformation cannot account for distortions at widely varying depths. homepages.inf.ed.ac.uk /rbf/HIPR2/affine.htm   (1396 words)

 cairo_matrix_t A cairo_matrix holds an affine transformation, such as a scale, rotation, or shear, or a combination of those. The primary use of transformation matrices in Cairo is as the current transformation matrix in a cairo_t. Affine transformations are position invariant, so the same vector always transforms to the same vector. fishsoup.net /tmp/cairo-docs/cairo-cairo-matrix-t.html   (399 words)

 Chapter 5 Introduction to Transformations Affine transformations are the fundamental cornerstone of computer graphics and are central to OpenGL as well as most other graphics systems. Affine transformations have a simple form: The coordinates of Q are linear combinations of those of P. That is, The specific transformation is that which would carry the camera from its positon in the scene to its "generic" position, with the eye at the origin and the view volume aligned with the z-axis, as shown in Part (c) of the figure. arapaho.nsuok.edu /~kingw/math4223/chapter5/chapter5.html   (3682 words)

 [No title] Compared with the affine transform, the extra point (or 2 parameters) allows specification of a keystone warp by the relative distances between pairs of points. Because the projective transform is not linear, it cannot be composed as a sequence of translations, shears, scalings and (optionally) rotations. As with the affine transform, both sampling and interpolation implementations are given of the transform, with the interpolation method being about 5x slower. www.leptonica.com /affine.html   (2270 words)

 Comparison of Portal and Simulator Radiographs A spatial transformation is a mapping function that establishes a relationship among all points in an image and its warped counterpart. The affine transformation can be characterized by it's action on three points, as shown in figure 1. A general affine transformation is composed of a rotation, a magnification of different amounts in the directions of the x and y axis, a shear and a translation. www.buzzard.me.uk /jonathan/masters/mathematics.html   (1597 words)

 UNIT 28 - AFFINE AND CURVILINEAR TRANSFORMATIONS Calculate the transformations necessary to show this area on a screen containing 640 columns and 480 rows, with the columns and rows numbered from the top left corner, assuming that positions on the screen are referred to by row and column numbers. Describe the transformations between x and y and UTM Northing and UTM Easting given in the text of the unit for the fire study example in terms of (a) translation of the origin, (b) scaling, (c) rotation and (d) reflection. Discuss the factors which contribute to lack of fit between maps of the same area, and failure of affine transformations to register maps perfectly using control points. www.geog.ubc.ca /courses/klink/gis.notes/ncgia/u28.html   (1111 words)

 Affine   (Site not responding. Last check: 2007-10-21) Affine Geometry is not concerned with the notions of circle, angle and distance. Every affine transformation is obtained by composing a scaling transformation with an isometry, or a shear with a homothety and an isometry. The I Ching as affine coordinates in a 4x4x4 cube: H. m759.freeservers.com www.affine.info   (318 words)

 TONC: The Affine Transformation Matrix The fl axes are the normal base vectors (note that y points down!), the blue axes are the transformed base vectors and the cyan variables are the arguments of the transformation. However, the affine routines in BIOS use a counter-clockwise rotation, and I thouht it'd be a good idea to use that as a guideline for my functions. The transformation matrix for this is A, which ties texel p to screen pixel q via q= A · p. user.chem.tue.nl /jakvijn/tonc/affine.htm   (2162 words)

 Highbeam Encyclopedia - Search Results for Affine transformation Bayesian inversion of piecewise affine operators in a Gaussian framework. Culture, Cognitive Models, and the Performance of Institutions in Transformation Countries. Nonparametric estimation of the transformation in the transform-both-sides regression model. www.encyclopedia.com /SearchResults.aspx?Q=Affine+transformation   (295 words)

 : Class Transform4   (Site not responding. Last check: 2007-10-21) holds the LU decomposition of matrix M for the reverse affine transformation. Generates a affine transformation which is the identity, i.e. Applies the transformation to a given vector of polynomials, i.e. www.esat.kuleuven.ac.be /~cwolf/jhfe/doc/HFE/Transform4.html   (242 words)

 Applying Affine Transformation to Images > What is Interpolation?   (Site not responding. Last check: 2007-10-21) This scenario illustrates that an application must be capable of performing manipulations in a random order in such a way that at every step operations are concatenated. Such capability would be difficult to implement without affine transformations. Because this chapter requires a thorough knowledge of affine transformations, you may want to read Chapter 4 first. www.informit.com /articles/article.asp?p=23667&seqNum=4   (1086 words)

 Affine Transformation Practicals   (Site not responding. Last check: 2007-10-21) The coefficients can be computed in one step. A reflection of the house in the X and then the Y axis (as a single transformation) This can be computed as a 180° degree clockwise rotation of the house. www.husdal.com /mscgis/affine3.htm   (126 words)

 Citebase - Quantum affine transformation group and covariant differential calculus   (Site not responding. Last check: 2007-10-21) Quantum affine transformation group and covariant differential calculus Authors: Aizawa, N. Sato, H. We discuss quantum deformation of the affine transformation group and its Lie algebra. We discuss quantum deformation of the affine transformation algebra. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9308049   (653 words)

 Citebase - Quantum Deformation of the Affine Transformation Algebra   (Site not responding. Last check: 2007-10-21) Authors: Aizawa, N. Sato, H. We discuss quantum deformation of the affine transformation algebra. Quantum affine transformation group and covariant differential calculus [ Abstract, 1 Cites, Cached PDF ] We discuss quantum deformation of the affine transformation group and its Lie algebra. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/9308010   (607 words)

 [Suif-talk] Loop/Data Transformation in SUIF2, affine transformation/omega unified transformation I can do what I intend to using this approach, but I would = prefer to use something (within SUIF1/2) based around maybe the SUIF = affine transformation framework (work in progress), the Omega = Project's affine transformation framework, Dave Wonnacott's Memory = Mappings. Does anyone know the implementation status and any impending release = dates of the affine transformation framework for SUIF2, and, or have any = experience of integrating Omega with SUIF2? I intend to do some iterative transformation work = using=20 SUIF-2 and SUIF-1. suif.stanford.edu /pipermail/suif-talk/2002-February/000966.html   (210 words)

 affine transformation from FOLDOC   (Site not responding. Last check: 2007-10-21) A linear transformation followed by a translation. Given a matrix M and a vector v, Nearby terms: af « AFAC « AFAIK « affine transformation » affordance » AFIPS » AFJ ftp.sunet.se /foldoc/foldoc.cgi?affine+transformation   (32 words)

 Homothetic Transformation   (Site not responding. Last check: 2007-10-21) Please look for Homothetic Transformation, Everyday Mathematics and Everyday Math to find more Homothetic Transformation information. Focused on improving the lives on children through literacy, members of Alpha Kappa Alpha Sorority, Inc. will convene at their 62nd National Convention in Detroit bringing millions of dollars to the city. 50) this.border=1; this.alt='Thumbnails by Thumbshots.org';">Homothetic transformation - RecipeFacts www.homothetictransformation.info   (312 words)

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