
	Where results make sense

Topic: Affine combination

Related Topics

affine transformations

affine geometry

affine space

affine scheme

Affine representation

Affine cipher

affine varieties

affine maps

affine connection

Affine combination

affine function

Affine morphism

In the News (Wed 23 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Affine - Wikipedia, the free encyclopedia
		An affine combination is a linear combination of vectors.
		An affine representation, as applied to a topological group G, is a continuous homomorphism from G to the automorphism group of an affine space, A.
		Affine connections are connections on the tangent bundle of a differentiable manifold.
	en.wikipedia.org /wiki/Affine (376 words)

	Affine transformation - Wikipedia, the free encyclopedia
		A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations.
		An affine combination is a linear combination in which the sum of the coefficients is 1.
		The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace.
	en.wikipedia.org /wiki/Affine_transformation (1053 words)

	Affine transformation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-10)
		An (additional info and facts about affine combination) affine combination is a linear combination in which the sum of the coefficients is 1.
		An affine subspace of a vector space is a (additional info and facts about coset) coset of a (additional 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.
	www.absoluteastronomy.com /encyclopedia/a/af/affine_transformation.htm (692 words)

	Dimension, Polyhedra, and Faces (Site not responding. Last check: 2007-09-10)
		Definition 8 The dimension of an affine space is the dimension of the corresponding subspace.
		Definition 9 The affine hull of a set is the set of all affine combinations of points in the set.
		Definition 10 The dimension of a polyhedron is the dimension of its affine hull.
	www.rpi.edu /~mitchj/matp6620/handouts/faces (197 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.
	www.mpi-sb.mpg.de /~kettner/courses/lib_design_03/notes/gkernel.html (1098 words)

	[No title]
		Affine space: We will usually be rather informal in our presentation of geometry, but it is good to start things off on a somewhat formal footing.
		Given any scalar a and two points p and q, define the affine combination Aff(p,q,a) to be: (1 - a) * p + a * q = p + a * (p - q) Note that the left-hand side of this equation is not legal given our list of operations.
		But this is how the affine combination is typically expressed, namely as the weighted average of two points.
	www.cse.wustl.edu /~pless/506/l2.html (1841 words)

	Mathematical Programming Glossary - A (Site not responding. Last check: 2007-09-10)
		A linear combination of vectors, Sum_k{a_k x^k}, whose coefficients sum to 1 -- i.e., Sum_k{a_k} = 1.
		Affine hull of a set, S. The intersection of all affine sets containing S. Equivalently, the set of all affine combinations of points in the set.
		Affine set (or affine manifold, affine variety, linear variety, flat.) One that contains the line through any two of its points; i.e., x, y in S implies ax + (1-a)y is in S for all real values, a.
	carbon.cudenver.edu /~hgreenbe/glossary/A.html (2792 words)

	Perspective as a Symmetry Transformation by Gyorgy Darvas in the Nexus Network Journal vol. 5 no. 1 (Spring 2003) (Site not responding. Last check: 2007-09-10)
		Affine projection is a symmetry transformation in which straight lines are transformed into other straight lines but angles are not conserved in this transformation (Figure 5).
		Thus they were able to give up the straight lines and fixed direction demanded by the affine projection, and replace this by a topology, making it possible for them to stress certain important features of the represented object.
		Therefore we should also accept the sophisticated forms of combined symmetries that appear in modern art, and which are products of a long development in multiplying and transforming the vanishing points and the artist's viewpoints, as manifestations of perspective representation.
	www.nexusjournal.com /Darvas.html (3848 words)

	Line at infinity
		In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to incidence properties of the resulting projective plane.
		However, this circle is actually like cross-cap, which is homeomorphic to a Möbius strip: diametrically opposite points of the circle are equivalent -- they are the same point.
		Topologically this is quite different, in that it is a Riemann sphere, which is therefore a 2-sphere, being added to a complex affine space of two dimensions over C (so four real dimensions), resulting in a four-dimensional compact manifold.
	encyclopedia.codeboy.net /wikipedia/l/li/line_at_infinity.html (714 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		The span of vectors is the union of all linear combinations of the vectors.
		Contrast a vector space to an Affine Space that is a collection of points with no inherent origin but that does have an associated vector space and two operations with associativity and an identity element.
		A convex combination is when the sum of the parametric coefficients is less than or equal to one.
	www.cs.helsinki.fi /group/goa/matem/matiks.html (341 words)

	affine - GameDev.Net Discussion Forums
		This allows a rotation to occur around a point that is not the origin and allows the shearing of an object (which is used in the perspective portion of the graphics pipeline).
		It can also be shown that an affine combination can be expressed as a sum of one point and some vectors.
		Geometrically, an affine combination of two points gives a point on the line segment joining the two points, an affine combination of three points gives a point in the triangle whose vertices are the three points, and so on.
	www.gamedev.net /community/forums/viewreply.asp?ID=1011074 (1515 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		Affine combination -- sum of the coefficients of a linear combination is 1
		Since this result is symmetrical in A, B and C, it must also be two-thirds of the way along the median from B and two-thirds of the way along the median from C. Hence the three medians meet there, and G is the centroid.
		This result generalizes nicely for a regular polygon with N sides: the centroid is simply the average of the location of the N vertices, another affine combination
	www.csci.csusb.edu /tongyu/courses/cs420/notes/viewing1.php (1546 words)

	detection and long term tracking of moving objects in aerial video
		The system uses a combination of affine image registration and local motion estimation to detect patches of the image that may be moving differently from the background.
		On a given frame, it computes the affine transformation from the previous frame to the current frame, and places that transformation in a circular array of results that covers the frames in the system frame span.
		Other modules request affine parameters for a particular frame t (which is the transformation of the background from the frame t-1 to the frame t), and the background estimator replies with a vector of the 6 parameters that define an affine transformation.
	www.cs.cornell.edu /Vision/wbell/identtracker (2221 words)

	Robotics Institute Seminar (Site not responding. Last check: 2007-09-10)
		The GLC model is both general and linear in the sense that, given any vector space where rays are represented as points, it describes all 2D affine subspaces (planes) formed by the affine combination of 3 rays.
		The incident radiance seen along the rays of these 2D affine subspaces are a precise definition of a projected image of a 3D scene.
		Since the GLC model provides a complete description of all 2D affine subspaces, it can be used as a tool for first-order differential analysis of arbitrary (higher-order) multiperspective imaging systems.
	www.cs.cmu.edu /~ri-seminar/archives/2004.fall/2004.Oct.1.html (266 words)

	HINT (Site not responding. Last check: 2007-09-10)
		This is called an affine combination of a, b, and c.
		are all nonnegative, then this is called a convex combination of a, b, and c and x in that situation is inside the triangle.
		Find the expression of x as an affine combination of the vertices of the triangle and see if it is a convex combination.
	www.mathphysics.com /spingarn/vec/flat_geometry/planes/Ptri-int2_h.html (72 words)

	lignumCAD Technical Reference (Site not responding. Last check: 2007-09-10)
		Affine spaces are at the heart of OpenGL, which also has excellent documentation.
		An affine transformation is a type of linear transformation which preserves the affine combination of two points:
		A rotation is performed by picking a vector as a reference direction and then twisting the affine space around that direction.
	lignumcad.sourceforge.net /doc/en/HTML/SOHTML/TechnicalReference.html (3092 words)

	Affine Spaces (Site not responding. Last check: 2007-09-10)
		S is an affine space if it is closed under affine combinations.
		An affine space is a translation of a subspace.
		Note that every point in a polytope is a convex combination of the extreme points.
	www.rpi.edu /~mitchj/matp6640/affine/affine.html (183 words)

	Affine - TheBestLinks.com - Affine transformation, Affine geometry, Affine group, Affine space, ...
		Affine - TheBestLinks.com - Affine transformation, Affine geometry, Affine group, Affine space,...
		Affine, Affine transformation, Affine geometry, Affine group, Affine space...
		This is a disambiguation page, i.e., a navigational aid which lists other pages that might otherwise share the same title.
	www.thebestlinks.com /Affine.html (110 words)

	Graphics lecture notes 21: Bezier and Spline curves (Ch.11)
		These forms may also be considered to be weighted averages (affine combinations) of the control points, where the weights are ratios of polynomials in t.
		Affine Invariance: Applying an affine transformation T to the control points and computing the corresponding Bezier curve, has the same result as taking a Bezier curve and transforming each of its points using T. This is because affine transformations preserve affine combinations (of points).
		Invariance properties: B-spline curves are invariant under only affine transformations, but NURBS curves are invariant under projective transformations, in which the fourth row of the matrix is not constrained to be (0 0 0 1); these include perspective transformations.
	www.cis.syr.edu /~mohan/gra21.html (1635 words)

	CS223 Vision Final Project: Jeremy Weinberger (Site not responding. Last check: 2007-09-10)
		This method allows for enhanced accuracy over affine tracking, and retargeting of the motion sequence to a different 2D or 3D model.
		The least-square affine deformation is still obviously based on the original contour.
		The idea behind this method is to identify key poses of the object to be tracked, and to estimate arbitrary motions and poses of the object as a combination of affine parameters and weighted sums of the key poses.
	mrl.nyu.edu /~jeremy/vision/final (743 words)

	The Stanford SUIF Compiler Group - Affine Transformations (Site not responding. Last check: 2007-09-10)
		Developed an algorithm that finds the optimal affine partitioning that maximizes the degree of parallelism while minimizing the degree of synchronization.
		Developed an algorithm that uses affine partitioning to improve the data locality of uniprocessor and multiprocessor programs.
		A combination of affine transformations to improve parallelization and locality has allowed a speed up of 20 times on a 32-processor, which represents over a three-fold improvement over the previously best results.
	suif.stanford.edu /research/affine.html (367 words)

	PPL: General Information on the PPL
		of the considered variable in the linear expression is non-zero, the affine transformation is invertible.
		Thus, in the sequel, by PPL representation of a polyhedra, we are referring to the corresponding representation of its corresponding polyhedral cone.
		The affine dimension of a facet is equal to
	www.cs.unipr.it /ppl/Documentation/devref/main.html (6864 words)

	Mathematical Background
		x(t) as a linear combination of (-t)a + tb followed by a translation a.
		Now notice that in the linear combination the coefficients of a and b sum up to 0.
		An important property of an affine map: Recall that an affine map is just a linear transformation followed by a translation and is given by
	www.math.hmc.edu /faculty/gu/math142/mellon/Application_to_CAGD/Mathematical_Background.html (636 words)

	Computer Graphics : Mathematics : 5 / 23 : Affine Space
		A affine space consists of a set, called the points of the affine space, an associated vector space and two operations which connect the affine and the vector space
		Given a point P and a vector u, we can add u to P and get another point in the affine space
		Affine combination of the points P and Q by the real number t : a point such as :
	escience.anu.edu.au /lecture/cg/Maths/affineSpace.en.html (250 words)

	s_convex
		, a barycentric combination is a linear combination in which the coefficients add to 1.
		(b) Show, conversely, that if a linear combination is preserved by all translations, or even by one nontrivial translation, then the linear combination is barycentric.
		Prove that the image of a convex set under an affine transformation is convex.
	www.math.ucla.edu /~baker/149.1.02w/handouts/s_convex/node7.html (1276 words)

	Vectors (Site not responding. Last check: 2007-09-10)
		An affine space consists a set of points, a derived vector space, and two operations viz.
		However, the weighted sums of points is called as barycentric combination, also called the affine combination is defined for points, where the weights sum to one.
		Every affine map can be composed of scalings, rotations, translations, shears.
	www.cs.fit.edu /wds/classes/cse5255/thesis/Vector/Vector.html (483 words)

	Use of Randomization in Program Analysis (Site not responding. Last check: 2007-09-10)
		First, both branches of a conditional are executed on each run and at joint points we perform an affine combination of the joining states.
		Secondly, in the branches of an equality conditional we adjust the data values on the fly to reflect the truth value of the guarding Boolean expression.
		The algorithm is simpler to implement than alternative deterministic versions, has better computational complexity, and has an extremely small probability of error for even a small number of runs.
	www.eecs.berkeley.edu /IPRO/Summary/03abstracts/gulwani.1.html (377 words)

	Geometric Construction Of The Levi-Civita Parallelism - Kock (ResearchIndex) (Site not responding. Last check: 2007-09-10)
		In terms of synthetic differential geometry, we give a variational characterization of the connection (parallelism) associated to a pseudo-Riemannian metric on a manifold.
		Introduction A basic result in differential geometry is: to a Riemannian metric on a manifold, there exists a unique symmetric affine connection compatible with it.
		Recall that an affine combination is a linear combination where the sum of the coefficients is 1; and it is a...
	citeseer.ist.psu.edu /kock98geometric.html (434 words)

	Chapter 4 Vector Tools for Graphics
		Face of a 3D polygon is a convex combination of its vertices.
		Linear combination of points is point iff it is affine
		Affine combination of points: P = A (1 - t) + B t
	www.eng.mu.edu /corlissg/151.04Sp/ch4.html (1054 words)

	C++ Point and Vector Class
		Although we try to be strict in defining appropriate point and vector operations, there is one area of difficulty; namely, points can only be added as affine sums, whereas any weighted sum works for vectors.
		It's too bad that C++ operator overloading does not permit detection of whether a sum is affine or not, and that's the rub.
		An affine sum of points defines another point object, which is in a unique spatial location no matter which coordinate frame of reference is used.
	geometryalgorithms.com /Archive/algorithm_0301/algorithm_0301.htm (2080 words)

Try your search on: Qwika (all wikis)