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Topic: Hyperplane at infinity

	Projective geometry vs. affine geometry
		is the subgroup of projective transformations that fix the hyperplane at infinity.
		It can be easily interpreted in projective geometry if we fix the hyperplane at infinity.
		are parallel if and only if all the points in their intersection lie on the hyperplane at infinity.
	www.math.poly.edu /courses/projective_geometry/chapter_four/node2.html (410 words)

	Reference.com/Encyclopedia/Hyperplane at infinity
		A pair of non-parallel affine hyperplanes intersect at an affine subspace of dimension n − 2, but a parallel pair of affine hyperplanes intersect at a projective subspace of the ideal hyperplane (the intersection lies on the ideal hyperplane).
		Thus, parallel hyperplanes, which did not meet in the affine space, intersect in the projective completion due to the addition of the hyperplane at infinity.
		In a projective space, any hyperplane may be chosen to be the hyperplane at infinity.
	www.reference.com /browse/wiki/Hyperplane_at_infinity (406 words)

	Plane at infinity - Wikipedia, the free encyclopedia
		In projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties.
		Note that since the (real) plane at infinity is a projective plane, it is homeomorphic to the surface of a "sphere modulo antipodes", i.e.
		In effect, what the plane at infinity does is to add a point at infinity to every line, converting it into a projective line, and to add a line at infinity to every plane, converting it into a projective plane.
	en.wikipedia.org /wiki/Plane_at_infinity (525 words)

	Erlangen programme
		The solution in abstract terms was to use symmetry as an underlying principle, and to state first that different geometries could co-exist, because they dealt with different types of propositions and invariances related to different types of symmetry and transformation.
		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).
		The affine group[?] will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen hyperplane at infinity.
	www.ebroadcast.com.au /lookup/encyclopedia/er/Erlangen_programme.html (586 words)

	Qhull format options (F)
		The hyperplane is a perpendicular bisector if the midpoint of the input sites lies on the plane, all Voronoi vertices in the ridge lie on the plane, and the angle between the input sites and the plane is ninety degrees.
		It is an offset from the facet's hyperplane.
		It corresponds to a facet of the dual polytope.
	www.msri.org /about/computing/docs/qhull/qh-optf.htm (2465 words)

	Classification with boosted dyadic kernel discriminants - Patent 7076473
		The method of claim 1 wherein a linear hyperplane is defined by a discriminant function of the form f(x}=+b, where w is the projection vector w, and b is a bias, and sgn f(x).di-elect cons.{-1, +1} denotes a binary classification.
		A linear hyperplane classifier according to the invention is defined by a discriminant function of the form f(x}=+b, (1) where sgn f(x).di-elect cons.{-1, +1} denotes the binary classification.
		The discrete-valued hyperplane classifier is given by a sign threshold h.sub.ij(x)=sgn (f.sub.ij(x)), (15) whereas a "confidence-rated" classifier with output in the range [-1, +1] can be obtained by passing f.sub.ij() through a bipolar sigmoidal non-linearity, such as a hyperbolic tangent h.sub.ij(x)=tan h(.beta.f.sub.ij(x)), where.beta.
	www.freepatentsonline.com /7076473.html (4375 words)

	Point at infinity - Wikipedia, the free encyclopedia
		The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line,
		Since the lines are parallel, they intersect at a point at infinity which lies on
		When a pair of projective lines are parallel they intersect at their common point at infinity.
	en.wikipedia.org /wiki/Point_at_infinity (236 words)

	IBM: Optimization on a Fractal Feasible Region
		Since the hypersphere encloses the fractal, and the union of its images under the various mappings also encloses the fractal, the fractal must lie entirely on the side of the hyperplane where the images of H are.
		Another hyperplane is obtained by choosing a point on the intersection of H and the second hyperplane, to the fractal side of both the first and second hyperplanes.
		Continue this process, always selecting the new point on the intersection of H and one of the hyperplanes, and always to the fractal side of all the hyperplanes.
	www.vm.ibm.com /devpages/GREER/FRACTOPT.HTML (1703 words)

	Homogeneous Transformation Matrices
		The "hyperplane" is the class of all such representations, but for convenience we may identify a particular representation as the hyperplane.
		Hyperplane matrices are represented by lower case letters, or by lower case superscripts.
		The normal of the hyperplane at infinity, w, is undefined.
	www.silcom.com /~barnowl/HTransf.htm (4174 words)

	PlanetMath: proof of Borsuk-Ulam theorem
		Proof of the Borsuk-Ulam theorem: I'm going to prove a stronger statement than the one given in the statement of the Borsak-Ulam theorem here, which is:
		By cellular approximation, this may be assumed to take the hyperplane at infinity (the
		Cross-references: contradiction, factors, restricted, injective, sequence, antipodal map, cycles, generated by, commutativity, isomorphism, exact sequences, morphism, homotopic, class, homotopy, covering, lifts, structure, cell, infinity, hyperplane, approximation, sphere, induction, degree, map, odd, Borsuk-Ulam theorem
	planetmath.org /encyclopedia/ProofOfBorsukUlamTheorem.html (228 words)

	UC Davis Math: Glossary (Site not responding. Last check: 2007-10-30)
		A convex body has constant width if any two parallel hyperplanes that touch the convex body on opposite sides are the same distance apart.
		A homeomorphism of the circle at infinity of the hyperbolic plane to itself, if it is sufficiently nice in a natural sense, extends uniquely to an earthquake.
		The general notion includes line fields and hyperplane fields; line fields are very similar to vector fields such as the magnetic field.
	www.math.ucdavis.edu /profiles/glossary.html (9932 words)

	Infinity Information Portal @ Infinitely.org (Site not responding. Last check: 2007-10-30)
		For example, in optics, an object which is much further away than the focal length of a lens is said to be "at infinity", as the distance of the image from the lens varies very little as the distance increases further.
		In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes.
		In mathematics, infinity is relevant to, or the subject matter of, limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite.
	www.infinitely.org (1491 words)

	[No title]
		A horosphere has infinite radius and meets the sphere at infinity at one point, and its geometric center is the same point.
		A surface (or a manifold) which locally minimizes surface area (or surface volume), which means that one cannot replace small patches of the surface and decrease the area.
		Riemann sphere A topological sphere consisting of the complex plane and the point at infinity; an example of a Riemann surface.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	Geometric morphometrics glossary (part 1) (Site not responding. Last check: 2007-10-30)
		For more than three landmarks, a given transformation has only one direction of covariants, but a full plane (four landmarks) or hyperplane (five or more landmarks) of invariants (see the Orange Book).
		Similarly, to declare an interpolation (such as a thin-plate spline) a "homology map" means that one intends to refer to its features as if they had something to do with valid biological explanations pertaining to the regions between the landmarks, about which we have no data.
		A hyperplane is typically characterized by the vector to which it is orthogonal.
	life.bio.sunysb.edu /morph/glossary/gloss1.html (4285 words)

	Basic concepts
		A projective subspace of dimension one is called a projective line and a projective subspace of codimension one is called a projective hyperplane.
		is called the hyperplane at infinity (visualize this in figure 1).
		As the next exercise shows, we can make any projective hyperplane be our hyperplane at infinity.
	www.math.poly.edu /courses/projective_geometry/chapter_four/node1.html (297 words)

	Options
		corresponds to the infinity constraint which is added for vertex/ray enumeration when the input system is not homogeneous.
		One can ignore the infinity plane for some purposes, but for analyzing the combinatorial structure of polyhedra, it is very important information.
		The 7th inequality is redundant because the first four facets intersects at a single infinity point (corresponding to a unique extreme ray) and hence the polyhedron has no infinity facet, although the polyhedron is not bounded.
	www.cs.mcgill.ca /~fukuda/soft/cddman/node4.html (2502 words)

	Distribution of sum of N continuous random variables - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-10-30)
		And intuitively I am guesisng the sum of N random variables as N approaches infinity is normal.
		The set of all points the coordinates of which have a sum of X. For instance, (0,1,1) is on the (sum_i x_i = 2) hyperplane, because 0 + 1 + 1 = 2.
		The density function is the area of the intersection of the hyperplane and the cube.
	www.gamedev.net /community/forums/topic.asp?topic_id=424672 (1480 words)

	Springer Online Reference Works
		The poles of proper hyperplanes are ideal points, and the proper points are the poles of ideal hyperplanes.
		There are several conformal interpretations of a Lobachevskii space, one of which is the Poincaré model.
		It is also possible to have a conformal interpretation of the space on one of its hyperplanes.
	eom.springer.de /L/l060050.htm (616 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'.
		This construct gives an invariant notion of parallelism because two lines are called parallel if they meet in 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.
	www.lns.cornell.edu /spr/2003-11/msg0056053.html (437 words)

	Amazon.com: "improper hyperplane": Key Phrase page (Site not responding. Last check: 2007-10-30)
		The plane that is parallel to the screen and passes through the projector is called the improper hyperplane.
		..,x")Er(k):xo:A 01, that is the complement of the improper hyperplane or hyperplane at infinity Xo = 0, and whose inverse is the map which associates to each projective point (xo,x1,...
		This hyperplane is called the ideal hyperplane or the hyperplane at infinity (or the improper hyperplane).
	www.amazon.com /phrase/improper-hyperplane (469 words)

	Qhull functions, macros, and data structures
		A hyperplane is defined by d normalized coefficients and an offset.
		A centrum is a point on the facet's hyperplane that is near the center of the facet.
		A nonsimplicial facet is an approximation that is defined by offsets from the facet's hyperplane.
	www.msri.org /about/computing/docs/qhull/qh-c.htm (4658 words)

	M6221 Lecture Notes 1e
		is called the hyperplane at infinity and its points are called points at infinity (sometimes improper points).
		The points of this plane (points at infinity) are those with last coordinate 0.
		Each line not parallel to a hyperplane H meets H in precisely one point of A.
	www-math.cudenver.edu /~wcherowi/courses/m6221/pglc1e.html (432 words)

	libecc: Elliptic Curve Cryptography C++ Library - Reference Manual
		As one of the goals of this project is to reach people who did not study mathematics, lets first explain what a hyperplane means.
		A hyperplane is a subspace of one dimension less than the space in which it is contained (mathematicians would say, of codimension 1) that is 'flat'.
		is a hyperplane and that n is the normal of that hyperplane.
	libecc.sourceforge.net /reference-manual/group__theory__aspace.html (1955 words)

	list of talks
		We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain families of boundary singularities along the hyperplane at infinity.
		We construct an effective algorithmic method to compute the homological monodromy of a complex polynomial in two variables which is good at infinity.
		Applications are given to compute the intersection matrix of Iomdine surfaces in distinguished basis of vanishing cycles and to prove the existence of conjugate polynomials in some number field which are not topologically equivalents.
	math.univ-lille1.fr /~tibar/Sing03/talks.htm (873 words)

	IMUJ: Acta Mathematica
		The generalization we have in mind allows that some of the zeros are on the hyperplane at infinity.
		Moreover a whole series of formulas will be considered, the first one being a classical formula of Jacobi [5] and the second one the above mentioned formula of B. Segre.
		This paper collects together formulae concerning singularities at infinity of plane algebraic curves.
	www.im.uj.edu.pl /badania/acta/spis39.html (1532 words)

	Amazon.com: "projective hyperplane": Key Phrase page (Site not responding. Last check: 2007-10-30)
		A projective hyperplane H is a proper subspace of a linear space S such that each line of S has a point in...
		Now P (E)' is, by definition, the set of projective hyperplanes in the projective space P (E).
		A projective hyperplane of IP (E) is the projectivization [S] of a hyperplane S C E. Key Phrases in this book: Using the Maurer-Cartan
	www.amazon.com /phrase/projective-hyperplane (456 words)

	The Miracle Octad Generator
		It so happens that the octad stabilizer, which may be viewed as acting separately on the stabilized octad and on the set of the remaining 16 points of the 24, is isomorphic to the automorphism group of the affine 4-space over the 2-element field.
		The group of the geometry of the affine n-space is the subgroup of the group of the projective n-space leaving the "hyperplane at infinity" invariant (as a set, not pointwise), as in Veblen's discussion above.
		One might argue that a stabilized octad in the Mathieu geometry plays a role in some way analogous to that of a "hyperplane at infinity."
	finitegeometry.org /sc/24/MOG.html (827 words)

	Canonical Injection of into
		=0 behaves like a hyperplane, called the hyperplane at infinity.
		In a general projective space any coordinate (or linear combination) can act as the homogenizing coordinate and all hyperplanes are equivalent -- none is especially singled out as the ``hyperplane at infinity''.
		These issues will be discussed more fully in chapter 4.
	www.cse.iitd.ernet.in /~suban/vision/tutorial/node13.html (106 words)

	generation5 - Perceptrons
		Perceptrons can only classify data when the two classes can be divided by a straight line (or, more generally, a hyperplane if there are more than two inputs) — this is called linear separation.
		To explain the concept of linear separation further, let us look at the function shown to the right.
		To look at something a little more complicated, take a look at Perceptrons being applied to optical character recognition or just play with the ONR applet.
	www.generation5.org /content/1999/perceptron.asp (1163 words)

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