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Topic: Projective space

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	Projective space - Wikipedia, the free encyclopedia
		Projective spaces are essential to algebraic geometry through the rich field of projective geometry developed in the nineteenth century, but also in the constructions of the modern theory (based on graded algebras).
		Projective spaces and their generalisation to flag manifolds also play a big part in topology, the theory of Lie groups and algebraic groups, and their representation theory.
		The use of projective spaces makes quite rigorous the talk about a 'line at infinity' (where parallel lines meet), or a 'plane at infinity' for three dimensions: a translation of the latter can be made as part of the projective space associated to a four-dimensional real vector space.
	en.wikipedia.org /wiki/Projective_space (696 words)

	Projective Hilbert space - Wikipedia, the free encyclopedia
		This is the usual construction of projective space, applied to a Hilbert space.
		The physical significance of the projective Hilbert space is that in quantum theory, the wave functions ψ and λψ represent the same physical state, for any λ ≠ 0.
		In the case H is finite-dimensional the set of projective rays may be treated just as any other projective space; it is a homogeneous space for a unitary group or orthogonal group, in the complex and real cases respectively.
	en.wikipedia.org /wiki/Projective_Hilbert_space (189 words)

	PlanetMath: projective space
		Projective space is defined to be the set of the corresponding equivalence classes.
		Projective space also admits a more conventional type of coordinate system, called affine coordinates.
		This is version 4 of projective space, born on 2001-12-21, modified 2002-07-24.
	planetmath.org /encyclopedia/ProjectiveSpace.html (326 words)

	PROJECTIVE SPACE: THE RACIAL OTHER
		One has here a microcosm of a racist society in the projections, degradations and self-denigrations of these couples, which are inevitably passed on to their children so that attempts at integration at an individual level also perpetuate dimensions of racism in the very process of seeking to overcome it.
		It is formed through the projection and displacement of the group member's infantile self, of the sadistic child who survives within even the most compassionate adult.
		Projective identification of split-off primitive parts are here run riot but based on oppressive inequalities in the heat of summer.
	human-nature.com /mental/chap6.html (10230 words)

	Z4=An Electronic Display
		Classical affine and projective spaces have a simple relationship: removing part of a projective space leaves an affine subspace of the same dimension as the original projective space.
		Such projective spaces are called desarguesian, in honor of the French mathematician, Girard Desargues (1591-1661), who first introduced the principal concepts of projective geometry.
		The projective planes of 7, 13 and 21 points are well known as are the vector spaces which connect their affine subplanes.
	www.geocities.com /horst1925/projtext.html (1273 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.
		A linear transformation in projective n-space will scale, rotate, or shear a point set in (n+1) dimensions, but the mapping back to affine space shows that the configuration may be changed quite a bit.
	www.science.gmu.edu /~jsteidel/806-prj/appendix.html (705 words)

	Question Corner -- Understanding Projective Geometry
		Projective geometry can be thought of as the collection of all lines through the origin in three-dimensional space.
		The remaining points in projective space are horizontal lines through the origin in 3-d space; these are the "points at infinity".
		What all this means is that, in projective space, the "line" corresponding to l is actually a family of lines through the origin consisting of: (1) the lines that pass through l, and (2) the limiting horizontal line.
	www.math.toronto.edu /mathnet/questionCorner/projective.html (2444 words)

	Hilbert Space
		Hilbert space is not a space of simple points, rather it is a space of functions at a higher level of mathematical abstraction.
		Rather, Hilbert space is a mathematical device for arranging pieces of information, with each complex coordinate representing a possibility, or probability amplitude, for a given quantum state that might correspond to a definite eigenvalue for energy, or position, or momentum, or spin, etc. Note, that not all of these observable properties can be definite simultaneously.
		The angular momentum of a particle relative to a direction in space is incompatible with the angle of rotation of that particle in a plane perpendicular to that direction in space.
	www.qedcorp.com /pcr/pcr/hilberts.html (2670 words)

	Tangent and Secant Varieties and Isomorphic Projections (Site not responding. Last check: 2007-11-05)
		This is achieved through finding points in the ambient projective space which don't lie on either the tangent or secant varieties of the scheme.
		If X is projective, the tangent variety can be computed projectively but it is usually quicker to compute the result for an affine patch and then take the projective closure.
		For a scheme X in affine or ordinary projective space over a field, the secant variety SX is a subscheme of the ambient space whose set of closed points is the closure of the union of all lines joining distinct pairs of closed points of X (secants).
	magma.maths.usyd.edu.au /magma/htmlhelp/text1164.htm (963 words)

	Welcome to Adobe GoLive 4
		Projective space, however, is not adequate for the animation at the right.
		The problem is that projective space curves around on itself in a way difficult to visualize and, as a result, a straight line does not divide the projective plane into two disjoint halves.
		In two-sided projective space we ignore positive multiples of sets of homogenous coordinates but condsider two sets of coordinates different if one is a negative multiple of the other.
	www.math.umd.edu /users/gah/Pages/Overviews02.html (1422 words)

	Projective Transformations
		There is no emphasis on projective spaces of any particular dimension in a purely mathematical study of projective geometry, but in computer vision some cases are of more interest than others.
		Perspective projection is a particular type of projectivity called a perspectivity, in which all rays of projection pass through a single point - this puts constraints on the form of the matrix P as described in [Mundy 1992].
		A projectivity from a projective plane to a projective plane is called a plane-to-plane projectivity, although it is often referred to by simply using the more general term of projectivity.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/BEARDSLEY/node3.html (775 words)

	M6221 Lecture Notes 2a
		The vector spaces that we start from are a bit more general than those typically studied in a linear algebra course, in that we will permit the scalars to come from an algebraic structure more general than a field.
		Given a vector space, there is concept of dimension of the vector space (for finite dimensional vector spaces, this is the number of elements in a basis) and this is NOT the same as the concept of dimension in a geometry.
		Then P(V) is a projective space which we refer to as the projective space coordinatized by F. Lemma 2.1.2: (a) Let V' be a subspace of the vector space V. Then P(V') is a subspace of P(V).
	www-math.cudenver.edu /~wcherowi/courses/m6221/pglc2a.html (714 words)

	Fundamental theorem of projective geometry - Wikipedia, the free encyclopedia
		is a projective space and F and F′ are frames of P
		In case n = 1 this comes down to saying that given two ordered triples of distinct points, there is a projective transformation of the projective line taking the first triple to the second.
		This is a basic result on Möbius transformations, saying that the group they form is "triply" transitive.
	www.bucyrus.us /project/wikipedia/index.php/Fundamental_theorem_of_projective_geometry (171 words)

	Constructing Schemes (Site not responding. Last check: 2007-11-05)
		As shown in the examples in the introduction to this chapter, schemes are defined inside some ambient space, either affine or projective space, by a collection of polynomials from the coordinate ring associated with that space.
		For example, in ordinary projective space, there is one illegal point with all coordinates 0 and this is defined by the redundant ideal (x_1,..., x_n).
		As the process of saturation may be quite expensive in higher dimensional ambient spaces, the ideal of X is not saturated until the saturation property is required and once saturation has been performed, this is recorded internally.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1152.htm (856 words)

	Introduction and First Examples
		Projective spaces are normally defined in the same way, but they can also be defined with weights.
		Linear systems in projective space are simply collections of hypersurfaces having a common degree which are parametrised linearly by a vector space.
		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.
	www.umich.edu /~gpcc/scs/magma/text1018.htm (2824 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)

	Linear Systems
		In the projective case, this is the space of all homogeneous polynomials of degree d on P, whereas in the affine case it includes all polynomials of degree no bigger than d.
		If P is a projective space and F is a sequence of homogeneous polynomials all of the same degree defined on P, or if P is an affine space and F is a sequence of polynomials defined on P this returns the linear system generated by these polynomials.
		The map from the coefficient space of the linear system L to the polynomial ring that is the parent of the sections of L. When evaluated at a vector v, this map will return the polynomial section of L whose coefficients with respect to the basis of L are v.
	www.math.niu.edu /help/math/magmahelp/text975.html (3293 words)

	Ambients
		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.
		As with ordinary projective space, polynomials of a given bidegree form a vector space with a favourite basis of monomials.
	www.math.lsu.edu /magma/text1151.htm (2167 words)

	[No title]
		Physically, minimal projections correspond to "pure states" - states of affairs in which the answer to some maximally informative question is "yes", like "is the z component of the angular momentum of this spin-1/2 particle equal to 1/2?" Geometrically, the space of minimal projections is just the space of "lines" in our Hilbert space.
		The quotient space is 16-dimensional - twice the dimension of the octonions.
		The quotient space is 32-dimensional - twice the dimension of the bioctonions.
	math.ucr.edu /home/baez/twf_ascii/week106 (4547 words)

	Introduction (Site not responding. Last check: 2007-11-05)
		Projective geometry models well the imaging process of a camera because it allows a much larger class of transformations than just translations and rotations, a class which includes perspective projections.
		Projective transformations preserve type (that is, points remain points and lines remain lines), incidence (that is, whether a point lies on a line), and a measure known as the cross ratio, which will be described in section 2.4.
		The purpose of this monograph will be to provide a readable introduction to the field of projective geometry and a handy reference for some of the more important equations.
	ai.stanford.edu /~birch/projective/node1.html (471 words)

	Robert T. Collins // Publications // AICV93 (Site not responding. Last check: 2007-11-05)
		In this paper, uncertainty in projective elements is represented and manipulated using probability density functions in projective space.
		Projective n-space can be visualized using the surface of a unit sphere in (n+1)-dimensional Euclidean space.
		This two-to-one map from the unit sphere to projective space enables probability density functions on the sphere to be interpreted as probability density functions over the points of projective space.
	www.cs.cmu.edu /People/rcollins/Pub/aicv93.html (191 words)

	The Math Forum - Math Library - Projective Geom. (Site not responding. Last check: 2007-11-05)
		Basics, path curves, counter space, pivot transforms, and some people involved in the development of projective geometry, which is concerned with incidences: where elements such as lines planes and points either coincide or not.
		It defines a set of geometric primitives in 3d space and allows for the construction of geometric models that can be manipulated interactively, while defined geometric relationships remain invariant.
		A brief definition of projective geometry, by the author of an honours dissertation covering ideas from the areas of projective geometry and group theory.
	mathforum.org /library/topics/projective_g (1073 words)

	55:148,55:247 Chapter 9, Part 2
		Points in the projective space are expressed in homogeneous (also projective) co-ordinates, which we will denote in bold with a tilde.
		The scene point ~X_w is expressed up to scale in homogeneous co-ordinates (recall that projection is expressed in the projection space) and thus all alpha, M are equivalent for alpha not equal to 0.
		The projection matrix M is estimated from the co-ordinates of points with known scene positions.
	www.icaen.uiowa.edu /~dip/LECTURE/3DVisionP1_2.html (3013 words)

	Projective Lines
		Projective lines are not very interesting from the viewpoint of axiomatic projective geometry, since they have only one line on which all the points lie.
		These nontrivial projections all have rank 1, so they are the points of our would-be projective space.
		Our would-be projective space has just one line, corresponding to the projection 1, and all the points lie on this line.
	math.ucr.edu /home/baez/octonions/node9.html (1085 words)

	Schemes
		If F is a sequence of schemes lying in a common ambient space whose base ring admits automatic coercion to K or is the domain of a ring map m then this returns the base change of the elements of F as a new sequence.
		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.
		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.math.niu.edu /help/math/magmahelp/text973.html (7384 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		For example if we intersect a hypersurface X in projective space with a linear space which does contain it we will obtain a hypersurface in the linear space.
		After a projective linear transformation we may take the linear space to be x_k=x_[k+1}=...
		The only Veronese variety which is a hypersurface is the rational normal curve in the projective plane (generally a Veronese variety is an embedding of projective n-space in projective N space where N exceeds n+1 except for the case of the conic in P^2).
	www.math.rutgers.edu /courses/535/535-f02/why.html (288 words)

	Projective space (Site not responding. Last check: 2007-11-05)
		This is a generator-only picture of the smallest projective space.
		Another remarkable fact about this picture is that the five generator lines partition the point set of the space.
		Furthermore, the 6 images of this spread are also spreads and all 7 spreads partition the line set of the space.
	www.maths.monash.edu.au /~bpolster/space.html (192 words)

	PROJECTIVE GEOMETRY (Site not responding. Last check: 2007-11-05)
		Projective geometry is a beautiful subject which has some remarkable applications beyond those in standard textbooks.
		Steiner's spiritual research showed that there is another kind of space in which more subtle aspects of reality such as life processes take place.
		Adams took his descriptions of how this space is experienced and found a way of specifying it geometrically, which is dealt with in the Counter Space Page.
	www.anth.org.uk /NCT (189 words)

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