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Topic: Projective plane

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	Klein Bottle & Projective plane (Site not responding. Last check: 2007-10-27)
		[2]: It is a cruciform projective plane on transparent Cross-cap.
		are disk-type projective planes that correspond to the upper and lower of doked Cross-cap respectively.
		The whole of projective plane is represented by a hemisphere or disk-type projective plane or Cross-cap.
	www1.kcn.ne.jp /~iittoo/us27i_klpr.htm (3773 words)

	Projective plane - Wikipedia, the free encyclopedia
		In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from the combinatorics of block designs.
		The most common projective plane is the real projective plane, which is a topological surface with surprising geometric properties; after that is the complex projective plane of algebraic geometry, a topological four-dimensional manifold.
		In the case of finite projective planes, the only proof known of the purely geometric statement that Desargues theorem then implies Pappus' theorem (the converse being always true and provable geometrically) is through this algebraic route, using Wedderburn's theorem that finite division rings must be commutative.
	en.wikipedia.org /wiki/Projective_plane (1005 words)

	Station Information - Projective plane
		It is easy to check that it obeys the rules required of projective planes: any pair of distinct great circles meet at a pair of antipodal points, and any two distinct pairs of antipodal points lie on a single great circle.
		In this representation of the Fano plane, the seven points are shown as small blobs, and the seven lines are shown as six line segments and a circle.
		The definition of projective plane by incidence properties is something special to two dimensions: in general projective space is defined via linear algebra.
	www.stationinformation.com /encyclopedia/p/pr/projective_plane.html (486 words)

	Z4=An Electronic Display
		The ideal plane is on exactly 7 ideal lines and each ideal line is on exactly two projective planes that intersect the 8 affine points in two parallel affine planes, each with exactly 4 points.
		With an additional postulate, some permutations of points have a geometric significance: they may also induce permutations of lines, planes, etc. In the case of a projective plane, when a permutation of points also induces a permutation of lines, it is called a collineation.
		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)

	The Real Projective Plane
		It is probably the simplest example of a closed non-orientable surface; removing a disc from the real projective plane may yield another familiar non-orientable surface, the Möbius band.
		The real projective plane was one of the first (if not the first) post-Enlightenment examples of a non-Euclidean geometry.
		The real projective plane is a central object in a classical subject known as "projective geometry", which was an outgrowth of the work of the Renaissance artists and some later geometrically-minded philosophers, especially Jean Victor Poncelet, who undertook to axiomatize its geometry.
	homepages.wmich.edu /~drichter/rptwo.htm (1076 words)

	Projective space (Site not responding. Last check: 2007-10-27)
		While the theory of projective planes has an aspect that belongs to combinatorics too, that is absent in the general case.
		Projective space is basic in 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).
		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.
	www.sciencedaily.com /encyclopedia/projective_space (522 words)

	PlanetMath: projective plane
		) and the affine plane is a grid of
		For finite planes both conditions are the same by Wedderburn's theorem, so Desarguesian and Pappian are synonyms.
		This is version 3 of projective plane, born on 2005-04-10, modified 2005-04-11.
	planetmath.org /encyclopedia/ProjectivePlane2.html (1356 words)

	projective plane
		The projective plane needs a fourth dimension, in addition to three we live in, to be fully realized.
		The idea of the projective plane arose from the study of perspective by mathematicians and painters in the Renaissance.
		The study of the geometry that adds this extra line of ideal points to the ordinary familiar plane came to be known as projective geometry, because of its use in studying projections of figures onto different lines.
	www.daviddarling.info /encyclopedia/P/projective_plane.html (274 words)

	PlanetMath: finite projective plane
		These two conditions, diameter 3 and girth 6, are not only necessary for the graph to represent a projective plane, they are also sufficient, and therefore form an alternative formulation of what it means to be a projective plane.
		There are skew fields, semifields, (left and right) near fields and quasifields, and their algebraic properties translate to symmetries of the projective planes co-oordinatised by them.
		This is version 2 of finite projective plane, born on 2005-04-11, modified 2005-08-31.
	planetmath.org /encyclopedia/FiniteProjectivePlane4.html (1726 words)

	Real projective plane - Wikipedia, the free encyclopedia
		In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space.
		It is often described intuitively, in relation with a Möbius strip: it would result if one could glue the single edge of the strip to itself in the correct direction.
		The resulting surface, a 2-dimensional compact non-orientable manifold, is a little hard to visualize, because it cannot be embedded in 3-dimensional Euclidean space without intersecting itself.
	en.wikipedia.org /wiki/Real_projective_plane (264 words)

	Projective plane -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-27)
		this is an example of the ((geometry) the interchangeability of the roles of points and planes in the theorems of projective geometry) duality of projective planes:
		For all known finite projective planes, the order is a prime power.
		The definition of projective plane by incidence properties is something special to two dimensions: in general (Click link for more info and facts about projective space) projective space is defined via (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra.
	www.absoluteastronomy.com /encyclopedia/p/pr/projective_plane.htm (504 words)

	AllRefer.com - projective geometry (Mathematics) - Encyclopedia
		The basic elements retain their character under projection; e.g., the projection of a line is another line, and the point of intersection of two lines is projected into another point that is the intersection of the projections of the two original lines.
		The concept of parallelism does not appear at all in projective geometry; any pair of distinct lines intersects in a point, and if these lines are parallel in the sense of Euclidean geometry, then their point of intersection is at infinity.
		The plane that includes the ideal line, or line at infinity, consisting of all such ideal points, is called the projective plane.
	reference.allrefer.com /encyclopedia/P/projctgeo.html (326 words)

	Problem 14: The Projective Plane
		Now that we have seen projective planes of orders 2, 3 and 4, we are curious as to what orders exist corresponding projective planes.
		While one may assume that projective planes exist for all orders greater than 1, this is not true.
		In the case of projective planes of order 9, there are four such non-isomorphic projective planes.
	home.wlu.edu /~mcraea/Finite_Geometry/NoneuclideanGeometry/Prob14ProjPlane/problem14.html (527 words)

	[No title]
		Subject: Freudenthal-Tits magic square and projective planes Date: Tue, 1 Feb 2000 13:05:59 -0800 (PST) Newsgroups: sci.math.research Summary: [missing] The Freudenthal-Tits magic square is a marvelous and somewhat mysterious method of relating the exceptional Lie groups to the normed division algebras R, C, H, and O: reals, complexes, quaternions and octonions.
		You can show the projective plane is the same as the affine plane together with extra points, which play the role of "points at infinity".
		We say that our projective plane is "Desarguesian" if whenever this happens, something else happens: the intersection of L and L', the intersection of M and M', and the intersection of N and N' all lie on the same line.
	www.math.niu.edu /~rusin/known-math/00_incoming/projplane (3734 words)

	Projective Transformations
		Perspective projection is a projectivity from projective 3-space to the projective plane.
		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)

	[No title]
		Projective geometry is the study of those aspects of geometry that are preserved by projective transformations.
		A "collineation" is a map from a projective plane to itself that preserves all lines.
		In particular, the nontrivial projections in J(F) correspond to the 1- and 2-dimensional subspaces of F^3, and thus to the points and lines in the projective plane PF^2.
	math.ucr.edu /home/baez/twf_ascii/week145 (2920 words)

	The History of the Problem
		There are several different definitions for a finite projective plane and this set of axioms is chosen to highlight the striking duality of lines and points.
		An early reference to a finite projective plane is in the paper by Veblen [32], which studied the axioms for geometry and used the plane of order 2 as an example.
		Projective planes are special cases of a class of combinatorial objects called symmetric block designs.
	www.cecm.sfu.ca /organics/papers/lam/paper/html/node2-an.shtml (1582 words)

	Complex projective plane (Site not responding. Last check: 2007-10-27)
		The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane.
		It is known that any non-singular rational variety is obtained from the plane by a sequence of blowing up transformations and their inverses ('blowing down') of curves, which must be of a very particular type.
		The group of birational automorphisms of the complex projective plane is the Cremona group.
	www.sciencedaily.com /encyclopedia/complex_projective_plane (263 words)

	Course Notes, Projective Geometry (Site not responding. Last check: 2007-10-27)
		Points in the projective plane are usually represented through the use of homogeneous coordinates.
		We shall associate the lines in P with the points in the projective plane and the and the planes in Q with the lines in the projective plane.
		Definition: A projective transformation P is a 1-1 onto map of the projective plane to the projective plane that is defined by an invertible linear map
	www.rpi.edu /~piperb/geometry/notes/set4 (998 words)

	Paper strip
		For the projective plane it does not matter whether one cuts the square vertically or horizontally.
		In order to convert a disk back into the original projective plane endpoints of every diameter should be identified because when we glued two halves together one was flipped in the horizontal axis.
		The reasons for this shape to be called Projective Plane is that it actually models geometry of the plane obtained through a perspecitve transformation.
	www.cut-the-knot.org /do_you_know/paper_strip.shtml (1581 words)

	Introduction
		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)

	The Math Forum - Math Library - Projective Geom. (Site not responding. Last check: 2007-10-27)
		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 is particularly suitable for the visualization of concepts of Projective Geometry.
		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)

	Steiner Surfaces
		The "real projective plane" is the set of lines through the origin in real 3-dimensional space; when each of these lines is represented by one point, the resulting set is, in an abstract way, a smooth, two-dimensional surface.
		This mathematical description of the projective plane is called a "homogeneous coordinate system," and it generalizes to any dimension: real projective n-space is the set of lines through the origin in n+1 dimensions.
		A classification of Steiner surfaces was known in the XIX century in the case where the coordinates and projective transformations are allowed to be complex.
	www.ipfw.edu /math/Coffman/steinersurface.html (2078 words)

	projective plane : Definition from the Online Dictionary at Datasegment.com (Site not responding. Last check: 2007-10-27)
		1 definition found projective plane - Free On-line Dictionary of Computing (19 Sep 2003) : projective plane The space of equivalence classes of vectors under non-zero scalar multiplication.
		Elements are sets of the form kv: k != 0, k scalar, v != O, v a vector where O is the origin.
		A projective plane is in no meaningful sense a plane and would therefore be (but isn't) better described as a "projective space".
	onlinedictionary.datasegment.com /word/projective+plane (124 words)

	The Projective Plane
		Tracing the projective plane from front to back, you may observe its most striking quality, one sidedness.
		If you run your hand palm down along the "top outside" of the projective plane, it will be turned 90 degrees by the time it gets to the back in figure 7.
		The projective plane is truly the mother of all nonorientable surfaces.
	sweb.uky.edu /~jcscov0/projective_plane.htm (951 words)

	Knitting One-Sided Surfaces (Site not responding. Last check: 2007-10-27)
		The model is based off of a particular immersion of the projective plane known as Boy's Surface.
		While there are simpler representations of the Projective Plane, I find that the lack of pinch points an immersion provides makes it much easier to visually understand what the surface is doing, which is why I chose Boy's Surface over the usual Cross-Cap representation (not an immersion).
		I've also written a pattern for the projective plane hat if you'd like to make one of your very own (It involves some pretty advanced knitting, so I don't recommend attempting it unless you're an experienced knitter).
	www.math.gatech.edu /~berglund/OneSided.html (728 words)

	Introduction and ToC: Affine and projective planes (Site not responding. Last check: 2007-10-27)
		A given plane automatically produces such a code and hence the theory is better suited to uncovering new planes using known planes than to building them out of whole cloth.
		These two ternary codes classify, if you will, the four projective planes, B classifying the translation planes and C the non-translation planes: any two affine planes linear over (this notion is defined in the Glossary at the end of the paper) B or over C can be obtained from one another by a derivation.
		The passage from the projective to the affine hull
	www.lehigh.edu /efa0/public/www-data/aandptoc.html (1162 words)

	The projective plane (Site not responding. Last check: 2007-10-27)
		First of all, the projective plane is a set of all straight lines in 3D space, passing through the origin (see fig.1a).
		Therefore, another model of the projective plane is a sphere in 3D space, on which the pairs of opposite points are identified, i.e.
		This obstacle is a reason of common opinion, that the projective plane cannot be presented as a ``good'' surface in 3D space.
	sim.ol.ru /~nikitin/vismat_html/node2.html (492 words)

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