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	Projective geometry - Wikipedia, the free encyclopedia
		Projective geometry is a non-metrical form of geometry that emerged in the early 19th century.
		Projective geometry is a non-Euclidean geometry that formalizes one of the central principles of art: that parallel lines meet at infinity and therefore are to be drawn that way.
		In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line".
	en.wikipedia.org /wiki/Projective_geometry (1990 words)

	Projective geometry (Site not responding. Last check: 2007-11-06)
		In a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the nameprojective geometry was a stepping stone from analytic geometry to algebraicgeometry.
		This period in geometry was rather overtaken by the research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretchedexisting techniques, and then by invariant theory.
		Some important work was done in enumerative geometry inparticular, by Schubert, that is now considered an anticipation of the theory of Chern classes in their guise as representing the algebraic topology of Grassmannians.
	www.therfcc.org /projective-geometry-69630.html (607 words)

	Basics (Site not responding. Last check: 2007-11-06)
		Projective geometry is concerned with incidences, that is, where elements such as lines planes and points either coincide or not.
		This is the Fundamental Theorem of projective geometry.
		Projective transformation in which it is demonstrated that parallelism is not conserved.
	www.anth.org.uk /NCT/basics.htm (1152 words)

	Projective geometry (Site not responding. Last check: 2007-11-06)
		In case of the projective plane alone the axiomatic approach may encounter that cannot be described via linear algebra.
		Some important work was in enumerative geometry in particular by Schubert is now considered an anticipation of the of Chern classes in their guise as representing the algebraic topology of Grassmannians.
		Projective geometry suffers from its absence in school teaching and that it is assumed to be trivial in algebraic geometry books.
	www.freeglossary.com /Projective_geometry (848 words)

	projective geometry. The Columbia Encyclopedia, Sixth Edition. 2001-05 (Site not responding. Last check: 2007-11-06)
		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.
		Projective geometry is more general than the familiar Euclidean geometry and includes the metric geometries (both Euclidean and non-Euclidean) as special cases.
	www.bartleby.com /65/pr/projctgeo.html (277 words)

	Various Geometries
		Another approach to defining and classifying various geometries was introduced, in 1872, by Felix Klein (1849-1925) in the inaugural address he gave upon appointment to the Faculty and Senate of the University of Erlanger.
		Affine Geometry is not concerned with the notions of circle, angle and distance.
		Projective Geometry originated in the works of Désargues (1593-1662), B.Pascal, G.Monge (1746-1818) and was further developed in the 19th century by J.V.Poncelet (1788-1867) and C.J.Brianchon (1785-1864).
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	Question Corner -- Understanding Projective Geometry
		Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such.
		Projective geometry can be thought of as the collection of all lines through the origin in three-dimensional space.
		In summary, then, projective geometry can be thought of as the study of points and "lines" (great circles) on a surface obtained by gluing the equator of a hemisphere to itself.
	www.math.toronto.edu /mathnet/questionCorner/projective.html (2444 words)

	Nineteenth Century Geometry
		Today projective geometry does not play a big role in mathematics, but in the late nineteenth century it came to be synonymous with modern geometry.
		one may say that the truth of the geometry of Euclid is not incompatible with the truth of the geometry of Lobachevsky, for the existence of a group is not incompatible with that of another group.
		Geometry distinguishes itself from other natural sciences because it obtains only very few concepts and laws directly from experience, and aims at obtaining from them the laws of more complex phenomena by purely deductive means.
	plato.stanford.edu /entries/geometry-19th (4782 words)

	projective geometry
		The branch of geometry that deals with properties of geometric figures that remain unchanged under projection.
		These elements retain their character under projection; for example, 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 doesn't 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.
	www.daviddarling.info /encyclopedia/P/projective_geometry.html (482 words)

	Cornell Mathematics- Robert Connelly- Math 452 Home Page (Site not responding. Last check: 2007-11-06)
		In a way, projective geometry is an example where one can apply the insight obtained from a simple set of axioms, unlike the situation of Euclid, where there were a very large number of hidden, subtle, complicated, axioms.
		Projective geometry has just three simple axioms, two are just mirror duals of each other, and the third does not really count.
		The properties of the projection of a cube, depending on how the cube sits with respect to the viewing plane, lead to some unexpected properties, as well as some very particular constraints, on how one is to draw a cube in realistic perspective.
	www.math.cornell.edu /%7Econnelly/452.stuff.html (629 words)

	The Math Forum - Math Library - Projective Geom. (Site not responding. Last check: 2007-11-06)
		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 (1059 words)

	Citations: An Outline of Projective Geometry - Garner (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Projective geometry is emerging as an attractive framework for computer vision [39] In this paper, we assume that the reader is familiar with some elementary projective geometry.
		Projective geometry is emerging as an attractive framework for computer vision [40] In this paper, we assume that the reader is familiar with some elementary projective geometry.
		Every (projective) geometry is characterized by a unique dimension d (finite or infinite) which is defined via a number of related geometrical notions.
	citeseer.ist.psu.edu /context/185647/0 (787 words)

	Projective Geometry And Modern Algebra - Contents (Site not responding. Last check: 2007-11-06)
		Starting with an engaging historical foreword, which develops the viewpoint that the modern theory of projective geometry comes from the fine arts and the classical Greek study of conics, the authors present the synthetic and analytic aspects of basic projective geometry.
		The techniques and concepts of modern algebra are introduced for their natural role in the study of projective geometry; groups appear as automorphism groups of configurations, division rings appear in the study of Desargues' theorem and the study of the independence of the seven axioms given for projective geometry.
		Projective planes over fields are characterized in terms of one of these axioms, commonly known as the fundamental theorem (equivalently, Pappus' theorem).
	www.id.cbs.dk /~mtk/pg.html (211 words)

	Section I. Geometric Algebra
		Major theorems of projective geometry are reduced to algebraic identities which apply as well to metrical geometry.
		Relations among Clifford algebras of different dimensions are interpreted geometrically as "projective and conformal splits." The conformal split is employed to simplify and elucidate the pin and spin representations of the conformal group for arbitrary dimension and signature.
		Abstract: Projective geometry is formulated in the language of geometric algebra, a unified mathematical language based on Clifford algebra.
	modelingnts.la.asu.edu /html/GeoAlg.html (717 words)

	Octonionic Projective Geometry
		Projective geometry is a venerable subject that has its origins in the study of perspective by Renaissance painters.
		We have already met one example of a projective plane in Section 2.1: the smallest one of all, the Fano plane.
		Projective geometry was very fashionable in the 1800s, with such worthies as Poncelet, Brianchon, Steiner and von Staudt making important contributions.
	math.ucr.edu /home/baez/octonions/node8.html (1306 words)

	Epipolar Geometry
		The application of projective geometry techniques in computer vision is most notable in the Stereo Vision problem which is very closely related to Structure-from-Motion.
		It defines the geometry of the correspondences between two views in a compact way, encoding intrinsic camera geometry as well as the extrinsic relative motion between the two cameras.
		The result is that it is not possible to determine the epipolar geometry between close consecutive frames and it cannot be determined from image correspondences alone.
	www1.cs.columbia.edu /~jebara/htmlpapers/SFM/node8.html (796 words)

	Havlicek/Projective Geometry
		Axioms of affine and projective planes, Connection between affine and projective planes, Principle of duality.
		Perspectivities, Projectivities, Axioms of Desargues and Pappos, Hessenberg's theorem, Perspective and projective collineations, Harmonic tetrads, Fano's axiom.
		Projective spaces over vector spaces, Fundamental theorem of projective geometry, Affine spaces over vector spaces, Fundamental theorem of affine geometry, Representation of collineations and affinities in terms of semilinear mappings, Projective and affine coordinates, Cross ratios and affine ratios.
	www.geometrie.tuwien.ac.at /havlicek/pgeom.html (300 words)

	55:148,55:247 Chapter 9, Part 2
		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.
		The ray CX represents all possible projections of the point X to the left image, and is also projected into the epipolar line l' in the right image.
	www.icaen.uiowa.edu /~dip/LECTURE/3DVisionP1_2.html (3013 words)

	Wiley::Affine and Projective Geometry
		In the second part, geometry is used to introduce lattice theory, and the book culminates with the fundamental theorem of projective geometry.
		Throughout, the text explores geometry's correlation to algebra in ways that are meant to foster inquiry and develop mathematical insights whether or not one has a background in algebra.
		Affine and Projective Geometry's broad scope and its communicative tone make it an ideal choice for all students and professionals who would like to further their understanding of things mathematical.
	eu.wiley.com /WileyCDA/WileyTitle/productCd-0471113158.html (331 words)

	Amazon.com: Geometry: Books: David A. Brannan,Matthew F. Esplen,Jeremy J. Gray (Site not responding. Last check: 2007-11-06)
		During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses).
		In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself.
		And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations.
	www.amazon.com /exec/obidos/tg/detail/-/0521597870?v=glance (2663 words)

	Collineation - Wikipedia, the free encyclopedia
		A collineation, roughly, is a map from one projective space to the other, preserving the geometric structure.
		Let V be a vector space over a field K (of dimension at least three) and W a vector space over a field L.
		This implies K and L are isomorphic fields, V and W have the same dimension, and there is a semilinear map φ such that φ induces α.
	en.wikipedia.org /wiki/Fundamental_theorem_of_projective_geometry (317 words)

	Projective Geometry (Site not responding. Last check: 2007-11-06)
		Projective Geometry is the superset of geometry that includes Euclidean and non-Euclidean geometries.
		One can see the example as either that the memberships can be graphed as an abstraction of geometry or what the book states: that geometry is about items and their logical relationship between each other.
		My instinct for space time geometry is that their is no physical space.
	home.att.net /~bob.rutkiewicz/projective_geometry.htm (386 words)

	Xah: Introduction to Real Projective Plane
		In affine (or Euclidean) geometry, the line p (through O) parallel to o would be exceptional, for it would have no corresponding point on o; but when we have extended the affine plane to the projective plane, the corresponding point P is just the point at infinity on o.
		In affine geometry the point X makes an infinite jump; but in projective geometry its motion, through the single point at infinity, is continuous.
		This questions sterms from studying projective geometry, but I think it is an elementary idea in topology, which I have not studied.
	xahlee.org /projective_geometry/projective_geometry.html (6405 words)

	projective geometry on Encyclopedia.com
		PROJECTIVE GEOMETRY [projective geometry] branch of geometry concerned with those properties of geometric figures that remain invariant under projection.
		Two properties that are invariant under projection are the order of three or more points on a line and the harmonic relationship, or cross ratio, among four points, A, B, C, D, i.e., AC / BC : AD / BD.
		Projective differential geometry old and new; from the Schwarzian derivative to the cohomology of diffeomorphism groups.(Brief Article)(Book Review)
	www.encyclopedia.com /html/p1/projctgeo.asp (414 words)

	Amazon.com: Projective Geometry: Books: H.S.M. Coxeter (Site not responding. Last check: 2007-11-06)
		Projective geometry is simpler: its constructions require only a ruler.
		The plane geometry of the first six books of Euclid's Elements may be described as the geometry of lines and circles: its tools are the straight-edge (or unmarked ruler) and the compasses.
		Linear Algebra and Projective Geometry by Reinhold Baer
	www.amazon.com /exec/obidos/tg/detail/-/0387406239?v=glance (762 words)

	projective geometry
		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
		differential geometry - differential geometry, branch of geometry in which the concepts of the calculus are applied to...
		geometry: Their Relationship to Each Other - Their Relationship to Each Other The different geometries are classified and related to one another...
	www.factmonster.com /ce6/sci/A0840240.html (348 words)

	Galois Geometry
		"Projective spaces over a finite field, otherwise known as Galois geometries, find wide application in coding theory, algebraic geometry, design theory, graph theory, and group theory as well as being beautiful objects of study in their own right."
		The simplest Galois geometries are the projective spaces of one, two, and three dimensions over the two-element Galois field...
		That is to say, the binary projective line, plane, and 3-space.
	log24.com /theory/GalG.htm (613 words)

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