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Topic: Incidence geometry

	Incidence geometry (structure) - Wikipedia, the free encyclopedia
		Incidence geometry is a mathematical structure composed of objects of various types and an incidence relation between them.
		Incidence geometries of rank two are also known as incidence structures.
		A map on a surface is a rank 3 incidence geometry.
	en.wikipedia.org /wiki/Incidence_geometry (125 words)

	PlanetMath: incidence geometry
		Incidence geometry is essentially geometry based on the first postulate in Euclid's Elements.
		In this entry, we will define incidence geometry using abstract notions of sets, functions, and relations (specifically, an incidence relation) and then briefly discuss how this definition is related to the axioms of incidence that we know from high school and college.
		This is version 19 of incidence geometry, born on 2005-08-01, modified 2005-10-02.
	planetmath.org /encyclopedia/IncidenceGeometry.html (1044 words)

	Category:Geometry - Wikipedia, the free encyclopedia
		Geometry is the branch of mathematics dealing with spatial relationships.
		From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry.
		Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.
	en.wikipedia.org /wiki/Category:Geometry (115 words)

	Incidence (geometry) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-08-18)
		In (The pure mathematics of points and lines and curves and surfaces) geometry, the (An abstraction belonging to or characteristic of two entities or parts together) relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L'), and 'intersects' (as in 'line L
		Historically, (The geometry of properties that remain invariant under projection) projective geometry was introduced in order to make the propositions of incidence true (without exceptions such as are caused by parallels).
		Then the propositions of incidence are derived from the following basic result on (Click link for more info and facts about vector space) vector spaces: given subspaces U and V of a vector space W, the dimension of their intersection is at least dim U + dim V − dim W.
	www.absoluteastronomy.com /encyclopedia/i/in/incidence_(geometry).htm (696 words)

	51: Geometry
		Solid geometry is placed here (actually in 51M05) because it mirrors elementary plane geometry, but spherical geometry is primarily on the page for general convex geometry.
		Cabri-geometry is used for teaching secondary school geometry, but, equally important, is its use for university level instruction and as a tool by mathematicians in their research work.
		A useful collection of Geometry Formulas and Facts is taken from the CRC Standard Mathematical Tables and Formulas, and available at the The Geometry Center.
	www.math.niu.edu /~rusin/known-math/index/51-XX.html (828 words)

	Residues (Site not responding. Last check: 2007-08-18)
		The residue of F is the coset geometry Gamma_F = Gamma(cap_(j in F) G_j; (G_iintersect (cap_(j in F)G_j))_(i in I\ t(F))).
		Given an incidence geometry D and a flag f of D, return the residue of the flag f as an incidence geometry.
		Given a coset geometry C and a subset f of the set of types of C, return the residue of the flag consisting in the maximal parabolics of C whose type is in f.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text1433.htm (119 words)

	Construction of Incidence and Coset Geometries
		Construct the incidence geometry IG from the coset geometry C. This is done using Tits' algorithm described in the introduction of this chapter.
		We assume that ig is the incidence geometry corresponding to the Hoffman-Singleton graph, as described in the previous example.
		The group G of the coset geometry CG is the automorphism group of D. Magma determines a chamber C of D, that is a clique of the incidence graph of D containing one element of each type.
	www.math.niu.edu /help/math/magmahelp/text1176.html (1137 words)

	Diagram of an Incidence Geometry
		The diagram of a firm, residually connected, flag--transitive geometry Gamma is a complete graph K, whose vertices are the elements of the set of types I of Gamma, provided with some additional structure which is further described as follows.
		Every vertex i of K is labelled with a sequence of two positive integers, the first one being s_i and the second one being the number of elements of type i in D. Every edge { i, j } is labelled with a sequence of three positive integers which are respectively d_(ij), g_(ij) and d_(ji).
		So when the user has to compute the diagram of an incidence geometry, it is strongly advised that he first converts it into a coset geometry and then computes the diagram of the corresponding coset geometry.
	www.math.lsu.edu /magma/text1473.htm (671 words)

	[No title]
		This leads us into the world of projective geometry, where all we have are points, lines, planes, etc., together with incidence relations like: "the point P lies on the line Q", "the line Q lies on the plane R", and so on, satisfying various axioms.
		Projective geometry goes way back to the Renaissance painters and their interest in perspective, and axiomatic projective geometry was very fashionable in the 19th century, but here we are seeing it in a more modern light, because we're seeing its relation to quantum logic.
		In a geometry with symmetry group G, different types of figure correspond to different subgroups of G. The idea is that for each type of figure, there is a space X of all figures of that type, upon which G acts.
	math.ucr.edu /home/baez/twf_ascii/week178 (3254 words)

	5.8 Radar Imagery Versus VIR Imagery (Site not responding. Last check: 2007-08-18)
		Imaging geometry and electromagnetic wave properties together produce the very different appearances of a radar image, an aerial photograph or a VIR satellite image.
		While the mountain slopes AB and BC are equal, the radar imaging geometry dictates that the radar-facing slope (AB) wil be imaged (B'A') as leaning toward the radar.
		Surfaces that return a strong signal and are bright in a radar image may return a weak signal in the VIR range of the spectrum and appear dark on a photograph, Landsat or SPOT image.
	www.ccrs.nrcan.gc.ca /ccrs/learn/tutorials/stereosc/chap5/chapter5_9_e.html (1332 words)

	Introduction (Site not responding. Last check: 2007-08-18)
		If the latter condition is satisfied, then an incidence geometry is a geometry in the sense of Buekenhout.
		It is possible to construct incidence geometries from a group and some of its subgroups using an algorithm first introduced by Jacques Tits.
		Again, a coset geometry is not a geometry in the sense of Buekenhout.
	wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text1429.htm (442 words)

	The Math Forum - Math Library - Geometry
		A collection of handouts for a two-week summer workshop entitled 'Geometry and the Imagination', led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 17-28, 1991.
		A short article designed to provide an introduction to geometry, including classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory.
		Some notes on a most general definition of "geometry," first elucidated by Felix Klein, which is based on a set of geometric invariants under a group of transformations.
	mathforum.org /library/topics/geometry (2304 words)

	[No title] (Site not responding. Last check: 2007-08-18)
		An axiomatic approach is used to develop the incidence properties and models of this geometry are considered.
		Euclidean geometry is developed from an abstract algebra point of view by using transformational geometry.
		Neutral geometry, with its origins in Euclid and its extension with Saccheri’s work, to the more recent development in transformational geometry will be reviewed from a historical perspective.
	www.runet.edu /~math-web/Math/math403.html (250 words)

	Foundations of geometry (Site not responding. Last check: 2007-08-18)
		Euclidean Geometry is the attempt to build geometry out of the rules of logic combined with some ``evident truths'' or axioms.
		Unlike the ``High School Geometry'' text books, this makes no reference to the ``Ruler Placement Postulate'' or a ``Protractor Placement Postulate'', both of which are anti-thetical to a purely geometric approach.
		The arithmetic aspects of geometry should grow out of it rather than be imposed from outside.
	www.imsc.ernet.in /~kapil/geometry/euclid/node1.html (284 words)

	Finite-Geometry Models
		In this note we describe four of the most useful construction principles for constructing pictures of small incidence geometries which capture large parts of the abstract beauty of the geometries they depict.
		The kind of pictures we want to concentrate on in this note are immediately accessible and can serve to lure students into studying incidence geometry and as a first step in teaching students pictorial thinking in geometry.
		Given a small, highly symmetrical geometry with n points, look for the same number of points arranged into a highly symmetrical spatial object.
	log24.com /theory/Polster.html (438 words)

	Homework, UMBC Math 306, Spr02
		Interpret "incidence" in the Euclidean sense of a point lying on a punctured circle.
		I: Incidence Geometry (3 axioms on pgs 50-51 of Greenberg)
		It is a theorem in hyperbolic geometry that there is a unique line l called the line of enclosure of this angle such that l is limiting parallel to both sides, ray OA' and ray OA.
	www.math.umbc.edu /~campbell/Math306Spr02/HW (1167 words)

	Euclid's Mathematical System
		Hilbert recognized that Euclid's proof for the side-angle-side criterion of congruence in triangles was based on an unstated assumption (the principle of superposition) and that this criterion had to be treated as an axiom.
		However, when we wish to look at a specific example of our geometry, called a model, we will have to establish how each of these undefined terms is to be interpreted.
		We could change our example though to be a set of seven points, where lines would be ordered pairs of points, incidence would be set inclusion and betweenness and congruence would be meaningless.
	www.math.uncc.edu /~droyster/math3181/notes/hyprgeom/node27.html (1110 words)

	The Math Forum - Math Library - Elliptic &Spherical Geom.
		Course notes that include: Origins of geometry; spherical geometry; logic and the axiomatic method; proof; Euclid's mathematical system; incidence geometry; betweenness axioms; congruence theorems; axioms of continuity; neutral geometry; theorems of continuity; the work of Saccheri and Gauss; hyperbolic geometry; classification of parallels; the pseudosphere; hyperbolic trigonometry and hyperbolic analytic geometry; and more.
		A unit written as an enrichment lesson for students in basic geometry or geometry; also, the section on spherical geometry can be used in an Algebra II Trigonometry class as an extension or an introduction to spherical geometry.
		An examination of some properties of triangles in elliptic geometry, which for this purpose are equivalent to geometry on a hemisphere.
	mathforum.org /library/topics/elliptic_g (1608 words)

	Bruce Cooperstein
		By incidence geometry is meant a pair (P,L) consisting of a set P whose members are called points, and a collection L of distinguished subsets of P whose members are called lines.
		Recently he has become interested in the “embeddings” of such geometries - ways in which these geometries can be modeled by point sets in projective space together with some full projective lines.
		Cooperstein: On a Connection Between Hyperbolic Ovoids on the Hyperbolic Quadric Q (10,q) and the Lie Incidence Geometry E6,1(q), 55-64 in Groups and Geometries, Birkhauser, 1998.
	www.math.ucsc.edu /Faculty/Coop.html (405 words)

	Properties of Incidence Geometries and Coset Geometries (Site not responding. Last check: 2007-08-18)
		An incidence geometry Gamma is flag--transitive if for every two flags x, y of the same type of Gamma, there exists an element g of Aut(Gamma) such that g(x) = y.
		Given an incidence geometry D, tests if this incidence geometry corresponds to a graph: D must be of rank two and such that for one of the two types, say e, all elements of this type are incident with exactly two elements of the other type.
		Given a coset geometry C, tests if this geometry corresponds to a graph: C must be of rank two and one of the two maximal parabolic subgroups, say G_e, must contain the Borel subgroup as a subgroup of index 2.
	www.umich.edu /~gpcc/scs/magma/text1246.htm (385 words)

	Final (Site not responding. Last check: 2007-08-18)
		Incidence Geometry (you should not remember the axioms but given axioms you should be able to prove propositions of Incidence geometry, see problem 6 p.64)
		Models of incidence geometry, projective and affine planes, Euclidean, hyperbolic and elliptic parallel properties (see problems 8-9 p.64)
		Neutral geometry is based on 14 axioms (Hilbert's system of axioms).
	www.math.ntnu.no /~eugenia/MA2401/finale.html (505 words)

	m133
		This course treats the elementary theory of affine and projective planes, finite geometries, Euclidean and Non-Euclidean geometries, groups of transformations and other algebraic structures related to geometry.
		Approaches to the study of geometry, examples from empirical geometry, an overview of the course.
		Background material from solid geometry and vector algebra, planes, incidence geometry of the sphere, the spherical triangle inequality, isometries of the sphere, Euler's formula, spherical triangles, congruence theorems and trigonometry, finite rotation groups and isometry groups of the sphere.
	math.ucr.edu /home/UndergradInfo/pages/m133 (263 words)

	Jeopardy Questions (Midterm)
		Use the words neutral, incidence, and hyperbolic to complete this sentence: “Every model of ________ geometry is a model of _________ geometry, which is itself a model of _________ geometry.”
		Up to isomorphism, this is the number of 4-point models of incidence geometry.
		This form of geometry is the mathematical foundation underlying “perspective painting” from the Renaissance era.
	www.thecoo.edu /~jadouma/WebDocs/303Jeopardy.midterm.htm (442 words)

	MichNExT 2005 Program
		For many students preparing to become secondary teachers, a college geometry course is their introduction to mathematical proof and to the process of doing mathematics.
		Incidence geometry, the beginning of an axiomatic development of geometry, provides opportunities for students to explore open-ended questions that suit their level of training.
		This talk is a report on a unit of incidence geometry, and the resulting work by students, presented in a course of geometry for secondary teachers at the University of Michigan.
	www.calvin.edu /~rpruim/next/mich/2005/program2005.shtml (595 words)

	AERADE subject listing for AEROFOILS AND WINGS - General. Aerofoils at subcritical speeds - lift, pitching moment.
		ESDU 72024 provides a means of estimating the lift-curve slope at zero lift, zero-lift incidence, zero-lift pitching moment coefficient and chordwise location of the aerodynamic centre for any aerofoil in compressible inviscid airflow from a knowledge of a number of parameters characterising the thickness distribution and camber line shape.
		The method predicts the increment in lift coefficient by which the maximum exceeds the value at zero incidence, and a prediction method for lift coefficient at zero incidence is included.
		The method of Pankhurst is used to calculate the inviscid zero-lift incidence and two simple factors presented graphically are used to correct that value for the effects of viscosity and compressibility; both depend on the camber-line geometry in the vicinity of the trailing edge and Reynolds number while the latter also depends on Mach number.
	aerade.cranfield.ac.uk /subject-listing/esdu/ES55.html (1609 words)

	Talks
		The RWPRI geometries of the symmetric group Sym(7).
		Magma workshop on Group Theory and Algebraic Geometry, University of Warwick, August 22-26, 2005.
		Incidence geometry, finite group theory and computational algebra.
	cso.ulb.ac.be /~dleemans/talks.html (218 words)

	Geometry Index
		This is a small, experimental collection of pages on incidence geometry, based on the lecture series Geometrie I given at the Christian Albrechts Universität Kiel.
		The intended audience are people trying to get an overview of some of the basic definitions and results in incidence geometry.
		Primary entry points are the page on general incidence structures or the page on projective planes.
	www.math.uni-kiel.de /geometrie/klein/math/geometry.html (204 words)

	[No title] (Site not responding. Last check: 2007-08-18)
		One cornerstone of Euclidean geometry is the parallel line postulate: For each line l and each point p that does not lie on l, there exists a unique line m through p parallel to l.
		For thousands of years, it was expected (but not proven!) that this follows from the other axioms of geometry.
		Concrete models of non-Euclidean geometry will be constructed, e.g., the surface of a sphere can be interpreted as a non-Euclidean `plane' with its great circles as `lines'.
	math.rice.edu /~hassett/teaching/366spring03/366advert.html (215 words)

	Footnotes to 'From rotating needles to stability of waves...'
		After discussing Tom Wolff's (n+2)/2 Kakeya result, I made the comment "However, there appears to be a limit to what can be achieved purely by applying elementary incidence geometry facts and standard combinatorial tools".
		This argument, and any other argument which uses as its fundamental ingredients incidence geometry and combinatorics, will also work for the complex analogue of the Kakeya problem in C^3, giving a lower bound of 5 for the real Minkowski or Hausdorff dimension of such sets.
		In particular, arguments which are based solely on incidence geometry (which cannot easily distinguish between parallel and non-parallel lines) are extremely unlikely to improve upon Wolff's bound.
	www.math.ucla.edu /~tao/notices.html (1190 words)

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