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Topic: Parabolic geometry

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Mark Stepnoski

	Wikipedia: Euclidean geometry
		In differential geometry, and in constrast to the main types of non-Euclidean geometry it is also called "flat" geometry, or "parabolic" geometry because it is between elliptic geometry which is positively curved, and hyperbolic geometry which is negatively curved.
		The traditional presentation of Euclidean geometry is as an axiomatic system, which hoped to prove all the "true statements" as theorems in geometry from a set of finite number of axioms.
		In particular, this postulate separates Euclidean geometry from hyperbolic geometry, where many parallel lines could be drawn through the point, and from elliptic and projective geometry, where no parallel lines exist.
	www.factbook.org /wikipedia/en/e/eu/euclidean_geometry.html (681 words)

	Geometry
		Euclidean and Non-Euclidean Geometries by Maria Helena Noronha (Prentice Hall) is to be used in undergraduate geometry courses at the junior‑senior level.
		Euclidean and hyperbolic geometries are constructed upon a consistent set of axioms, as well as presenting the analytic aspects of their models and their isometries.
		A difficulty usually found at the beginning of courses that build a geometry upon a set of postulates is to have the students understand that results derived from these postulates hold in some of the non‑Euclidean geometries as well, and therefore their proofs cannot rely on facts obtained from their drawings.
	www.wordtrade.com /science/mathematics/geometry.htm (5443 words)

	Euclidean geometry
		Euclidean geometry, also called "flat" or "parabolic" geometry, is named after the Greek mathematician Euclid.
		This is the kind of geometry familiar to most people, since it is the kind usually taught in high school.
		Euclidean geometry is distinguished from other geometries by the parallel postulate, which is usually phrased as follows: Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
	www.ebroadcast.com.au /lookup/encyclopedia/eu/Euclidean.html (317 words)

	GEOMETRY OF THE PARABOLA
		This has occurred with the parabolic and pointed arches, whose geometry is present in the section of a body existing in the nature that has a system of great stability due to the harmony between its parts.
		Known since a very remote antiquity, in the plane geometry Vitruvio proposed that "so that a space divided into unequal parts results pleasant and aesthetic, the relation between the smallest and the largest part must be the same as between the largest and the whole".
		GEOMETRY OF THE PARABOLA ACCORDING TO All parabolas are similar: while the size varies, the constants of their configuration are the same for all them.
	members.tripod.com /vismath7/rojas (555 words)

	Chapter 8
		Since parabolic geometries strongly dominate solar concentrators, a rather thorough examination of the analytical description of parabolic geometry is presented in this chapter.
		For a spherical or a parabolic dish, the plane of curvature is rotated to generate the dish geometry.
		For a parabolic mirror to focus sharply, therefore, it must accurately track the motion of the sun to keep the axis (or plane) of symmetry parallel to the incident rays of the sun.
	www.powerfromthesun.net /Chapter8/Chapter8new.htm (3499 words)

	Research interests of Andreas Cap
		Parabolic geometries are Cartan geometries of type (G,P), where G is a (real or complex) semisimple Lie group and P is a parabolic subgroup of G. The homogeneous models G/P are the so-called generalized flag manifolds.
		In the case of parabolic geometries, this leads to a new type of geometric objects, which are called tractors.
		For general parabolic geometries, the description in terms of tractor bundles and connections was developed in two joint articles, [22] and [19], with Rod Gover (University of Auckland).
	www.mat.univie.ac.at /~cap/research.html (2536 words)

	solar4.htm
		Parabolic geometry is well known, and it was probably the very first type of solar cooker.
		The reason for its popularity was the focus which was much better and sharper than that of other types of reflectors, but at the same time it was very sensitive to even a slight change in the position of the sun and hence the use of such reflectors meant constant tracking.
		An asymmetrical parabolic reflector enables a cook to be as close as possible to the cooking vessel.
	ashokk_3.tripod.com /solar4.htm (3150 words)

	Foundations of Mathematics
		Hyperbolic and parabolic geometry maintain these properties, but differ from Euclidean geometry with respect to a third property: According to Euclid's axioms, [3] one and only one line L' that does not intersect a given line L can be drawn through any given point P exterior to L.
		Parabolic geometry originated with B. Riemann in 1848, and was part of a comprehensive treatment of geometry that included parabolic, hyperbolic and Euclidean geometry as special cases.
		The degree to which geometry and algebra evolved independently of each other is initially surprising, but is probably due to the fundamental difference in the nature of the respective intuitions that generate them.
	bahai-library.com /?file=hatcher_foundations_mathematics.html (14860 words)

	A brief review of geometries wrt GR (Site not responding. Last check: 2007-11-02)
		My point is, which you have not proved incorrect, is that you are discussing the same object, whether you are doing it by using the particular geometry "intrinsic" to it, or whether you are embedding it in a higher dimension and viewing it as a curvature or whatever.
		Geometry is a tool often used in working with shapes.
		Because of explanatory power of shapes, and by claiming one geometry is the "right" geometry to work with that shape, the explanatory power is transferred to the geometry.
	www.mukesh.ws /georev5.html (612 words)

	Citebase - Correspondence spaces and twistor spaces for parabolic geometries (Site not responding. Last check: 2007-11-02)
		Conversely, for a parabolic geometry of type (G,Q) on a smooth manifold M, we construct a distribution corresponding to P, and find the exact conditions for its integrability.
		We find equivalent conditions for the existence of a parabolic geometry of type (G,P) on the twistor space N such that M is locally isomorphic to the correspondence space Cal CN, thus obtaining a complete local characterization of correspondence spaces.
		We show that all these constructions preserve the subclass of normal parabolic geometries (which are determined by some underlying geometric structure) and that in the regular normal case, all characterizations can be expressed in terms of the harmonic curvature of the Cartan connection, which is easier to handle.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0102097 (218 words)

	Grade 1 Issue 12/2003 Mathematics Magazine
		In the Euclidean geometry, also called parabolic geometry, the fifth Euclidean postulate that there is only one parallel to a given line passing through an exterior point, is kept or validated.
		While in the Riemannian geometry, called elliptic geometry, the fifth Euclidean postulate is also invalidated as follows: there is no parallel to a given line passing through an exterior point.
		It seems that Smarandache Geometries are connected with the Theory of Relativity (because they include the Riemannian geometry in a subspace) and with the Parallel Universes (because they combine separate spaces into one space only) too.
	www.mathematicsmagazine.com /1-2004/Sm_Geom_1_2004.htm (698 words)

	Inductive heating
		According to this, tree geometry’s were studied, the first one is given in figure 4a and corresponds to the coil supplied by SAET with the converter.
		Now as the geometric conditions (air gap, cross-section and number of turns of the coil) of the heating furnace are established, we shall dedicate afterward to establish the optimal heating parameters of the inductive heating process, e.
		As the aims of the project are restricted to basic research the main goal is the temperature uniformity in the slug and thus a uniform liquid fraction which determine the thixotropic deformation behavior of the steel billet.
	elap.montefiore.ulg.ac.be /services/thixo/ELAP-SAET_html/ELAP-SAET.htm (2579 words)

	iSight Guide
		In the projective and conformal models, the "elliptic", "parabolic" and "hyperbolic" options menu chooses the geometry.
		In the Möbius model, this do not affect the geometry, but modify the mouse bindings so that translations are elliptic, parabolic or hyperbolic.
		All geometry objects can be seen or hidden in all windows.
	www.gang.umass.edu /software/guide/isight/isight07.html (451 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		Notes on the Parabolic Zigzag-Strut Tensegrity Dome Spencer Hunter, 2003 This document, unaltered, is in the public domain.
		Because of the parabolic geometry, the templates are irregular trapezoids with differing diagonal measurements.
		Spherical geometry (as in geodesic domes) would yield easier-to-calculate regular symmetrical trapezoids and quarter the strut-length variation of this dome.
	www.u.arizona.edu /~shunter/znotes.txt (267 words)

	The 23rd Winter School GEOMETRY AND PHYSICS (Site not responding. Last check: 2007-11-02)
		A triumph in the (riemannian) geometry of submanifolds of euclidean space is the classical Bonnet Theorem characterizing such submanifolds in terms of the Gauss--Codazzi--Ricci equations satisfied by the intrinsic and extrinsic geometry of the submanifold.
		The goal of these lectures is to explore the geometry of submanifolds of generalized flag varieties $G/P$ with emphasis on the conformal $n$-sphere and other examples of classical interest, such as real projective space $RP^n$ (where $G=SL(n+1,R)$).
		The recently developed machinery of parabolic geometry provides a wonderful opportunity to understand and extend many classical considerations in submanifold geometry, illuminating such objects as the central sphere congruence, Ribaucour transformations and the Darboux cubic form.
	www.math.muni.cz /~srni/calderbank03.html (197 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		Analogously, the simplest model for a Carnot--Caratheodory geometry is a nilpotent Lie group $N$ say, with a few additional properties.
		The simplest model for a parabolic geometry is a nilpotent Lie group with somewhat different properties.
		This talk is a survey of various results on the differential geometry of nilpotent Lie groups.
	www.math.wisc.edu /~seeger/cowlingabs (78 words)

	Symmetries and Overdetermined Systems of Partial Differential Equations
		By definition, these are the vector fields that preserve the structure in question—the Killing fields of Riemannian differential geometry, for example.
		Parabolic differential geometry provides a synthesis and generalization of various classical geometries including conformal, projective, and CR.
		Even in the flat model G/P, for G a semisimple Lie group with P a parabolic subgroup, there is much to be gleaned from the representation theory of G. In particular, the Bernstein-Gelfand-Gelfand (BGG) complex is a series of G-invariant differential operators, the first of which is overdetermined.
	www.ima.umn.edu /2005-2006/SP7.17-8.4.06 (1446 words)

	Geometry - Cunnan
		The most familiar geometry is known to mathematicians as Euclidean geometry, also called "flat" or "parabolic" geometry, is named after the Greek mathematician Euclid.
		The Elements is a collection of definitions, postulates, and proofs from Euclidean geometry, named after Euclid.
		Euclid based his work on 23 definitions, such as point, line and surface, five postulates and five "common notions" (today they are called axioms).
	cunnan.sca.org.au /index.php?title=Geometry&redirect=no (172 words)

	DC MetaData for: Correspondence Spaces and Twistor Spaces for Parabolic Geometries (Site not responding. Last check: 2007-11-02)
		P\subset G$, we associate to a parabolic geometry of type $(G,P)$ on a
		parabolic geometries (which are determined by some underlying
		Keywords: twistor space, normal parabolic geometry, twistor correspondence, Cartan connection, Lagrangian contact structure, CR structure, projective structure, conformal structure
	www.esi.ac.at /Preprint-shadows/esi989.html (231 words)

	Re: Parabolic, Elliptical, and Hyperbolic Geometry.
		> According to http://members.aol.com/jeff570/mathword.html : > > The terms HYPERBOLIC GEOMETRY, ELLIPTIC GEOMETRY, and PARABOLIC > GEOMETRY were introduced by Felix Klein (1849-1925) in 1871 in "Über > die sogenannte Nicht-Euklidische Geometrie" (On so-called > non-Euclidean geometry), reprinted in his Gesammelte mathematische > Abhandlungen I (1921) p.
		The reference has been translated into English, in John Stillwell, "Sources of Hyperbolic Geometry," 1996, p.72 (3rd paragraph and footnote).
		I'd rather not type it all now, but anyway Stillwell's little book is well worth looking at.
	www.usenet.com /newsgroups/sci.math/msg18607.html (133 words)

	Tractor Bundles for Irreducible Parabolic Geometries (ResearchIndex)
		We use the general results on tractor calculi for parabolic geometries obtained in [3] to give a simple and eective characterisation of arbitrary normal tractor bundles on manifolds equipped with an irreducible parabolic geometry (also called almost Hermitian symmetric{ or AHS{structure in the literature).
		Moreover, we also construct the corresponding normal adjoint tractor bundle and give explicit formulae for the normal tractor connections as well as the fundamental D{operators on...
		5 Invariants and calculus for conformal geometry (context) - Gover - 1998
	citeseer.ist.psu.edu /301056.html (317 words)

	Citebase - Rational curves and parabolic geometries (Site not responding. Last check: 2007-11-02)
		The twistor transform of a parabolic geometry has two steps: lift up to a geometry of higher dimension, and then drop down again, to a geometry of lower dimension.
		We uncover necessary and sufficient global conditions for complex analytic geometries: rationality of curves defined by certain differential equations.
		We apply the theorems to second and third order ordinary differential equations to determine whether their solutions are rational curves.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0603276 (169 words)

	DC MetaData for: On the Geometry of Chains (Site not responding. Last check: 2007-11-02)
		DC MetaData for: On the Geometry of Chains
		Both the parabolic contact structure and the system of
		geometry associated to the family of chains can be obtained in that
	www.esi.ac.at /Preprint-shadows/esi1651.html (246 words)

	/usr/local/etc/httpd/htdocs/archive/sem_coll_2003
		The corresponding geometries are called parabolic geometries; there are a number of classically interesting special cases.
		Contact geometry is a venerable subject that arose out of the study of Geometric Optics in the 1800's.
		In the recent past there have been many breakthroughs in the study of contact geometry that hint at surprising connections with low dimensional topology, complex geometry and dynamics.
	www.math.buffalo.edu /archive/sem_coll_2003.html (1360 words)

	Correspondence Spaces and Twistor Spaces for Parabolic Geometries (ResearchIndex)
		Correspondence Spaces and Twistor Spaces for Parabolic Geometries (2001)
		Conversely, for a parabolic geometry of type (G; Q) on a smooth manifold M, we construct a canonical distribution corresponding to P, and nd the exact...
		8 The geometry of hyperbolic and elliptic CR{manifolds of codi..
	citeseer.ist.psu.edu /433497.html (372 words)

	Amazon.com: The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching (Maa Spectrum.): ... (Site not responding. Last check: 2007-11-02)
		Amazon.com: The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching (Maa Spectrum.): Books: Chris Pritchard
		The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching (Maa Spectrum.) (Paperback)
		discusses deductive geometry, cyclic quadrilateral, regular dodecagon, higher geometry, parabolic segment, synthetic geometry
	www.amazon.com /gp/product/sitb-next/0521531624 (386 words)

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