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	Pseudosphere -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		This makes it very hard to represent a pseudosphere in the (A space in which Euclid's axioms and definitions apply; a metric space that is linear and finite-dimensional) Euclidean space of drawings.
		By increasingly shrinking the pseudosphere as it goes further out towards the edge, it will fit into a circle, called the Poincaré disk; with the "edge" representing infinity.
		The name "pseudosphere" comes about because it is a (Click link for more info and facts about two dimensional) two dimensional (The outer boundary of an artifact or a material layer constituting or resembling such a boundary) surface of constant curvature.
	www.absoluteastronomy.com /encyclopedia/p/ps/pseudosphere.htm (371 words)

	3D-XplorMath Surface Gallery (Site not responding. Last check: 2007-10-22)
		Pseudosphere Hermann Karcher The Pseudosphere is a surface of revolution (of the Tractrix) and has Gaussian curvature minus one, or in other words, the product of its principal curvatures is -1.
		The Pseudosphere is best known because its intrinsic geometry is hyperbolic, the meridians are a family of asymptotic geodesics and the orthogonal latitudes are therefore a geodesically parallel family of "horocycles", i.e.
		This Pseudosphere is obtained by the construction which relates solutions of the Sine-Gordon equation to surfaces of Gaussian curvature -1, here the solution is a one-soliton solution: q(u,v) := 4 arctan(exp(u)).
	rsp.math.brandeis.edu /3D-XplorMath/Surface/pseudosphere/pseudosphere.html (255 words)

	Tractrix (Site not responding. Last check: 2007-10-22)
		Imagine a horizontal bar held at the vertex of the catenary and the point of contact marked as P. When the bar is rolled against the cantenary without slipping, the path of P will be a tractrix.
		A pseudosphere is a model of part of a hyperbolic (non-Euclidean) space.
		The pseudosphere is a surface with constant negative curvature, as opposed to a sphere which has constant positive curvature.
	www.pballew.net /tractrix.html (466 words)

	Pseudosphere - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-22)
		In geometry, a pseudosphere or tractricoid in the traditional usage, is the result of revolving a tractrix about its asymptote.
		A section that leads to the infinite edge, ends up becoming "wrapped" around and joined at its opposite sides, yielding the aforementioned "tractricoid" shape, which is also called a "Gabriel's Horn" (since it resembles a horn with the mouthpiece lying at infinity).
		The name "pseudosphere" comes about because it is a two dimensional surface of constant curvature.
	xahlee.org /_p/wiki/Pseudosphere.html (371 words)

	NonEuclid: The Pseudosphere
		Precisely because the pseudosphere is actually bigger than the plane, it is very hard to represent it in the normal Euclidean Geometry of our drawings.
		But there is a special trick for shrinking a pseudosphere to fit inside a circular boundary.
		Shrinking the pseudosphere to fit into a bounded disk distorts the pseudosphere, but it distorts it in a very careful way.
	cs.unm.edu /~joel/NonEuclid/pseudosphere.html (624 words)

	Unifying Synchronous and Asynchronous Message-Passing Models (Site not responding. Last check: 2007-10-22)
		The key idea is the concept of a pseudosphere, a new combinatorial structure in which each process from a set of processes is independently assigned a value from a set of values.
		Pseudospheres have a number of nice combinatorial properties, but their principal interest lies in the observation that the behavior of protocols in the three models can be characterized as simple unions of pseudospheres, where the exact structure of these unions is determined by the timing properties of the model.
		We use this pseudosphere construction to derive new and remarkably succinct proofs of bounds on consensus and k-set agreement in the asynchronous and synchronous models, as well as the first lower bound on wait-free k-set agreement in the semi-synchronous model.
	www.matem.unam.mx /~rajsbaum/podc98hrt.html (148 words)

	Pseudosphere (Site not responding. Last check: 2007-10-22)
		The pseudosphere is a surface of constant negative curvature.
		This program allows the user to construct pseudospheres of various resolutions, and to view them from various angles, to place points on the surface, and to connect the points with lines (ie, geodesics in the pseudosphere).
		Lines on the pseudosphere are geodesics that start at the edge, move outwards, and eventually return to the edge, usually circling the axis one or more times.
	kohlrabi.cs.umanitoba.ca /Students/Pseudosphere.html (154 words)

	[No title]
		It is the generalized helicoid of slant b generated by a tractrix.
		The case b=0 is the standard parametrization of a pseudosphere.
		To plot try pseudocrosscap[-1,0.5,0,2Pi]." pseudosphere::usage="{u,v}->pseudosphere[a][u,v] is the standard parametrization of a pseudosphere of constant negative curvature -1/a^2.
	www.ma.umist.ac.uk /kd/mmaprogs/SURFS.m (2950 words)

	The Pseudosphere
		The surface that represents this region is called a pseudosphere .
		The pseudosphere is a surface of constant negative curvature.
		Since the pseudosphere represents a portion of the hyperbolic plane isometrically, we can compare Gauss' formula with the formula for the area in the hyperbolic plane.
	www.math.uncc.edu /~droyster/math3181/notes/hyprgeom/node69.html (796 words)

	Hypotenews Issue 11
		The type of surface required for this geometry is called a pseudosphere- a diagram of which is shown.
		This type of pseudosphere can be generated by rotating a type of curve known as a tractrix about the x axis.
		On the surface of a pseudosphere it is possible to have an unlimited number of lines which are parallel to a first line, all passing through the same point.
	www-xray.ast.cam.ac.uk /~jgraham/hypo/geometry.html (1148 words)

	Xah: Special Plane Curves: Tractrix
		The tractrix is orthogonal to a set of circles of constant radius whose centers are on the asymptote of the tractrix; conversely, the curve that cuts 90 degrees into a series of identical circles on a line is the tractrix.
		The surface of revolution of tractrix around its asymptote is called pseudosphere.
		It is a surface of constant negative Gaussian curvature.
	xahlee.org /SpecialPlaneCurves_dir/Tractrix_dir/tractrix.html (300 words)

	Free College Essays.com - Free Essays, Term Papers and Book Reports.
		The pseudosphere is the two-dimensional object, which is related to the normal sphere used in Spherical Geometry.
		However, the pseudosphere is smaller then the plane it lies on and tends to “bend back” on itself.
		Since the pseudosphere is bigger than the plane it lies on, it is hard to be drawn out and to be visualized.
	www.free-college-essays.com /Mathematics/15504-Geometry.html (1535 words)

	he curvature: from geometry to cosmology
		The pseudosphere shown at right is a homogeneous space but is not isotropic: for all the points the principal curvatures are variable, holding their product equal to -1.
		It seems impossible to represent the pseudosphere in a portion of a plane, because of its infiniteness.
		The real pseudosphere (right) is a partial model: it has a hole, but the hyperbolic surface have not.
	www.vialattea.net /curvatura/eng (4578 words)

	Hyperbolic geometry 2 (Site not responding. Last check: 2007-10-22)
		The pseudosphere is a trumpet-shaped surface that extends infinitely at both the pointed end and the bell edge.
		He designed this hyperbolic model as the interior of a circle -- without any "edge." The dotted line on the model shows that the surface goes to infinity in all directions.
		A hands-on model can be made by crocheting a surface that becomes more "curly" as it approaches the infinite edge.
	math.youngzones.org /Non-Egeometry/hyperbolic2.html (191 words)

	ppp.html
		by Roger L. Bagula 17 March 2001(C) Abstract:An hyperbolic Pseudosphere based projective plane parametric equation for the surface as derived in Mathematica is presented.
		It was by chance in going over old surface models that the time differential of the Pseudosphere was plotted as a 3d parametric and found to be a sphere.
		projection of the Pseudosphere when squared gave a new projective plane that has a torus with four holes topology overall.
	www.homestead.com /tftn/ppp.html (755 words)

	PSEUDOSPHERE (Site not responding. Last check: 2007-10-22)
		A pseudosphere is a Riemannian manifold with constant negative intrinsic curvature.
		It can be defined with any number of dimensions >=2; the rest of the article will discuss the two-dimensional one.
		The pseudosphere is the space described by hyperbolic geometry.
	www.websters-online-dictionary.org /Ps/Pseudosphere.html (388 words)

	sciforums.com - Gravity and Spacetime (Site not responding. Last check: 2007-10-22)
		As a result, a sphere has a closed surface and a finite area, while a pseudosphere has an open surface and an infinite area.
		One way to think of this is that a pseudosphere is more intensely infinite then the plane.
		Another result of the pseudosphere's negative curvature is that the angles of a triangle drawn on its surface add up to less than 180°
	www.sciforums.com /showthread.php?p=771003 (790 words)

	Mathematical research (from Nikolay Ivanovich Lobachevsky) -- Britannica Concise Encyclopedia - The online ...
		Lobachevsky called his work “imaginary geometry,”; but as a sympathizer with the empirical spirit of Francis Bacon (1561–1626), he attempted to determine the “true” geometry of space by analyzing astronomical data obtained in the measurement of the parallax of stars.
		A physical interpretation of Lobachevsky's geometry on a surface of negative curvature (see the figure of a pseudosphere) was discovered by the Italian mathematician Eugenio Beltrami, but not until 1868.
		From 1835 to 1838 Lobachevsky published “Imaginary geometry,”; “New foundations of geometry with the complete theory of parallels,” and “Application of geometry to certain integrals.” In 1842 his work was noticed and highly praised by Gauss, at whose instigation Lobachevsky was elected that year as a corresponding member of the Royal Society of Göttingen.
	www.britannica.com /ebc/article-214813?tocId=214813 (1114 words)

	Cabinet Magazine Online - Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina (Site not responding. Last check: 2007-10-22)
		Crocheted model of pseudosphere (the hyperbolic equiabvalent of a cone) by Daina Taimina.
		In 1868, the Italian mathematician Eugenio Beltrami had described a surface called a pseudosphere, which is the hyperbolic equivalent of a cone.
		I have crocheted a number of these models and what I find so interesting is that when you make them you get a very concrete sense of the space expanding exponentially.
	www.cabinetmagazine.org /issues/16/crocheting.php (2537 words)

	Chapter 5
		The term "pseudosphere" seems to have originated with Hermann von Helmholtz (1821-1894, German) who was contrasting spherical space with what he called pseudospherical space.
		Eugenio Beltrami (1835-1900, Italian) actually constructed the surface, which is called the pseudosphere, and showed that its geometry is locally the same as (locally isometric to) the hyperbolic geometry constructed by Lobatchevsky.
		We can also crochet a pseudosphere by starting with 5 or 6 chain stitches and continue in spiral fashion increasing as when crocheting the hyperbolic plane.
	www.math.cornell.edu /~dwh/books/eg00/00EG-05 (3493 words)

	non-euclidean geometry (Site not responding. Last check: 2007-10-22)
		The first (unsuccessful) attempt at a model of hyperbolic geometry was the pseudosphere.
		Below is a picture of a triangle on the pseudosphere.
		One of ideas this model does capture is that of triangles having angle sum less than 180 degrees.
	www.ma.utexas.edu /users/heather/non_eucl.html (137 words)

	Notes on Differential Geometry by B. Csikós
		Vector fields along hypersurfaces, tangential vector fields, derivations of vector fields with respect to a tangent direction, the Weingarten map, bilinear forms, the first and second fundamental forms of a hypersurface, principal directions and principal curvatures, mean curvature and the Gaussian curvature, Euler's formula.
		Umbilical, spherical and planar points, surfaces consisting of umbilics, surfaces of revolution, Beltrami's pseudosphere, lines of curvature, parameterizations for which coordinate lines are lines of curvature, Dupin's theorem, confocal second order surfaces; ruled and developable surfaces: equivalent definitions, basic examples, relations to surfaces with K=0, structure theorem.
		Gauss frame of a parameterized hypersurface, formulae for the partial derivatives of the Gauss frame vector fields, Christoffel symbols, Gauss and Codazzi-Mainardi equations, fundamental theorem of hypersurfaces, "Theorema Egregium", components of the curvature tensor, tensors in linear algebra, tensor fields over a hypersurface, curvature tensor.
	www.cs.elte.hu /geometry/csikos/dif/dif.html (588 words)

	Saddle Shape Universe (Site not responding. Last check: 2007-10-22)
		The saddle model is an imperfect analogy for an open universe because it possesses a center.
		The best representation is an infinite surface called a pseudosphere which is impossible to represent in three dimensional space.
		For an artistic visualization, the painting Circle Limit IV by M. Escher is in fact the projection of a pseudosphere onto a plane.
	www.fas.harvard.edu /~scdiroff/lds/AstronomyAstrophysics/SaddleShapeUniverse/SaddleShapeUniverse.html (255 words)

	Smolyaninov Igor I.
		Smolyaninov, Igor I. 30 Issue 4 Page 382 Igor I. Smolyaninov, Christopher C Plasmon-polaritons on the surface of a pseudosphere Plasmon-polaritons on the surface of a pseudosphere Plasmon-polariton behavior on the surfaces of constant negative curvature is considered.
		Smolyaninov, Igor I. Smolyaninov, Igor I. and Davis, Christopher C. Linear and nonlinear optics of surface plasmon toy-models of fl holes and wormholes.
		Smolyaninov, Igor I. Plasmon-polaritons on the surface of a pseudosphere Plasmon-polaritons on the surface of a pseudosphere Plasmon-polariton behavior on the surfaces of constant negative curvature is considered.
	nanotechnology-nutrition.com /funding-nanotechnology/Smolyaninov-Igor-I..html (423 words)

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