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Topic: Elliptic geometry

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Parallel Postulate

Euclidean geometry

Hyperbolic space

Hyperbolic geometry

Riemannian geometry

Tangent

Angle

Euclid

Riemannian metric

Euclidean space

Special relativity

Bernhard Riemann

Riemann hypothesis

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	Elliptic geometry - Wikipedia, the free encyclopedia
		Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry, in which, given a line L and a point p outside L, there exists no line parallel to L passing through p.
		Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly one line parallel to L passing through p.
		Elliptic geometry is sometimes called Riemannian geometry, in honor of Bernhard Riemann, but this term is usually used for a vast generalization of elliptic geometry.
	www.wikipedia.org /wiki/Elliptic_geometry (237 words)

	Non-Euclidean geometry - Wikipedia, the free encyclopedia
		In Euclidean geometry, however, the lines remain at a constant distance, while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular.
		Sometimes he is unjustly credited with only discovering elliptic geometry; but in fact, this construction shows that his work was far-reaching, with his theorems holding for all geometries.
		Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same).
	en.wikipedia.org /wiki/Non-Euclidean_geometry (1062 words)

	Encyclopedia: Elliptic geometry
		A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments.
		Spherical geometry is the geometry of the two-dimensional surface of a sphere.
		In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions.
	www.nationmaster.com /encyclopedia/Elliptic-geometry (585 words)

	AllRefer.com - non-Euclidean geometry : Elliptic Geometry (Mathematics) - Encyclopedia
		In elliptic geometry there are no parallels to a given line L through an external point P, and the sum of the angles of a triangle is greater than 180°.
		Riemann's geometry is called elliptic because a line in the plane described by this geometry has no point at infinity, where parallels may intersect it, just as an ellipse has no asymptotes.
		An idea of the geometry on such a plane is obtained by considering the geometry on the surface of a sphere, which is a special case of an ellipsoid.
	reference.allrefer.com /encyclopedia/N/nonEucli-elliptic-geometry.html (255 words)

	Non-Euclidean Geometry
		Elliptic geometry also says that the shortest distance between two points is an arc on a great circle (the “greatest” size circle that can be made on a sphere’s surface).
		Elliptical geometry, which describes the surface of a sphere, is used by pilots and ship captains as they navigate around the spherical Earth, which we live.
		Much like elliptic geometries, the area of a triangle is proportional to its angle sum and of course this implies that there are no similar triangles as well.
	www.geocities.com /CapeCanaveral/7997/noneuclid.html (2640 words)

	Elliptic geometry
		Elliptic geometry is a non-euclidean geometry developed by German geometer Bernhard Riemann.
		It is based on a change in Euclid's fifth postulate, that through a point outside a line, there is one line parallel to the other line.
		Two right angles on the equator, a quarter of the circumference away from each other, make the base, and the angle on the pole is also 90 degrees.
	www.ebroadcast.com.au /lookup/encyclopedia/el/Elliptic_geometry.html (121 words)

	Elliptic geometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		For example, the sum of the (The space between two lines or planes that intersect; the inclination of one line to another; measured in degrees or radians) angles of any (A three-sided polygon) triangle is always greater than 180°.
		This gives us a triangle with an angle sum of 270°, which would be impossible in (Geometry based on Euclid's axioms: e.g., only one line can be drawn through a point parallel to another line) Euclidean geometry.
		Elliptic geometry is sometimes called (A non-Euclidean geometry that regards space is like a sphere and a line is a great circle) Riemannian geometry, in honor of (Pioneer of non-Euclidean geometry (1826-1866)) Bernhard Riemann, but this term is usually used for a vast generalization of elliptic geometry.
	www.absoluteastronomy.com /encyclopedia/e/el/elliptic_geometry.htm (197 words)

	MATH / PHYSICS Research Group Courses
		Geometry and String Theory, I and II are taught every other year and Anomalies in Quantum Field Theory and Strings, I and II are taught the alternate years.
		Geometry and String Theory I is the first term in a two semester interdisciplinary course taught jointly by Burt Ovrut of the Physics Department and Ron Donagi of the Mathematics Department.
		Geometry and String Theory II is the second term in a two semester interdisciplinary course taught jointly by Burt Ovrut of the Physics Department and Ron Donagi of the Mathematics Department.
	dept.physics.upenn.edu /mprg/edcourses.html (1108 words)

	Learn more about Non-Euclidean geometry in the online encyclopedia. (Site not responding. Last check: 2007-10-08)
		The term non-Euclidean geometry describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry.
		In Euclidean geometry, if we start with a point A and a line l, then we can only draw one line through A that is parallel to l.
		About the same time, the Hungarian Janos Bolyai also wrote a treatise on hyperbolic geometry, which was published in 1832 as an appendix to a work of his father's.
	www.onlineencyclopedia.org /n/no/non_euclidean_geometry_1.html (1089 words)

	elliptic geometry -- Britannica Concise Encyclopedia - The online encyclopedia you can trust!
		Though many of elliptic geometry's theorems are identical to those of Euclidean geometry, others differ (e.g., the angles in a triangle add up to more than 180°).
		In plane elliptic geometry there are no parallels to a given line through a given point; it may be viewed as the geometry of a spherical surface on which antipodal points have been identified and all lines are great circles.
		In particular, geometry may be thought of as offering (1) precise definitions of many different figures; (2) construction methods for drawing figures; (3) a wealth of facts about the figures; and, most important, (4) ways to prove the facts.
	www.britannica.com /ebc/article-9363631?tocId=9363631 (917 words)

	elliptic geometry (Site not responding. Last check: 2007-10-08)
		Elliptic Geometry -- from MathWorld Elliptic Geometry -- from MathWorld A constant curvature non-Euclidean geometry which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a...
		Elliptic geometry (sometimes known as Riemannian geometry) is a non-Euclidean geometry...
		Elliptic geometry, like hyperbolic geometry, violates Euclid's parallel postulate, which asserts that there is exactly...
	learning-gd.com /articles/282/elliptic-geometry.html (222 words)

	PlanetMath: elliptic curve (Site not responding. Last check: 2007-10-08)
		These pictures are in some sense not representative of most of the elliptic curves that people work with, since many of the interesting cases tend to be of elliptic curves over algebraically closed fields.
		The points on an elliptic curve have a natural group structure, which makes the elliptic curve into an abelian variety.
		This is version 28 of elliptic curve, born on 2001-12-12, modified 2005-05-21.
	planetmath.org /encyclopedia/EllipticCurve.html (565 words)

	Euclid's Elements, Book I, Proposition 16
		Plane elliptic geometry is closely related to spherical geometry, but it differs in that antipodal points on the sphere are identified.
		Elliptic geometry fails Post.5, the parallel postulate, as well, since any two "straight lines" in an elliptic plane meet.
		The proof of this particular proposition fails for elliptic geometry, and the statement of the proposition is false for elliptic geometry.
	aleph0.clarku.edu /~djoyce/java/elements/bookI/propI16.html (823 words)

	The Math Forum - Math Library - Elliptic &Spherical Geom. (Site not responding. Last check: 2007-10-08)
		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)

	Parallel - Part 4
		Elliptic Geometry is one of the non-Euclidean geometries, which completely reject the validity of Euclid's fifth postulate and modify his second postulate.
		In Riemannian geometry, a straight line of finite length can be extended continuously without bounds, but all straight lines are of the same length.
		In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant.
	www.bsu.edu /web/cvjones/AlgBridge/parallel3.htm (241 words)

	Math 410
		The geometry obtained from Absolute Geometry by omitting the separation axiom and adding RPP is called single elliptic geometry.
		The geometry obtained from Absolute Geometry by changing the incidence axiom about line intersection to: two lines intersect in exactly two points is called double elliptic geometry.
		Single Elliptic -- Geometry of the hemisphere where points on the boundary are associated as one.
	mathserv.monmouth.edu /coursenotes/kuntz/math317/m31712.htm (634 words)

	Elliptic Geometry
		An elliptic straight line is an arc of a great circle.
		Determine the area of the region obtained in the intersection of the unit disk in
		The area of a triangle in the single elliptic geometry model is equal to its angular excess,
	www.msci.memphis.edu /~botelhof/XXII.html (131 words)

	The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (Site not responding. Last check: 2007-10-08)
		Using general theorems of algebraic geometry instead of explicit polynomial calculation simplifies discussion, and at the same time paved the way for the reader towards the higher dimensional version of elliptic curves --- abelian varieties, whose geometry and arithmetic predate much of modern number theory research.
		The theory of elliptic curves has to rank as one of the most fascinating fields in all of mathematics.
		The author reserves the cases of elliptic curves in characteristics 2 and 3 to the appendix.
	www.nuris.us /The-Arithmetic-of-Elliptic-Curves-Graduate-Texts-in-Mathematics-106-06742730091087065847.jsp (1304 words)

	[No title]
		Elliptic and hyperbolic geometry The Euclidean, hyperbolic, and elliptic plane geometries obtained from variations of Hilbert's axioms (see [4] and [3]) would correspond to surfaces (Riemannian 2-manifolds) with constant Gauss curvature.
		Geometric formulas in the different geometries One characteristic difference between the three geometries is reflected in the angle sum of a triangle.
		Impulse curvatures Probably the most important aspect of non-Euclidean geometry that is not obvious from the projections is that lines are straight in both hyperbolic and elliptic geometry.
	www.esu.edu /math/sshema/proceed02/Iseri.doc (4120 words)

	Maybe this Explains the Economic Cycle... best Elliptic Geometry (Site not responding. Last check: 2007-10-08)
		Triangles in Elliptic Geometry Triangles in Elliptic Geometry An examination of some properties of triangles in elliptic geometry, which for this purpose are equivalent to geometry on a hemisphere.
		This document is a Review of Non-Euclidean Geometry with LOGO by Helen Sims-Coomber and Ralp Martin prepared by Pam Bishiop of CTI Mathematics which appeared in the November 91 edition of the CTI Maths quarterly newsletter MathsandStats.
		Elliptic geometry or Riemannian geometry is a non-Euclidean geometry, in which, given...
	ascot.pl /th/Fourier3/Elliptic-Geometry.htm (607 words)

	Elliptic Geometry: Lines and Triangles (Site not responding. Last check: 2007-10-08)
		The simplest model for elliptic geometry is the surface of the earth with the lines of longitude on its surface.
		Thus, elliptic geometry is used daily in aviation navigation.
		A line of latitude that is not the equator of a sphere represents what other concept in elliptic geometry in the model of the sphere?
	www.beva.org /math323/asgn7/dec5.htm (196 words)

	AllRefer.com - non-Euclidean geometry (Mathematics) - Encyclopedia
		non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates.
		The second alternative, which allows no parallels through any external point, leads to the elliptic geometry developed by the German Bernhard Riemann in 1854.
		The results of these two types of non-Euclidean geometry are identical with those of Euclidean geometry in every respect except those propositions involving parallel lines, either explicitly or implicitly (as in the theorem for the sum of the angles of a triangle).
	reference.allrefer.com /encyclopedia/N/nonEucli.html (241 words)

	PlanetMath: the $j$-invariant classifies elliptic curves up to isomorphism (Site not responding. Last check: 2007-10-08)
		-invariant classifies elliptic curves up to isomorphism" is owned by alozano.
		Cross-references: satisfies, curve, elliptic curves, algebraic closure, fixed, field, isomorphism of varieties, isomorphism
		-invariant classifies elliptic curves up to isomorphism, born on 2005-03-01, modified 2005-03-01.
	planetmath.org /encyclopedia/JInvariantClassifiesEllipticCurvesUpToIsomorphism.html (96 words)

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