Non-Euclidean geometry.docx

5/19/2018 Non-Euclidean geometry.docx

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University of Sulaymaniyah

College of engineering

Architecture department

2012 - 2013

Non-Euclidean geometry

Prepared by : checked by:

Ismael omer zhazad jamal

3rd

stage


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Non-Euclidean geometry:

Inmathematics,non-Euclidean geometryis a small set ofgeometries based onaxioms closely related to thosespecifyingEuclidean geometry.As Euclidean geometry lies at theintersection ofmetric geometry andaffine geometry,non-Euclidean geometry arises when either the metric requirement isrelaxed, or theparallel postulate is set aside. In the latter caseone obtainshyperbolic geometry andelliptic geometry,thetraditional non-Euclidean geometries. When the metricrequirement is relaxed, then there are affine planes associatedwith theplanar algebras which give rise tokinematicgeometries that have also been called non-Euclidean geometry.

The essential difference between the metric geometries is thenature ofparallel lines.Euclid's fifth postulate, theparallelpostulate,is equivalent toPlayfair's postulate, which states that,within a two-dimensional plane, for any given line and a point A,which is not on , there is exactly one line through Athat does notintersect . In hyperbolic geometry, by contrast, thereareinfinitely many lines through Anot intersecting , while inelliptic geometry, any line through Aintersects (see the entriesonhyperbolic geometry,elliptic geometry,andabsolutegeometry for more information).

Behavior of lines with a common perpendicular in each of the three types of geometry
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Another way to describe the differences between thesegeometries is to consider two straight lines indefinitely extendedin a two-dimensional plane that are bothperpendicular to a thirdline:

In Euclidean geometry the lines remain at aconstant distance from each other even if extended to infinity,and are known as parallels.

In hyperbolic geometry they "curve away" from each other,increasing in distance as one moves further from the points ofintersection with the common perpendicular; these lines areoften called ultraparallels.

In elliptic geometry the lines "curve toward" each other andintersect

Creation of non-Euclidean geometry

The beginning of the 19th century would finally witness decisive

steps in the creation of non-Euclidean geometry. Around1818,Schweikart

and then, around 1830,

theHungarianmathematicianJnos Bolyai andtheRussian mathematicianNikolai IvanovichLobachevsky separately published treatises on hyperbolicgeometry. (Schweikart did not publish his work, his contributionwas discovered much later.) Consequently, hyperbolic geometry

is called Bolyai-Lobachevskian geometry, as bothmathematicians, independent of each other, are the basic authorsof non-Euclidean geometry.Gauss mentioned to Bolyai's father,when shown the younger Bolyai's work, that he had developedsuch a geometry several years before,

though he did not publish.

While Lobachevsky created a non-Euclidean geometry bynegating the parallel postulate, Bolyai worked out a geometry

where both the Euclidean and the hyperbolic geometry arepossible depending on a parameter k. Bolyai ends his work bymentioning that it is not possible to decide through mathematicalreasoning alone if the geometry of the physical universe is
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Euclidean or non-Euclidean; this is a task for the physicalsciences.

Bernhard Riemann,in a famous lecture in 1854, founded the fieldofRiemannian geometry,discussing in particular the ideas nowcalledmanifolds,Riemannian metric,andcurvature.Heconstructed an infinite family of geometries which are notEuclidean by giving a formula for a family of Riemannian metricson the unit ball inEuclidean space.The simplest of these iscalledelliptic geometry and it is considered to be a non-Euclideangeometry due to its lack of parallel lines

.

Non-Euclidean Geometry

In three dimensions, there are three classes of constantcurvaturegeometries.All are based on the first four ofEuclid'spostulates,but each uses its own version of theparallelpostulate.The "flat" geometry of everyday intuition iscalledEuclidean geometry (or parabolic geometry), and the non-Euclidean geometries are calledhyperbolic geometry (orLobachevsky-Bolyai-Gauss geometry) andelliptic geometry (orRiemannian geometry).Spherical geometry is a non-Euclideantwo-dimensional geometry. It was not until 1868 that Beltramiproved that non-Euclidean geometries were as logicallyconsistent asEuclidean geometry.
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Models of non-Euclidean geometry

Elliptic geometry

The simplest model forelliptic geometry is a sphere, where linesare "great circles" (such as theequator or themeridians on

aglobe), and points opposite each other (called antipodal points)are identified (considered to be the same). This is also one of thestandard models of thereal projective plane.The difference isthat as a model of elliptic geometry a metric is introducedpermitting the measurement of lengths and angles, while as amodel of the projective plane there is no such metric.

In the elliptic model, for any given line and a point A, which is

not on , all lines through Awill intersect .Hyperbolic geometry

Even after the work of Lobachevsky, Gauss, and Bolyai, thequestion remained: "Does such a model exist forhyperbolicgeometry?". The model for hyperbolic geometrywas answeredbyEugenio Beltrami,in 1868, who first showed that a surfacecalled thepseudosphere has the appropriate curvature to modela portion ofhyperbolic space and in a second paper in the sameyear, defined the Klein model which models the entirety ofhyperbolic space, and used this to show that Euclidean geometryand hyperbolic geometry wereequiconsistent so that hyperbolicgeometry waslogically consistent if and only if Euclideangeometry was. (The reverse implication follows fromthehorosphere model of Euclidean geometry.)

In the hyperbolic model, within a two-dimensional plane, for anygiven line and a point A, which is not on , thereareinfinitelymany lines through Athat do not intersect .

In these models the concepts of non-Euclidean geometries arebeing represented by Euclidean objects in a Euclidean setting.This introduces a perceptual distortion wherein the straight lines

of the non-Euclidean geometry are being represented byEuclidean curves which visually bend. This "bending" is not a
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property of the non-Euclidean lines, only an artifice of the waythey are being represented

References :

http://en.wikipedia.org

http://mathworld.wolfram.com
http://en.wikipedia.org/http://mathworld.wolfram.com/http://mathworld.wolfram.com/http://mathworld.wolfram.com/http://en.wikipedia.org/


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Non-Euclidean geometry.docx