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1、3/16/12Euclideanspace-Wikipedia,thefreeencyclopediaEuclideanspaceFromWikipedia,thefreeencyclopediaInmathematics,EuclideanspaceistheEuclideanplaneandthree-dimensionalspaceofEuclideangeometry,aswellasthegeneralizationsofthesenotionstohigherdimensions.Theterm“Eu
2、clidean”distinguishesthesespacesfromthecurvedspacesofnon-EuclideangeometryandEinstein'sgeneraltheoryofrelativity,andisnamedfortheGreekmathematicianEuclidofAlexandria.ClassicalGreekgeometrydefinedtheEuclideanplaneandEuclideanthree-dimensionalspaceusingcertainp
3、ostulates,whiletheotherpropertiesofthesespaceswerededucedastheorems.Inmodernmathematics,itismorecommontodefineEuclideanspaceusingCartesiancoordinatesandtheideasofanalyticgeometry.Thisapproachbringsthetoolsofalgebraandcalculustobearonquestionsofgeometry,andEve
4、rypointinthree-dimensionalhastheadvantagethatitgeneralizeseasilytoEuclideanEuclideanspaceisdeterminedbythreespacesofmorethanthreedimensions.coordinates.Fromthemodernviewpoint,thereisessentiallyonlyoneEuclideanspaceofeachdimension.Indimensiononethisistherealli
5、ne;indimensiontwoitistheCartesianplane;andinhigherdimensionsitistherealcoordinatespacewiththreeormorerealnumbercoordinates.ThusapointinEuclideanspaceisatupleofrealnumbers,anddistancesaredefinedusingtheEuclideandistanceformula.Mathematiciansoftendenotethen-dim
6、ensionalEuclideanspaceby,orsometimesiftheywishtoemphasizeitsEuclideannature.Euclideanspaceshavefinitedimension.Contents1Intuitiveoverview2Realcoordinatespace3Euclideanstructure4TopologyofEuclideanspace5Generalizations6Seealso7ReferencesIntuitiveoverviewOneway
7、tothinkoftheEuclideanplaneisasasetofpointssatisfyingcertainrelationships,expressibleintermsofdistanceandangle.Forexample,therearetwofundamentaloperationsontheplane.Oneistranslation,whichmeansashiftingoftheplanesothateverypointisshiftedinthesamedirectionandbyt
8、hesamedistance.Theotherisrotationaboutafixedpointintheplane,inwhicheverypointintheplaneturnsaboutthatfixedpointthroughthesameangle.OneofthebasictenetsofEuclideangeometryisthattwofigures(t
