Eccentricity (mathematics) - Wikipedia

In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. All types of conic sections, ... Eccentricity(mathematics) FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Characteristicofconicsections Fortheeccentricityofavertexinagraph,seeEccentricity(graphtheory). Alltypesofconicsections,arrangedwithincreasingeccentricity.Notethatcurvaturedecreaseswitheccentricity,andthatnoneofthesecurvesintersect. Inmathematics,theeccentricityofaconicsectionisanon-negativerealnumberthatuniquelycharacterizesitsshape. Moreformallytwoconicsectionsaresimilarifandonlyiftheyhavethesameeccentricity. Onecanthinkoftheeccentricityasameasureofhowmuchaconicsectiondeviatesfrombeingcircular.Inparticular: Theeccentricityofacircleiszero. Theeccentricityofanellipsewhichisnotacircleisgreaterthanzerobutlessthan1. Theeccentricityofaparabolais1. Theeccentricityofahyperbolaisgreaterthan1. Theeccentricityofapairoflinesis ∞ {\displaystyle\infty} Contents 1Definitions 2Alternativenames 3Notation 4Values 5Ellipses 5.1Otherformulaefortheeccentricityofanellipse 6Hyperbolas 7Quadrics 8Celestialmechanics 9Analogousclassifications 10Seealso 11References 12Externallinks Definitions[edit] planesectionofacone Anyconicsectioncanbedefinedasthelocusofpointswhosedistancestoapoint(thefocus)andaline(thedirectrix)areinaconstantratio.Thatratioiscalledtheeccentricity,commonlydenotedase. Theeccentricitycanalsobedefinedintermsoftheintersectionofaplaneandadouble-nappedconeassociatedwiththeconicsection.Iftheconeisorientedwithitsaxisvertical,theeccentricityis[1] e = sin ⁡ β sin ⁡ α ,     0 < α < 90 ∘ ,   0 ≤ β ≤ 90 ∘   , {\displaystylee={\frac{\sin\beta}{\sin\alpha}},\\0 b {\displaystylea>b} 1 − b 2 a 2 {\displaystyle{\sqrt{1-{\frac{b^{2}}{a^{2}}}}}} a 2 − b 2 {\displaystyle{\sqrt{a^{2}-b^{2}}}} Parabola x 2 = 4 a y {\displaystylex^{2}=4ay} 1 {\displaystyle1} undefined( ∞ {\displaystyle\infty} ) Hyperbola x 2 a 2 − y 2 b 2 = 1 {\displaystyle{\frac{x^{2}}{a^{2}}}-{\frac{y^{2}}{b^{2}}}=1} or y 2 a 2 − x 2 b 2 = 1 {\displaystyle{\frac{y^{2}}{a^{2}}}-{\frac{x^{2}}{b^{2}}}=1} 1 + b 2 a 2 {\displaystyle{\sqrt{1+{\frac{b^{2}}{a^{2}}}}}} a 2 + b 2 {\displaystyle{\sqrt{a^{2}+b^{2}}}} Here,fortheellipseandthehyperbola,aisthelengthofthesemi-majoraxisandbisthelengthofthesemi-minoraxis. Whentheconicsectionisgiveninthegeneralquadraticform A x 2 + B x y + C y 2 + D x + E y + F = 0 , {\displaystyleAx^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,} thefollowingformulagivestheeccentricityeiftheconicsectionisnotaparabola(whichhaseccentricityequalto1),notadegeneratehyperbolaordegenerateellipse,andnotanimaginaryellipse:[2] e = 2 ( A − C ) 2 + B 2 η ( A + C ) + ( A − C ) 2 + B 2 {\displaystylee={\sqrt{\frac{2{\sqrt{(A-C)^{2}+B^{2}}}}{\eta(A+C)+{\sqrt{(A-C)^{2}+B^{2}}}}}}} where η = 1 {\displaystyle\eta=1} ifthedeterminantofthe3×3matrix [ A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ] {\displaystyle{\begin{bmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{bmatrix}}} isnegativeor η = − 1 {\displaystyle\eta=-1} ifthatdeterminantispositive. Ellipseandhyperbolawithconstantaandchangingeccentricitye. Ellipses[edit] Theeccentricityofanellipseisstrictlylessthan1.Whencircles(whichhaveeccentricity0)arecountedasellipses,theeccentricityofanellipseisgreaterthanorequalto0;ifcirclesaregivenaspecialcategoryandareexcludedfromthecategoryofellipses,thentheeccentricityofanellipseisstrictlygreaterthan0. Foranyellipse,letabethelengthofitssemi-majoraxisandbbethelengthofitssemi-minoraxis. Wedefineanumberofrelatedadditionalconcepts(onlyforellipses): Name Symbol intermsofaandb intermsofe Firsteccentricity e {\displaystylee} 1 − b 2 a 2 {\displaystyle{\sqrt{1-{\frac{b^{2}}{a^{2}}}}}} e {\displaystylee} Secondeccentricity e ′ {\displaystylee'} a 2 b 2 − 1 {\displaystyle{\sqrt{{\frac{a^{2}}{b^{2}}}-1}}} e 1 − e 2 {\displaystyle{\frac{e}{\sqrt{1-e^{2}}}}} Thirdeccentricity e ″ = m {\displaystylee''={\sqrt{m}}} a 2 − b 2 a 2 + b 2 {\displaystyle{\frac{\sqrt{a^{2}-b^{2}}}{\sqrt{a^{2}+b^{2}}}}} e 2 − e 2 {\displaystyle{\frac{e}{\sqrt{2-e^{2}}}}} Angulareccentricity α {\displaystyle\alpha} cos − 1 ⁡ ( b a ) {\displaystyle\cos^{-1}\left({\frac{b}{a}}\right)} sin − 1 ⁡ e {\displaystyle\sin^{-1}e} Otherformulaefortheeccentricityofanellipse[edit] Theeccentricityofanellipseis,mostsimply,theratioofthedistancecbetweenthecenteroftheellipseandeachfocustothelengthofthesemimajoraxisa. e = c a . {\displaystylee={\frac{c}{a}}.} Theeccentricityisalsotheratioofthesemimajoraxisatothedistancedfromthecentertothedirectrix: e = a d . {\displaystylee={\frac{a}{d}}.} Theeccentricitycanbeexpressedintermsoftheflatteningf(definedas f = 1 − b / a {\displaystylef=1-b/a} forsemimajoraxisaandsemiminoraxisb): e = 1 − ( 1 − f ) 2 = f ( 2 − f ) . {\displaystylee={\sqrt{1-(1-f)^{2}}}={\sqrt{f(2-f)}}.} (Flatteningmaybedenotedbyginsomesubjectareasiffislineareccentricity.) Definethemaximumandminimumradii r max {\displaystyler_{\text{max}}} and r min {\displaystyler_{\text{min}}} asthemaximumandminimumdistancesfromeitherfocustotheellipse(thatis,thedistancesfromeitherfocustothetwoendsofthemajoraxis).Thenwithsemimajoraxisa,theeccentricityisgivenby e = r max − r min r max + r min = r max − r min 2 a , {\displaystylee={\frac{r_{\text{max}}-r_{\text{min}}}{r_{\text{max}}+r_{\text{min}}}}={\frac{r_{\text{max}}-r_{\text{min}}}{2a}},} whichisthedistancebetweenthefocidividedbythelengthofthemajoraxis. Hyperbolas[edit] Theeccentricityofahyperbolacanbeanyrealnumbergreaterthan1,withnoupperbound.Theeccentricityofarectangularhyperbolais 2 {\displaystyle{\sqrt{2}}} . Quadrics[edit] Ellipses,hyperbolaswithallpossibleeccentricitiesfromzerotoinfinityandaparabolaononecubicsurface. Theeccentricityofathree-dimensionalquadricistheeccentricityofadesignatedsectionofit.Forexample,onatriaxialellipsoid,themeridionaleccentricityisthatoftheellipseformedbyasectioncontainingboththelongestandtheshortestaxes(oneofwhichwillbethepolaraxis),andtheequatorialeccentricityistheeccentricityoftheellipseformedbyasectionthroughthecentre,perpendiculartothepolaraxis(i.e.intheequatorialplane).But:conicsectionsmayoccuronsurfacesofhigherorder,too(seeimage). Celestialmechanics[edit] Mainarticle:Orbitaleccentricity Incelestialmechanics,forboundorbitsinasphericalpotential,thedefinitionaboveisinformallygeneralized.Whentheapocenterdistanceisclosetothepericenterdistance,theorbitissaidtohaveloweccentricity;whentheyareverydifferent,theorbitissaidbeeccentricorhavingeccentricitynearunity.Thisdefinitioncoincideswiththemathematicaldefinitionofeccentricityforellipses,inKeplerian,i.e., 1 / r {\displaystyle1/r} potentials. Analogousclassifications[edit] Thissectionneedsexpansion.Youcanhelpbyaddingtoit.(March2009) Anumberofclassificationsinmathematicsusederivedterminologyfromtheclassificationofconicsectionsbyeccentricity: ClassificationofelementsofSL2(R)aselliptic,parabolic,andhyperbolic–andsimilarlyforclassificationofelementsofPSL2(R),therealMöbiustransformations. Classificationofdiscretedistributionsbyvariance-to-meanratio;seecumulantsofsomediscreteprobabilitydistributionsfordetails. Classificationofpartialdifferentialequationsisbyanalogywiththeconicsectionsclassification;seeelliptic,parabolicandhyperbolicpartialdifferentialequations.[3] Seealso[edit] Keplerorbits Eccentricityvector Orbitaleccentricity Roundness(object) Conicconstant References[edit] ^Thomas,GeorgeB.;Finney,RossL.(1979),CalculusandAnalyticGeometry(fifthed.),Addison-Wesley,p.434.ISBN 0-201-07540-7 ^Ayoub,AyoubB.,"Theeccentricityofaconicsection",TheCollegeMathematicsJournal34(2),March2003,116-121. ^"ClassificationofLinearPDEsinTwoIndependentVariables".Retrieved2July2013. Externallinks[edit] WikimediaCommonshasmediarelatedtoEccentricity. MathWorld:Eccentricity vteGravitationalorbitsTypesGeneral Box Capture Circular Elliptical/Highlyelliptical Escape Horseshoe Hyperbolictrajectory Inclined/Non-inclined Kepler Lagrangepoint Osculating Parabolictrajectory Parking Prograde/Retrograde Synchronous semi sub Transferorbit Geocentric Geosynchronous Geostationary Geostationarytransfer Graveyard HighEarth LowEarth MediumEarth Molniya Near-equatorial OrbitoftheMoon Polar Sun-synchronous Tundra Aboutotherpoints Mars Areocentric Areosynchronous Areostationary Lagrangepoints Distantretrograde Halo Lissajous Lunar Sun Heliocentric Earth'sorbit Marscycler Heliosynchronous Other Lunarcycler ParametersShapeSize e  Eccentricity a  Semi-majoraxis b  Semi-minoraxis Q, q  Apsides Orientation i  Inclination Ω  Longitudeoftheascendingnode ω  Argumentofperiapsis ϖ  Longitudeoftheperiapsis Position M  Meananomaly ν,θ,f  Trueanomaly E  Eccentricanomaly L  Meanlongitude l  Truelongitude Variation T  Orbitalperiod n  Meanmotion v  Orbitalspeed t0  Epoch Maneuvers Bi-elliptictransfer Collisionavoidance(spacecraft) Delta-v Delta-vbudget Gravityassist Gravityturn Hohmanntransfer Inclinationchange Low-energytransfer Obertheffect Phasing Rocketequation Rendezvous Transposition,docking,andextraction Orbitalmechanics Astronomicalcoordinatesystems Characteristicenergy Escapevelocity Ephemeris Equatorialcoordinatesystem Groundtrack Hillsphere InterplanetaryTransportNetwork Kepler'slawsofplanetarymotion Lagrangianpoint n-bodyproblem Orbitequation Orbitalstatevectors Perturbation Retrogradeandprogrademotion Specificorbitalenergy Specificangularmomentum Two-lineelements Listoforbits Authoritycontrol:Nationallibraries Germany Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Eccentricity_(mathematics)&oldid=1112677149" Categories:ConicsectionsAnalyticgeometryHiddencategories:ArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataArticlestobeexpandedfromMarch2009AllarticlestobeexpandedArticlesusingsmallmessageboxesCommonscategorylinkisonWikidataArticleswithGNDidentifiers Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommons Languages AfrikaansAlemannischالعربيةAsturianuБеларускаяБеларуская(тарашкевіца)БългарскиBosanskiCatalàČeštinaDanskDeutschEestiΕλληνικάEspañolEsperantoEuskaraفارسیFrançaisGalego한국어ՀայերենHrvatskiItalianoעבריתქართულიLatinaLëtzebuergeschLietuviųമലയാളംNederlands日本語NorskbokmålOʻzbekcha/ўзбекчаPlattdüütschPolskiPortuguêsRomânăРусиньскыйРусскийScotsSlovenščinaکوردیСрпски/srpskiSrpskohrvatski/српскохрватскиSuomiSvenskaதமிழ்TaqbaylitไทยTürkçeУкраїнськаTiếngViệt中文 Editlinks