ellipse - wikipedia

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1、Ellipse-WikipediaEllipseFromWikipedia,thefreeencyclopediaAnellipseobtainedastheintersectionofaconewithaplane.Inmathematics,anellipse(fromGreekἔλλειψιςelleipsis,a"fallingshort")isthefiniteorboundedcaseofaconicsection,thegeometricshapethatresultsfromcuttingacircularconicalorcylindricalsurfacew

2、ithanobliqueplane(theothertwocasesbeingtheparabolaandthehyperbola).Itisalsothelocusofallpointsoftheplanewhosedistancestotwofixedpointsaddtothesameconstant.Ellipsesalsoariseasimagesofacircleorasphereunderparallelprojection,andsomecasesofperspectiveprojection.Indeed,circlesarespecialcasesofell

3、ipses.Anellipseisalsotheclosedandboundedcaseofanimplicitcurveofdegree2,andofarationalcurveofdegree2.ItisalsothesimplestLissajousfigure,formedwhenthehorizontalandverticalmotionsaresinusoidswiththesamefrequency.Contents1Elementsofanellipse2Drawingellipses2.1Thepins-and-stringmethod2.2Othermeth

4、ods3Ellipsesinphysics3.1Ellipticalreflectorsandacoustics3.2Planetaryorbits3.3Harmonicoscillators3.4Phasevisualization3.5Ellipticalgears3.6Optics4Mathematicaldefinitionsandproperties4.1InEuclideangeometry4.1.1Definition4.1.2Eccentricity4.1.3Directrix4.1.4Ellipseashypotrochoid1PDFcreatedwithpd

5、fFactorytrialversionwww.pdffactory.comEllipse-Wikipedia4.1.5Area4.1.6Circumference4.2Inprojectivegeometry4.3Inanalyticgeometry4.3.1Generalellipse4.3.2Canonicalform4.4Intrigonometry4.4.1Generalparametricform4.4.2Parametricformincanonicalposition4.4.3Polarformrelativetocenter4.4.4Polarformrela

6、tivetofocus4.4.5Gauss-mappedform4.4.6Angulareccentricity4.5Degreesoffreedom5Ellipsesincomputergraphics6Seealso7References8ExternallinksElementsofanellipseTheellipseandsomeofitsmathematicalproperties.Anellipseisasmoothclosedcurvewhichissymmetricaboutitscenter.Thedistancebetweenantipodalpoints

7、ontheellipse,orpairsofpointswhosemidpointisatthecenteroftheellipse,ismaximumandminimumalongtwoperpendiculardirections,themajoraxisortransverse[1]diameter,andtheminoraxisorconjugatediameter.Thesemimajoraxis(denotedbyainthefigure)andthesemiminoraxis(