24xyz3524rcoscosrsincosrsin35 ThesearethesameformulasforsphericalcoordinatesforR3exceptthatforsphericalcoordinatesrisavariablenota xedradiusCurvatureThecurvaturethatistheGaussiancurvature ID: 134393
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Theequationinrectangularcoordinatesforasphereofradiusrcenteredattheoriginisx2+y2+z2=r2:Astandardparameterizationofthesphereisintermsoflongitudeandlatitude.Thelongitudecanvaryfrom0to2(orfrom�toifyouprefer),andthelatitude canvaryfrom�=2to=2.(Sometimessomeothervariablethanlatitudeisused,forinstance,theanglefromapoleinwhichcasetheanglevariesfrom0to.)Forasphereofradiusrcenteredattheorigintheparameterizationis 24xyz35=24rcoscos rsincos rsin 35 ThesearethesameformulasforsphericalcoordinatesforR3exceptthatforsphericalcoor-dinates,risavariable,notaxedradius.Curvature.Thecurvature,thatistheGaussiancurvature,ofasurfaceisanintrinsicpropertyofthesurface.Thecurvatureofmostsurfacesvariesfrompointtopoint,butbecauseofthesymmetryofthesphereisitthesameeverywhere.Thesphereisasurfacewithconstantpositivecurvature.TheCylinder.Likethesphere,thecylinderisancient.SeeEuclid'sXI.Def.14. Figure2:ThecylinderTheequationinrectangularcoordinatesforacylinderofradiusrandwhoseaxisisthez-axisisx2+y2=r2:2 24xyz35=24ap t2�1cosbp t2�1sinct35 Thehyperboloidofonesheetisadoublyruledsurface.Througheachitspointstherearetwolinesthatlieonthesurface.Bothkindsofcircularhyperboloidsaswellastheconecanbeincludedinonefamilyofsurfacesbymodifyingtheirdeningequationsslightly.Considertheequationsx2+y2=z2+ewhereeisaconstant.Yougetaconewhene=0,ahyperbolaofonesheetwhene�0,andahyperbolaoftwosheetswhene0.TheEllipticParaboloid.Someofthecrosssectionsoftheellipticparaboloidareellipses,othersareparaboloids. Figure8:TheEllipticParaboloidTheellipticparaboloidisdenedbytheequationz c=x2 a2+y2 b2:Whena=b,it'sacircularparaboloid,alsocalledaparaboloidofrevolution.Itcanbeparameterizedby 24xyz35=24atcosbtsinct235 7 Thecircularparaboloidhasaninterestingfocalproperty.Ifthesurfaceismirroredandalightsourceplacedatitsfocus,thenthelightrayswillformaparallelbeamoflight.Likewiseabeamoflightcanbecollectedbyaparaboloidtoitsfocus.TheHyperbolicParaboloid.Someofthecrosssectionsofthehyperbolicparaboloidarehyperbolas,othersareparaboloids. Figure9:TheHyperbolicParaboloidIthastheequationz c=x2 a2�y2 b2andcanbeparameterizedseveralwaysincluding 24xyz35=24asbtc(s2�t2)35 Likethehyperboloidofonesheet,thehyperbolicparaboloidisadoublyruledsurface.Througheachitspointstherearetwolinesthatlieonthesurface.Thehyperbolicparaboloidisasurfacewithnegativecurvature,thatis,asaddlesurface.That'sbecausethesurfacedoesnotlieononesideofthetangentplaneatapointlikeitwouldforasurfacewithpositivecurvature;insteadpartofthesurfaceliesononesideofthetangentplay,andpartliesontheother.8 Figure11:TheHelicoidTheMobiusstrip.AMobiusstripisanonorientablesurface.Thatistosay,noorientationcanbeassignedconsistentlytotheentiresurfaceunlikeallthesurfacesmentionedabove.Acylindercanbeconstructedfronarectanglebyattachingonesidetotheother,butifahalftwistisappliedbeforetheattachment,thenaMobiusstripresults.ThisparticularMobiusstripisparameterizedby 24xyz35=24cos+scos 2cosusin+scos 2sinussin 235 wheresvariesfrom�1 4to1 4,andvariesfrom0to2.Math131HomePageathttp://math.clarku.edu/~djoyce/ma131/10