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PAGE 1 CURVEDAXISREVOLUTIONS by TODDHUCKABY AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegreeof BachelorofArtsinMathematics UndertheSponsorshipof ProfessorDavidMullins,Ph.D. Sarasota,Florida May2013 PAGE 2 CURVEDAXISREVOLUTIONS ToddHuckaby NewCollegeofFlorida,2013 Abstract Motivatedbystandardsolidsofrevolutioncomputablebyelementary calculusmethodsthisthesisdevelopsaconstructionofsolidsofrevolutionwithacurvedaxis.Weshowthatthevolumesobtaineddonot dependonthecurvatureoftheaxis.In H 3 ,weshowbyconstruction thereisnostraight-forwardgeneralization.Apartialgeneralization isconjecturedtoexistforcurvesinahorizontalplanebecausethe naturalmapbetweenthehorizontalcylindertothehorizontaltorus isseentopreservevolumeinanexample. DavidMullins,Ph.D. DivisionofNaturalSciences PAGE 3 Contents Contentsii ListofFiguresiv 1Introduction1 2SurfacesandSolidsofRevolution4 2.1SurfacesofRevolution........................4 2.2SolidsofRevolution..........................6 2.3SlantedVolumesofRevolution....................10 2.4Summary...............................13 3CurvedAxisRevolutions14 3.1Introduction..............................14 3.2CurvedAxisSurfacesofRevolution.................16 3.3CurvedAxisSolidsofRevolution..................17 4HyperbolicSpace26 4.1TheHyperbolicPlane........................26 4.2HyperbolicLines,Geodesics,Isometries H 2 .............29 ii PAGE 4 CONTENTS 4.3HyperbolicSpace...........................30 5HyperbolicRevolutionsin H 3 32 5.1EquidistantRegionsin H 2 ......................32 5.2VolumeofanEquidistantTube...................36 5.3HorizontalAxisExamplesHyperbolicCurve...........38 5.4VolumeofHorizontalCylinder....................39 5.5VolumeofHorizontalTorus.....................41 6Conclusions43 7References44 iii PAGE 5 ListofFigures 2.1SurfaceofRevolution.FigureFrom[1]...............4 2.2SlantedRegion............................10 3.1Planeviewofprolecurve = + n whichrevolvesaroundthe curvedaxis .............................14 3.2Thescalednormalvector ~n denesistheradiusofcirclecentered attheorigin.Thecircleshownistransformedasindicatedbythe arrowsendingtheverticalradiusvectorto ~n ............16 3.3Planeviewofprolecurve = + n whichrevolvesaroundthe curvedaxis togettheshapeontheright.............18 3.4TorusofRevolution..........................19 3.5TorusPiece..............................20 3.6Somethingunusual.Easytocompute.................21 3.7Transformation: S 1 S 2 .....................23 4.1Hyperboliclinesintheupperhalf-planemodel H 2 .........29 5.1EquidistantRegion..........................32 5.2HyperbolicDiskcenteredwithhyperboliccenter C h = i ......38 iv PAGE 6 LISTOFFIGURES 5.3Horizontalcylinder,hyperboliccurvedaxis..............39 v PAGE 7 Chapter1 Introduction ThepresentworkwassetinmotionbyaCalculuswithTheoryProblemgiven byProfessorDavidMullins.Theproblemasksforaformulaforcomputingthe volumeofthesolidofrevolutionobtainedbyrotatingafunctionaroundtheslant axisofrotation y = x .Theproblemcanbesolvedbyachangeofvariablesappliedtothemethodofdisks.Alternatively,a )]TJ/F24 7.9701 Tf 10.494 4.707 Td [( 4 rotationalignstheslantaxis withthestandardaxisallowingfortheusualmethodofdisks. Questions: 1.Isthereawaytocomputethevolumeofanysolidwithslantedaxis y = mx + b ?SeeChapter2 2.Isthereawaytocomputethevolumeofanysolidwithcurvedaxisof rotation y = x ?assumingthereexistsawell-denedconstructionSee Chapter3 Volumeofasolidobtainedbyrotatingacurvearoundacurvedaxisofrotation 1 PAGE 8 formsthenucleusofthisthesis. Surfacesandsolidswithcurvedaxisofrevolutionareparameterizedinterms oftheunitnormal ~n ofanaxisofrotation ~ ,andfunction ,calledtheradial function,whichdenestheradiusoftherevolutionateachpointoftheaxis.The curveundergoingrotationisdenedtobe ~ = ~ + ~n Theradialfunction inducesanaturaldieomorphismbetweentheradial functionofacurvedaxisrevolutionandastandardrevolutionwhichpreserves volume.Thisshowsthevolumeofacurvedaxissolidofrevolutionmaybe computedusingthediskmethodformula: Vol = Z b a f x 2 dx .1 Where f iscontinuous. Thecurvature k ofanaxisprovidesarestrictiononthegeneralityof and, therefore,anintrinsicrestrictionontheradiusoftherevolutionateachpointon theaxis. Question: 1.Arethereextrinsicpropertiessuchasspacecurvatureaectingtheconstructionofarevolutionaboutacurvedorstraightaxis?Howdoesthis aectthevolume?SeeChapter5 Afewexamplesofrevolutionshowthatstraight-forwardgeneralizationofvolume preservationisnotpossibleinhyperbolicspace. However,partialgeneralizationisconjecturedinhorizontalplaneswherethe metricisalwaysscaledeuclideanbecauseofvolumeispreservedunderthenat2 PAGE 9 uralmapbetweenthecylinderandtorusofthesameaxislengthinthesame horizontalplane z constant. 3 PAGE 10 Chapter2 SurfacesandSolidsofRevolution 2.1SurfacesofRevolution ThefollowingconstructionofsurfacesofrevolutionfollowsthenotationinPressley[1]. Figure2.1:SurfaceofRevolution.FigureFrom[1] 4 PAGE 11 Denition1. Asurfaceofrevolutionisthesurfaceobtainedbyrotatingaplane curvecalledtheprolecurvearoundastraightlineintheplane.Thecircles obtainedbyrotatingaxedpointontheprolecurvearoundtheaxisofrotation arecalledtheparallelsofthesurfaceandthecurvesonthesurfaceobtainedby rotatingtheprolecurvethroughaxedanglearecalleditsmeridians. AsinFigure2.1,supposetheaxisofrotationisthe z )]TJ/F23 11.9552 Tf 11.51 0 Td [(axis ,theplaneisthe xz )]TJ/F23 11.9552 Tf 12.031 0 Td [(plane or y =0,andtheprolecurveis : D R R 3 ,asmoothplane curvewiththefollowingcomponentform: u = f u ; 0 ;g u .1 Everypoint P onthesurfaceliesonameridianandmaybeobtainedbyrotating apoint Q ontheprolecurvethroughanangle v aroundthe z )]TJ/F23 11.9552 Tf 12.012 0 Td [(axis .Thatis, if Q = u isapointontheprolecurve,then P isobtainedbyarotationof the xy )]TJ/F23 11.9552 Tf 11.955 0 Td [(plane asfollows: P = 2 6 6 6 6 4 cos v )]TJ/F23 11.9552 Tf 9.298 0 Td [(sin v 0 sin v cos v 0 001 3 7 7 7 7 5 2 6 6 6 6 4 f u 0 g u 3 7 7 7 7 5 = 2 6 6 6 6 4 f u cos v f u sin v g u 3 7 7 7 7 5 .2 Theimageofallrotationsisaparametricsurface : U R 2 R 3 for v 2 ; 2 and u 2 D ,where D R istheopenintervaldomainof ,forall u;v 2 U .Surfaceandsolidparametrizationsin R 3 willbewritteninhorizontal 5 PAGE 12 vectorform.Forexample: u;v = f u cos v ;f u sin v ;g u .3 Surfacesofrevolutionappearinsinglevariablecalculusunderthefollowing premise: If : D R R isacontinuousfunction,asolidofrevolutionmaybe obtainedbyrotatingthegraphof z aroundthe z )]TJ/F23 11.9552 Tf 11.955 0 Td [(axis ." Thegraphoftheprolecurveisgivenby z = z ; 0 ;z .If u 2 D and v 2 ; 2 andthefunction isgiven,thenaturalparametrizationcanbewritten explicitlyas: @S = u cos v ; u sin v ;u .4 2.2SolidsofRevolution Denition2. Asolidofrevolutionistheinteriorofasurfaceofrevolution. @S denestheboundaryofasolidofrevolution.Thelevelcurvesofthe surfaceofrevolution @S alongthe z )]TJ/F23 11.9552 Tf 11.955 0 Td [(axis areitsparallelsgivenby: L z = f x;y;z : z 2 D ; x 2 + y 2 = z 2 g .5 Everypoint P of @S iscontainedinalevelcurvedescribedbyacircleequation withradius atsomeheight z .Ifthecrosssections C z ofaset S areexactly theinteriorsofthelevelcurves L z ofasurfaceofrevolution,then S isa solid ofrevolution .Thecrosssectionsofasolidofrevolutionaredescribedbychanging 6 PAGE 13 theprevioussettoaninequality: C z = f x;y;z : z 2 D ; x 2 + y 2 z 2 g .6 Asolidofrevolution S istheinteriorofasurfaceofrevolutionFigure2.1. Theparametrizationofthesolid S onthreevariablesisgivenby: S t;u;v = t u cos v ;t u sin v ;u .7 t 2 ; 1 u 2 Dv 2 ; 2 .8 2.2.1VolumeofSolidsofRevolution ThefollowingdenitionsandtheoremsonvolumemaybefoundinApostol[2]. Weassumethereexistcertainsets S ofpointsinthree-dimensionalspace, whichwecall measurablesets ,andasetfunction Vol ,calledthe volumefunction whichassignstoeachmeasurableset S anumber Vol S ,calledthevolumeof S .Thesymbol A denotestheclassofallmeasureablesetsinthree-dimensional space,andwecalleachset S in A a solid Denition3. CavalieriSolidSupposeSisagivensolidandLagivenline.If aplaneFisperpendiculartoL,theintersection F [ S iscalledacross-section perpendiculartoL.Ifeverycross-sectionperpendiculartoLisameasurableset initsownplane,wecallSa CavalieriSolid Denition4. AXIOMATICDEFINITIONOFVOLUME:Weassumethere existsaclass A ofsolidsandasetfunctionvol,whosedomainis A ,withthe followingproperties: 7 PAGE 14 1.Nonnegativeproperty:Foreachset S in A wehave Vol S 0 2.Additiveproperty:If S and T arein A ,the S [ T and S T arein A ,and wehave: Vol S [ T = Vol S + Vol T )]TJ/F23 11.9552 Tf 11.956 0 Td [(Vol S T : .9 3.DierenceProperty:If S and T arein A with S T ,then T )]TJ/F23 11.9552 Tf 12.021 0 Td [(S isin A andwehave Vol T )]TJ/F23 11.9552 Tf 11.955 0 Td [(S = Vol T )]TJ/F23 11.9552 Tf 11.956 0 Td [(Vol S : 4.Cavalieri`sprinciple:If S and T aretwoCavelrisolidsin A with area S F area T F foreveryplaneFperpendiculartoagivenline,then Vol S Vol T 5.Choiceofscale:Everyconvexsetisin A Cavalieri'sprincipleassignsequalvolumestotwoCavalierisolids, S and T ,if area S F = area T F foreveryplane F perpendiculartothegivenline L Theorem1. LetRbeaCavalierisolidin A withacross-sectionalareafunction Awhichisintegrableonaninterval[a,b].ThenthevolumeofRisequaltothe integralofthecross-sectionalarea: Vol R = Z b a A z dz .10 Proof. SeeTheorem2.7Apostol[2] Theorem2. Thevolumeofthesolidofrevolutionboundedbythesurface : u;v = f u cos v ;f u sin v ;g u .11 8 PAGE 15 isgivenby Vol = Z b a f u 2 g 0 u du .12 Proof. ThisproofisachangeofvariableschainruleappliedtoTheorem1.The areaof C z willbe A z = r z 2 ,where r = f u and z = g u .Bythechain rule, A z dz = f u 2 dg du du dz dz Z z = b z = a A z dz = Z z = b z = a f u 2 dg du du dz dz = Z u = g )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 b u = g )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 a f u 2 g 0 u du .13 Theorem3. Thevolumeofastandardsolidofrevolutionis Vol S = Z b a z 2 dz .14 Proof. Itisboundedbythesurface @S = u cos v ; u sin v ;u .Apply Theorem3with g u = u f u = u Denition5. The volumeintegral isatripleintegraloverthreecoordinates givingthevolumewithinsomeregion X : Vol X = ZZZ X dV .15 Where dV isthestandardEuclideanvolumeelementgivenby dV = dxdydz in rectangularcoordinates.Theorem1andDenition5willbeusedinterchangably whennecessary. 9 PAGE 16 2.3SlantedVolumesofRevolution Example1. byaChangeofVariablesSuppose f :[ a;b ] R isasmooth functionwhosegraphliesbeneaththeline y = x i.e., f x x and f 0 x > )]TJ/F18 11.9552 Tf 9.298 0 Td [(1. Furthermore,suppose R Figure2.2isaregionintheplaneboundedby y = x andthegraphof f x describedbythefollowingsystem: R = 8 > > > > > > > < > > > > > > > : y x y )]TJ/F23 11.9552 Tf 21.917 0 Td [(x + a + f a y )]TJ/F23 11.9552 Tf 21.917 0 Td [(x + b + f b y f x .16 Figure2.2:SlantedRegion Theregion R mayberotatedaround y = x toobtainasolidofrevolution S Itwillbeshownthatthevolumeofthesolid S maybecomputedwithaformula 10 PAGE 17 involvingasingleintegral. Theaxisofrotationisaboundaryof R .Thecrosssectionsperpendicularto theaxisintersect f uniquely,establishingtheexistenceofawell-denedradial function.Toseethis,let u;u beanypointontheaxisofrotation.Weshow thatthereisexactlyonepoint x;f x onthegraphof f thatliesontheline withslope )]TJ/F18 11.9552 Tf 9.298 0 Td [(1andcontains u;u u maybecomputedwiththeslopeformulaas follows: u )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x u )]TJ/F23 11.9552 Tf 11.956 0 Td [(x = )]TJ/F18 11.9552 Tf 9.299 0 Td [(1 u = x + f x 2 .17 Thesubstitution u asafunctionof x isrealizedinthesystemdening R {it iswelldenedattheendpoints.Inbetweenwesupposetherearetwopoints x 1 ;f x 1 and x 2 ;f x 2 whichlieontheperpendicularcrosssectioncontaining u inordertoderiveacontradiction. x 1 + f x 1 2 = x 2 + f x 2 2 f x 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x 1 x 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(x 2 = )]TJ/F18 11.9552 Tf 9.299 0 Td [(1.18 Bythemeanvaluetheorem,if x 1 6 = x 2 thereexists c 2 x 1 ;x 2 suchthat f 0 c = )]TJ/F18 11.9552 Tf 9.298 0 Td [(1.Thiscontradictsthehypothesis,therefore x 1 ;f x 1 = x 2 ;f x 2 Now u = d u;u ; x;f x iswelldened.Asubstitution u = x + f x 2 simpliesasfollows: 11 PAGE 18 x = r x + f x 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(x 2 + x + f x 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(f x 2 .19 = p 2 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x .20 Let ` = d ; 0 ; u;u bethedistancefromtheorigintothepoint u;u ` = d ; 0 ; u;u = p 2 u = p 2 2 x + f x .21 ` asafunctionof x providesachangeofvariablesmapfromthe x )]TJ/F23 11.9552 Tf 12.062 0 Td [(axis to theaxisofrotation.Thederivativeofthismapis: d` = p 2 2 + f 0 x dx .22 Eachdiskwillhavevolume: ` x 2 d` = 2 p 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x 2 + f 0 x dx .23 Finally,wehaveshown: Vol S = Z b a p 2 4 x )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x 2 + f 0 x dx .24 Example2. byRotation 12 PAGE 19 Considerarotationofthe xy )]TJ/F23 11.9552 Tf 11.956 0 Td [(plane : 2 6 6 6 6 4 cos 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [(sin 4 0 sin 4 cos 4 0 001 3 7 7 7 7 5 2 6 6 6 6 4 x f x 0 3 7 7 7 7 5 = 2 6 6 6 6 4 p 2 2 x )]TJ/F23 11.9552 Tf 11.956 0 Td [(f x p 2 2 x + f x 0 3 7 7 7 7 5 .25 Therightsideisaplanecurve ~ x = p 2 2 x )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x ; p 2 2 x + f x ; 0.26 Theareaunderaplanecurve ~ x = 1 x ; 2 x ; 0iscalculatedby Area = Z D 1 x 0 2 x dx .27 Thismaybeinterpretedasachangeofvariablesontheidentityfunctionequal to1.Bythecorollary: Z b a 1 x 2 0 2 x dx = Z b a p 2 4 x )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x 2 + f 0 x dx .28 2.4Summary Asolidofrevolutionisparametrizedby S t;u;v = t u cos v ;t u sin v ;u anditsboundaryisthesurface: @S = u cos v ; u sin v ;u Figure2.1. Thevolumeofasolidofrevolutionwithaslantedstraightlineaxisofrotation diersfromarevolutionaboutthe z )]TJ/F23 11.9552 Tf 12.03 0 Td [(axis byarigidmotion,oranorthogonal transformationwithjacobiandeterminant 1,sothevolumeisunaected. 13 PAGE 20 Chapter3 CurvedAxisRevolutions 3.1Introduction Figure3.1:Planeviewofprolecurve = + n whichrevolvesaroundthe curvedaxis Incontrastwiththeusualsurfacesandsolidsobtainedbystraightaxisrevolutionspreviouschapter,thesurfacesandsolidspresentedinthischapterare 14 PAGE 21 obtainedbyrevolutionsaboutacurvedaxis. Curvedaxisrevolutionscanbeunderstoodintuitivelyasfollows.Imaginea curvedaxis ~ simpleexample:apieceofacircle,andaprolecurve ~ which willundergorevolutionabout ~ simpleexamples:circleorstraightline.The revolutionofthepointsonthecurve ~ traceoutcirclescenteredat ~ u ,foreach u 2 D .Incontrastwiththeusualstraightaxisrevolutions,itisnolongertrue thattherotationsofpointsoccurinparallelplanes.Itremainstrue,however, thatthecross-sectionalplanesofrotationwillbeperpendiculartothecurvedaxis ~ astheyareforstraightaxisrevolutions. Theperpendicularnatureoftherevolutionssimpliesparametricconstructionsofcurvedaxissurfacesandsolidsbecause,assuming ~ issmooth,theradius ofrevolutionofapointon ~ canbethoughtofasarevolutionoftheunitnormal ~n scaledbyaradialfunction Inordertoensureone-to-oneness,itmustbeassumedthattherevolutionsof anytwopointsarenon-overlapping.Inotherwords,everycross-sectionalplaneof ~ mustintersect ~ atauniquepoint. Theone-oneassumptionensuresthatthereexistsawell-denedfunctionon D =[ a;b ] : D R suchthatthedistancefrom ~ u to ~ alongtheunit normal ~n isgivenby u asinthegureabove3.1 If ~ u and u aregiven,notethat: ~ u = ~ u + u ~n u .1 Threechoiceswilldeterminetheshapeandvolumeofacurvedaxissurface orsolidofrevolution: 15 PAGE 22 1.Axisgivenby ~ : D R 3 ,anarc-lengthparametrized plane curve. 2.AxisLengthgivenby ` ,arealnumber.And, 3.RadialLengthsgivenby : D R ,afunctiondenedonthesameinterval domain D as ~ 3.2CurvedAxisSurfacesofRevolution Figure3.2:Thescalednormalvector ~n denesistheradiusofcirclecentered attheorigin.Thecircleshownistransformedasindicatedbythearrowsending theverticalradiusvectorto ~n Supposewehaveavector ~n = n 1 ;n 2 ; 0inthe xy )]TJ/F23 11.9552 Tf 12.464 0 Td [(plane deningthe radiusofacircletracedoutbyrevolvingabouttheorigin,passingthroughthe z )]TJ/F23 11.9552 Tf 11.956 0 Td [(axis seeFigure3.2. Wemayobtainaparametrizationofthiscirclebyapplyingarotationxing z andsendingthepoint ;; 0tothepoint n 1 ;n 2 ; 0. 16 PAGE 23 Thisisdonewiththetransformationmatrixappliedtothecircle v = ;cos v ;sin v ,where = ;; 0willbesentto n 1 ;n 2 ; 0: 2 6 6 6 6 4 n 2 n 1 0 )]TJ/F23 11.9552 Tf 9.298 0 Td [(n 1 n 2 0 001 3 7 7 7 7 5 2 6 6 6 6 4 0 cos v sin v 3 7 7 7 7 5 = n 1 u cos v ;n 2 u cos v ;sin v = ~ N u;v ~ N u;v isaparametriccircleofradius centeredattheoriginwhichisobtainedabovebyarotationofthe yz )]TJ/F23 11.9552 Tf 12.161 0 Td [(plane containingthecircle sothatthe radiiofthecircleafterthetransformationlieinthesameplaneas ~n If ~ isthecurvedaxiswithunitnormal ~n ,thenthesurfaceparametrization isobtainedbycenteringeachcircle ~ N u;v atthepoint ~ u ontheaxisforall u 2 D .Itisgivenby: u;v = ~ u + u ~ N u;v .2 Thisconcludestheconstructionofthesurfaceparametrization u;v ofacurved axissurfaceofrevolution. 3.3CurvedAxisSolidsofRevolution Acurvedaxissolidofrevolutionistheinteriorofacurvedaxissurfaceofrevolutionsuchas u;v = ~ u + u ~ N u;v .Itcanbeparametrizedbythree variables: S t;u;v = ~ u + t ~ N u;v u 2 D ; v 2 [0 ; 2 ]; t 2 [0 ; u ] 17 PAGE 24 Thenewvariable t accountsforallradiilessthan u ,andcanbethought tollouttheinteriorofacurvedaxissurfaceofrevolutiontoobtainasolid.It isthoughtofasthesolidsobtainedbyrevolvingaregionaroundacurvedaxis ~ inFigure3.3togetthesolidontheright. Figure3.3:Planeviewofprolecurve = + n whichrevolvesaroundthe curvedaxis togettheshapeontheright 18 PAGE 25 Figure3.4:TorusofRevolution 3.3.1VolumeExample:CircleRevolvedAroundCircle Torus Thisexamplegivesanintegralformulaforthevolumeofatorusofrevolution doughnutshapeintermsofacurvedaxis andaprolecurve asinFigure3.4 above.Thevolumeofthetoruscanbecomputedinoneofmanyways,including standardsinglevariablecalculusmethods.If R themajorradiusisthedistance fromthecenterofthecirclestotheaxis and r minorradiusistheradiusof thetube,thenthevolumeofthetorusisknowntobe r 2 R Thisexampleestablishesatoruspieceapproximationformulawhichisanalagous tothediskmethodforconstantcurvatureaxes constantcurvatureifandonly ifitisacircle.Infact,thevolumewillbeequivalenttothediskmethodformula ofthecylinderaboveFigure3.4whichhasastraightaxisthesamelengthas Beginbytakingthecirclestobethepolarequationsnote:theyareconstant 19 PAGE 26 Figure3.5:TorusPiece functions isthecurvedaxiscircleand istheprolecurve: = R = R + r .3 Fromthevolumeofthetoruswededucethatthevolumeofasegmentsee Figure3.5ofthetorusofangle is: Vol segment = r 2 R .4 Bypartitioningthedomainintoangles d andnoticing r = )]TJ/F23 11.9552 Tf 12.367 0 Td [( ,thevolume ofthetoruscanbethoughtofasthesumoftoruspieceswithvolumeequalto )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 d for 2 ; 2 .Thus: Vol torus = Z 2 0 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 d = r 2 R .5 Notethatthevolumeobtainedisequivalenttothevolumeforthecylinderin 20 PAGE 27 Figure3.6:Somethingunusual.Easytocompute. thegurewiththesamelengthaxisandtheformulacanbethoughtofasthe diskmethodforthisstraightaxiscylinderofrevolution. Thistechniqueusesthevolumeofatorus,buttheapproximationtechnique whichisalwaysexactinthiscasecanbeextendedtorevolvingothercurves aboutacurvedaxiswithconstantcurvatureor,curvedaxisbeingasegment ofcircles.Thoughitisnotconsideredhere,itmaybepossibletoextendthis disk-likeapproximationmethodtoanaxisofrotationwithnon-constantcurvatureprovidedthatthearc-lengthofthecurvecanbeapproximatedlocallywith segmentsofacirclesimilartopolygonalapproximationgivenincalculus. 3.3.2VolumeExample:LineRevolvedAroundCircle Inthisexample,wecomputethevolumeofthesolidobtainedbyrevolvinga verticallineaboutasegmentofacircleintherstquadrant.Thisexampletakes 21 PAGE 28 placeinpolorcoordinatesaswell.For0 4 ,dene: = R = p 2 Rsec .6 Since )]TJ/F23 11.9552 Tf 12.547 0 Td [( isincreasingonthisinterval,atorusapproximationthatusesthe largestpointoneachintervalofthepartitionwillprovideastrictupper-sum estimateofthevolumeandtheonethatusesthesmallestpointwillprovidea strictlower-sumestimate.Asthesizeoftheintervalsofthepartitionapproach 0,thelower-andupper-sumswillconvergetothevolumeintegral: Vol = Z 4 0 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 d .7 Hence,thevolumeofthesolidis: Vol = Z 4 0 R )]TJ 11.955 10.473 Td [(p 2 Rsec 2 Rd .8 = R 3 3 4 )]TJ 11.955 10.473 Td [(p 2 ln + p 2 .9 Furthermore,if = R isacirclesegmentand isapolarcurvesuchthat )]TJ/F23 11.9552 Tf 12.016 0 Td [( for 1 2 ,thevolumeofthecurvedaxissolidofrevolutioncan becomputedby: Vol = Z 2 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [( 2 d .10 22 PAGE 29 Figure3.7:Transformation: S 1 S 2 3.3.3TheoremonCurvedAxisSolidsofRevolution Wewillprovethefollowingtheoremonvolumesofcurvedaxissolidsofrevolution: Theorem4. Considerastraightaxissolidofrevolution S 1 withradialfunction :[ a;b ] R andacurvedaxissolidofrevolution S 2 generatedbythecurve :[ a;b ] R 2 parametrizedbyarclength,radialfunction ,andthenatural map = ~ + y~n + z ^ k betweenthem.Incomponentform, isgivenby: x;y;z = 1 x + yn 1 x ; 2 x + yn 2 x ;z .11 Then seeFigure3.7preservestotalvolume: Vol S 1 = Vol S 2 = Z b a x 2 dx .12 23 PAGE 30 Proof. Webeginthisproofbyndingthejacobianofasfollows: j J x;y;z j = 0 1 x + yn 0 1 x n 1 x 0 0 2 x + yn 0 2 x n 2 x 0 001 .13 = j ~ t x + y ~ n 0 x ~n x j .14 = j ~ t ~n + y ~ n 0 ~n j .15 = j < 0 ; 0 ; 1 > + y ~ n 0 ~n j .16 = j < 0 ; 0 ; 1 > + y )]TJ/F23 11.9552 Tf 9.299 0 Td [( x ~ t + x ~ t ~n ~n j .17 = j < 0 ; 0 ; 1 > )]TJ/F23 11.9552 Tf 9.299 0 Td [(y x < 0 ; 0 ; 1 > j .18 =1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y x .19 Since ~ isparametrizedbyarclength,theFrenet-Serretformulasapplyand ~ 0 = t istheunittangent. ~ t ~n = < 0 ; 0 ; 1 > sincetheyarebothunitlength andorthogonal.Notethattorsion =0because ~ isaplanecurve. x isthe curvatureof ~ Vol S 1 = R ` 0 x 2 dx Thevolumeintegralofacurvedaxissolidofrevolutionmaybecomputed withthechangeofvariablestransformationasfollows: Vol S 2 = ZZZ S 2 dV .20 = ZZZ S 1 j J x;y;z j dV .21 24 PAGE 31 Thus: Vol S 2 = ZZZ S 1 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(yk x dV .22 = Z b a Z x )]TJ/F24 7.9701 Tf 6.587 0 Td [( x Z p x 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(z 2 )]TJ/F26 11.9552 Tf 6.587 8.028 Td [(p x 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(z 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(yk x dydzdx .23 = Z b a Z x )]TJ/F24 7.9701 Tf 6.586 0 Td [( x 2 p x 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(z 2 dzdx .24 = Z b a Z x )]TJ/F24 7.9701 Tf 6.586 0 Td [( x Z p x 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(z 2 )]TJ/F26 11.9552 Tf 6.587 8.027 Td [(p x 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(z 2 dydzdx .25 = ZZZ S 1 dV .26 = Vol S 1 .27 = Z b a x 2 dx .28 Thus,totalvolumeispreserved.Itcanbeseenthatlocalvolumeisnotpreserved byexaminingthejacobianfactor1 )]TJ/F23 11.9552 Tf 11.971 0 Td [(yk x .Thesimplefactthatthisisnever1 showsthatthisisnotanorthogonaltransformation,andthereforenotaeuclidean rigidmostion,sotheregionsinFigure3.7arenotcongruent.Signedcurvatureis positiveinonedirectionandnegativeintheother.Itcanbeseenthat,aspositivecurvaturedecreasesthejacobianfactorincreases,andasnegativecurvature increasesthejacobianfactordecreases. 25 PAGE 32 Chapter4 HyperbolicSpace Thegeometryofhyperbolicspace,likeeuclideanspace,iseasilyunderstoodas anextensionofitsrespectiveplanegeometry.Forabasicintroductionsee[1], [4],[5].See[4],[6]forhyperbolicspace H 3 4.1TheHyperbolicPlane H 2 istheupperhalf-planemodelofthehyperbolicplane. H 2 = f x;y 2 R 2 : y> 0 g .1 = f 2 C : Im > 0 g .2 Therstrepresentationistheusualcoordinateplanerestrictedto y> 0,and thesecondistheusualcomplexplanerestrictedto withpositiveimaginary componentcorrespondingtothecoecientof i = p )]TJ/F18 11.9552 Tf 9.298 0 Td [(1.If = a + bi isa pointin H 2 then b = Im 2 R + .Thetworepresentationsof H 2 willbeused 26 PAGE 33 interchangeably.Thehyperboliclengthofapiecewisesmoothcurvein H 2 canbe computedwiththehyperbolic elementofarclength givenby: ds H = ds E y .3 Example3. Theelementofarclengthcanbeusedtocalculatethehyperbolic lengthofeuclideanlines f :[ t o ;t 1 ] H 2 suchthat f t = x t ;y t = t;mt + b in H 2 : length H f = Z f t 1 f t o ds E y .4 = Z t 1 t o p x 0 t 2 + y 0 2 t dt y t .5 Case1. Suppose m;b 6 =0. length H f = Z t 1 t o p 1+ m 2 dt mt + b .6 = log mt 1 + b mt o + b r 1+ m 2 m 2 .7 Case2. Suppose m =0 ;b 6 =0. f t = t;b length H f = Z t 1 t o 1 b dt .8 = t 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(t o 1 b .9 Thus,if m =0, length H f = 1 b R t t o dt = 1 b length E f .Thisshowsthat everyhorizontalEuclideanlineinthehyperbolicplaneiseitheradilation orcontractionofeuclidean1-space E = R withmetric d H x 1 ;b ; x 2 ;b = 27 PAGE 34 1 b d E x 1 ;b ; x 2 ;b = 1 b j x 2 )]TJ/F23 11.9552 Tf 11.964 0 Td [(x 1 j .Thisimportantpropertywillsimplifyhyperbolicareaandvolumecomputationaswellsinceitisaxablesubset whichbehaveslikeitseuclideananaloguptoconstantscalingunderthe hyperbolicmetric. Case3. If b =0,then f t = t;mt and: length H f = Z t t o p 1+ m 2 mt dt = r 1+ m 2 m 2 ln t t o .10 Case4. If f t = c;mt averticalline,andthereforehyperbolicline,then length H f = Z t t o 1 t dt = ln t t o .11 f isa hyperbolicline suchthat length H f isminimalgeodesicoverthe setofallpathsconnecting t o to t 1 inthehyperbolicplane.Aninteresting pointisthatthislengthcanberecognizedasthelengthin Case3 asthe slopeofthelineapproachesinnity: m !1 Summary. Thehyperbolicplaneisthemetricspace H 2 ;d H ,where H 2 = f x;y 2 R 2 : y> 0 g = f 2 C : Im > 0 g and d H p;q = inf f length H f : f t o = p ; f t 1 = q g .Explicitcalculationofhyperbolicdistanceinthehyperbolicplaneisdonewiththeelementofarclength ds H = ds E y 28 PAGE 35 Figure4.1:Hyperboliclinesintheupperhalf-planemodel H 2 4.2HyperbolicLines,Geodesics,Isometries H 2 Amongallcurvesjoining p to q inthehyperbolicplane,thecirclearccentered onthe x )]TJ/F23 11.9552 Tf 11.42 0 Td [(axis possiblyaverticallinesegmentistheuniquedistancerealizing path. A geodesic isacurve g suchthatforevery q 2 g sucientlycloseto p ,the sectionof g joining p to q istheshortestcurvejoining p to q Thegeodesicsin H 2 ,therefore,arethehalfcirclespossiblyinniteinradius centeredonthe x )]TJ/F23 11.9552 Tf 9.325 0 Td [(axis .Figure4.1showsseveralexamplesof hyperbolicgeodesics alsoknownas hyperboliclines An isometry equal"+distance"ofametricspace X;d isamap m : X X suchthatthedistancebetweenanytwopoints p;q 2 X ispreserved undertransformationby m : d m p ;m q = d p;q .12 Theorientationpreservingisometriesofthehyperbolicplane H 2 arethe 29 PAGE 36 M o biustransformations ,denedas PSL 2 R ,andaregivenbythetransformations m = a + b c + d : a;b;c;d 2 R ; ad )]TJ/F23 11.9552 Tf 11.955 0 Td [(bc> 0.13 Thehyperbolicelementofareaisgivenby dA H = dA E y 2 .14 Intuitively,itisthearclengthelementappliedoversetsinboththe x and y directionsintheplane. 4.3HyperbolicSpace H 3 istheupperhalf-spacemodelofhyperbolicspace: H 3 = f x;y;z : z> 0 g R 3 g .15 Thehyperbolicstructureontheupperhalf-spacecanbeconstructedfromthe structureofthehyperbolicplane.Specically,ifthe y )]TJ/F23 11.9552 Tf 12.07 0 Td [(axis in H 2 isidentied withthe z )]TJ/F23 11.9552 Tf 12.068 0 Td [(axis in H 3 ,thenthehyperbolicstructureof H 3 canbeunderstood asallrotationsofthehyperbolicplanewhichxthe z )]TJ/F23 11.9552 Tf 12.047 0 Td [(axis theaxiswhichis aectedbyspacecurvature. Consequently,thehyperbolicdistanceinhorizontalplanesthoseparallelto the xy )]TJ/F23 11.9552 Tf 11.348 0 Td [(plane andperpendiculartotheaxisaectedbyspacecurvatureisseen tobescaledeuclideandistancebasedontheheightoftheplane.However,a 30 PAGE 37 rotationofgeodesicsshowsthatthetruehalfplanesof H 3 arethetophalvesof spherescenteredinthe xy )]TJ/F23 11.9552 Tf 11.955 0 Td [(plane Iftheeuclideanvolumeelementisdenoted dV E thenthevolumeelementin H 3 ,denoted dV H ,canberewritten: dV H = 1 z 3 dV E .16 Thus,wemaycalculatevolumeinthisspacewiththefollowingformula: Vol H D = ZZZ D 2 H 3 dV H = ZZZ D 2 H 3 1 z 3 dV E .17 31 PAGE 38 Chapter5 HyperbolicRevolutionsin H 3 5.1EquidistantRegionsin H 2 Figure5.1:EquidistantRegion Thelocusofpointsdistance R h R h meaninghyperbolicradius.Thisnotation willbeusedforthehyperbolicradiusofahyperboliccircleaswellastheequidis32 PAGE 39 tantregionhere.fromaverticalhyperboliclinesegmentformaboundarywhich is not ageodesicof H 2 i.e.,theboundaryisnotcontainedinahyperbolicline. Theboundarywillbeshowntotaketheformofaeuclideanstraightlinein H 2 Twosuchcurvesoneoneachsideoftheaxisandtwohyperboliclinesintersectingtheaxisatarightangle,thereforeattheirmaximumsenclosearegion SeeFigure5.1whichwillbereferredtoasaequidistantregionbecauseitcan beconstructedbyextendingaxeddistancealongtheshortestpathsgeodesics perpendiculartoahyperboliclinesegmentheretheaxisistakentobevertical sinceitisalsoaeuclideanline,butcanbemappedisometricallytoanyhyperbolic lineaxis. Theorem5. Non-GeodesicBoundaryTheboundaryedgesofanequidistant tubeareeuclideanstraight,asinFigure5.1. Proof. Thisprooftakesplaceinthehyperbolicplane H 2 withcoordinatesidentiedwith C m = t isanisometrywhichmapstheunitcircletothecircleofradius t centeredattheorigin.Wetake t tobetheradiusofthecirclewhichcontainsthe upperboundaryoftheequidistantregion. Let d bethedistancefromapointontheaxis 1 = i toapoint 2 = e i lying onthegeodesicwhichintersectstheimaginaryaxisatarightangleatthepoint i ,andthuscontains 1 .Then m 2 = ti and m 1 = te i Since m isanisometry: d H i;e i = d = d H ti;te i .1 Thisshowsthatbothpointsdistance d fromtheaxislieontheuniquelinemaking 33 PAGE 40 angle withtherealaxisequivalently,the x )]TJ/F23 11.9552 Tf 11.268 0 Td [(axis .Since m isanisometryfor anyrealnumber t> 0,allpointsdistance d fromtheimaginaryaxislieonthe sameeuclideanlinein H 2 Theorem6. Angle-RadiusTrigonometryIf R h isthedistancefrom 1 = i to 2 = e i where 0 << alongthehyperboliclinecontainingboth 1 and 2 i.e.,theradiusoftheequidistanttube,then cos = tanh R h sin = sech R h tan = csch R h .2 Proof. If R h denotesthedistancealongthethegeodesicwithapex i tothepoint e i ,wemayndtheboundaryofanequidistanttubebycalculating R h .We dothisbyconstructinganisometrywhichxes i andtakesthegeodesictothe imaginaryaxis-.Thedistance R h isthenthedistancefrom i totheimageof e i whichmaybecomputedexplicitly.Weconstruct m accordingly: m = +1 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( .3 Noticethat m xes i : m i = i +1 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(i = i +1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(i + i .4 34 PAGE 41 Thus, R h canbecomputedasfollows: R h = ln m e i i .5 Compute: m e i = 1+ e i 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(e i = + e i )]TJ/F23 11.9552 Tf 11.955 0 Td [(e )]TJ/F24 7.9701 Tf 6.586 0 Td [(i )]TJ/F23 11.9552 Tf 11.955 0 Td [(e i )]TJ/F23 11.9552 Tf 11.955 0 Td [(e )]TJ/F24 7.9701 Tf 6.586 0 Td [(i = 1 )]TJ/F18 11.9552 Tf 11.955 0 Td [(1+ e i )]TJ/F23 11.9552 Tf 11.956 0 Td [(e i 1+1 )]TJ/F18 11.9552 Tf 11.955 0 Td [( e i + e )]TJ/F24 7.9701 Tf 6.586 0 Td [(i .6 Notethat: e i )]TJ/F23 11.9552 Tf 11.955 0 Td [(e i =2 isin .7 e i + e )]TJ/F24 7.9701 Tf 6.586 0 Td [(i =2 cos .8 Therefore, m e i = 2 isin 2 )]TJ/F18 11.9552 Tf 11.955 0 Td [(2 cos = isin 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(cos = i p 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(cos 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(cos .9 Wecomputetheradius: R h = ln s 1+ cos 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(cos .10 Exponentiate,namevariable S : S = e R h = s 1+ cos 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(cos .11 35 PAGE 42 Solvefor cos cos = S 2 )]TJ/F18 11.9552 Tf 11.955 0 Td [(1 S 2 +1 = S )]TJ/F23 11.9552 Tf 11.955 0 Td [(S )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 S + S )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = e R h )]TJ/F23 11.9552 Tf 11.956 0 Td [(e )]TJ/F24 7.9701 Tf 6.587 0 Td [(R h e R h + e )]TJ/F24 7.9701 Tf 6.587 0 Td [(R h .12 Concludebynoting cos = tanh R h .13 Allnecessaryformulascanbecomputedbycompletingthetrianglesfor and 0 = 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [( 5.2VolumeofanEquidistantTube AnequidistanttubeisobtainedbyrevolvingtheregioninFigure5.1aboutthe z )]TJ/F23 11.9552 Tf 12.211 0 Td [(axis in H 3 .ThesolidappearstobeaEuclideanconewiththeatcircular facesreplacedwithsphericalfaces.Thefacesexpandinnitelyinonedirection andshrinktozerointheother,althoughthesurfaceareawillremainxed. Thefollowingisacalculationofthevolumeoftheequidistanttube T : Vol H T = ZZZ T dV H = ZZZ T 1 z 3 dV E .14 Let 0 betheanglebetweenthegeodesicandtheboundary, 0 = 2 )]TJ/F23 11.9552 Tf 12.913 0 Td [( where istheacuteangletheboundaryofanequidistantregionmakeswiththe x )]TJ/F23 11.9552 Tf 12.257 0 Td [(axis in H 2 mentionedinthelastsection.Weapplyasphericalchangeof 36 PAGE 43 coordinates: x = y = z = sin cos v sin sin v cos .15 2 [1 ;t ]; v 2 [0 ; 2 ]; 2 [0 ; 0 ].16 RecallthattheJacobianofthischangeofvariablesis 2 sin .Thus: ZZZ T 1 z 3 dV E = Z t 1 Z 0 0 Z 2 0 1 cos 3 2 sinddd .17 = Z t 1 Z 0 0 Z 2 0 1 tan sec 2 ddd .18 = ln t tan 2 o = Vol H T .19 Theonlynon-trivialintegrationaboveisa u = sec substitutionforthe d Bythetrig-relationships: cos = cos 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 0 = sin 0 = tanh R h .20 andbycompletingthetriangle: tan 2 o = sinh 2 R h .21 Inconclusion: Vol H Equid:Tube = ln t sinh 2 R h .22 37 PAGE 44 Theareaofadiskwithhyperbolicradius R h inanyplaneperpendiculartothe imaginaryaxisintheupperhalfspacewillhavearea A =4 sinh 2 R h 2 .This showsthatCaveleri'sPrinciplefails,sincewewouldexpectthevolumeofthe equidistanttubetobe4 ln t sinh 2 R h 2 .Infact,theexactdierencebetween theexpectedvolumeandtheactualvolumeisreadilyseenwhenwerewritethe volumeformulaasfollows: 4 ln t sinh 2 R h 2 cosh 2 R h 2 : .23 5.3HorizontalAxisExamplesHyperbolicCurve Ahyperbolicdiskistheinteriorofahyperboliccircleofagivenhyperboliccenter C h andhyperbolicradius R h seeFigure5.2.Formally,a hyperboliccircle isthe Figure5.2:HyperbolicDiskcenteredwithhyperboliccenter C h = i 38 PAGE 45 set f 2 H 2 : d H ;C h = R h g anda hyperbolicdisk isdescribedbythesameset except="isreplacedby ". Wewillfocusoncirclescenteredat x;y = ; 1inthehyperbolicplaneor =0+1 i = i in C .Generalizingtoothercenterscanbedonebyisometric mappings. Theorem7. Let S beahyperboliccirclecenteredat i withhyperbolicradius R h seeFigure5.2. S isaeuclideancirclecenteredat C e = cosh R h with euclideanradius R e = sinh R h Perimeter S =2 sinh R h and Area S = R R h 0 2 sinh r dr =4 sinh 2 R h 2 Proof. See[1] 5.4VolumeofHorizontalCylinder Figure5.3:Horizontalcylinder,hyperboliccurvedaxis. 39 PAGE 46 Cyl = f x;y;z 2 H 3 : y 2 + z )]TJ/F23 11.9552 Tf 11.955 0 Td [(c 2 = r 2 ; )]TJ/F23 11.9552 Tf 10.964 8.087 Td [(` 2 x ` 2 g .24 Where c;r refertotheeuclideancenterandradius,respectively. Vol H Cyl: = R ` 2 )]TJ/F25 5.9776 Tf 8.059 3.258 Td [(` 2 R r )]TJ/F24 7.9701 Tf 6.587 0 Td [(r R p r 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(y 2 + c )]TJ/F26 11.9552 Tf 6.586 8.248 Td [(p r 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(y 2 + c dV z 3 .25 = RR )]TJ/F21 7.9701 Tf 37.68 4.707 Td [(1 2 p r 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(y 2 + c 2 + 1 2 )]TJ/F26 11.9552 Tf 6.586 8.249 Td [(p r 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(y 2 + c 2 dydx .26 = RR 2 c p r 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(y 2 c 2 )]TJ/F26 11.9552 Tf 6.586 8.249 Td [(p r 2 )]TJ/F24 7.9701 Tf 6.586 0 Td [(y 2 2 dydx .27 Substitute y = rsin suchthat dy = rcos d and )]TJ/F24 7.9701 Tf 10.494 4.707 Td [( 2 y 2 : = ZZ )]TJ/F23 11.9552 Tf 29.604 8.088 Td [(r 2 cos 2 c 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(r 2 cos 2 2 d .28 Nowsubstitute = tan )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 t d = dt 1+ t 2 and cos 2 = 1 1+ t 2 .Thus, PAGE 47 Substitute t = q c 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(r 2 c 2 tan dt = q c 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(r 2 c 2 sec 2 d .Thus, )]TJ/F24 7.9701 Tf 10.494 4.708 Td [( 2 2 and: = ZZ 2 cr 2 q c 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(r 2 c 2 sec 2 c 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(r 2 tan 2 +1 2 ddx .31 = ZZ r 2 c 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r 2 3 = 2 cos 2 d .32 = ZZ r 2 c 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r 2 3 = 2 + 1 2 sin ddx .33 = Z r 2 c 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(r 2 3 = 2 .34 Wendthat r = ctanh R h = Z c 2 tanh 2 R h c 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(tanh 2 R h 3 = 2 .35 = sinh 2 R h cosh R h ` .36 5.5VolumeofHorizontalTorus Thenalconsiderationisatoruswhich,likethecylinderabove,liesbetweentwo horizontalplanesin H 3 .Again,assumethehyperboliccenterofeverycircleon thesurfaceisatheight z =1andlet R h bethehyperbolicradiusofeach. Thevolumeofthehorizontaltorusmaybecomputedwithatransformation likethatoftheeuclideancurvedaxissolidsofrevolution. Let x;y;z = R + y cos x R + y ; R + y sin x R + y ;z beachangeofvariablestakingthecylinder Cyl = f x;y;z 2 H 3 : y 2 + z )]TJ/F23 11.9552 Tf 13.288 0 Td [(c 2 = r 2 ; x 2 [ )]TJ/F23 11.9552 Tf 9.298 0 Td [(R;R ] g tothetorus Tor = f x;y;z : R )]TJ/F29 11.9552 Tf 11.78 10.365 Td [(p x 2 + y 2 2 + z )]TJ/F23 11.9552 Tf 11.78 0 Td [(c 2 = r 2 ; x 2 41 PAGE 48 [ )]TJ/F23 11.9552 Tf 9.298 0 Td [(R;R ] g .Wemaycomputethehyperbolicvolumeasfollows: Vol H Tor = ZZZ Tor 1 z 3 dV .37 = ZZZ C j J x;y;z j z 3 dV .38 = Z R )]TJ/F24 7.9701 Tf 6.587 0 Td [(R Z r )]TJ/F24 7.9701 Tf 6.587 0 Td [(r 1 z 3 Z + y z;x )]TJ/F24 7.9701 Tf 6.587 0 Td [(y z;x j J x;y j dydzdx .39 = Z R )]TJ/F24 7.9701 Tf 6.586 0 Td [(R Z r )]TJ/F24 7.9701 Tf 6.587 0 Td [(r 2 y z;x z 3 dydzdx .40 = Z R )]TJ/F24 7.9701 Tf 6.586 0 Td [(R Z r )]TJ/F24 7.9701 Tf 6.587 0 Td [(r Z + y z;x )]TJ/F24 7.9701 Tf 6.586 0 Td [(y z;x 1 z 3 dydzdx .41 Thestepstakenthusfarareanalogoustothestepstakeninthevolume equivalenceofeuclideancurvedaxissolids,sincethisdieomorphismsatisesthe previouslylaidoutconditionsoncethevolumeelementissetaside.Nowitcan beseenthatthistorusandcylinderhavethesamevolume: Vol H Tor = Z R )]TJ/F24 7.9701 Tf 6.587 0 Td [(R Z r )]TJ/F24 7.9701 Tf 6.587 0 Td [(r Z + y z;x )]TJ/F24 7.9701 Tf 6.586 0 Td [(y z;x 1 z 3 dydzdx .42 = ZZZ Cyl 1 z 3 dV .43 = Vol H Cyl .44 Finally,borrowingfromthecylindercomputation: Vol Cyl = sinh 2 R h cosh R h R h .45 42 PAGE 49 Chapter6 Conclusions Wehaveconstructedageneralparametrizationsforcurvedaxissurfacesofrevolutiongivenanaxisandaradialfunction.Theparametrizationissimpleenough thatitseemslikelythattheelementaryresultsfromthedierentialgeometry ofsurfacessuchastherstandsecondfundamentalformscouldbechecked easily.Wealsoobtainedavolumeparametrizationandatheoremonthevolumepreservationofthenaturalmapbetweenstraightandcurvedaxissolidsof revolution. Inhyperbolicspace,itbecameclearthatstraightforwardgeneralizationisnot possible.Afterobtainingaformulaforthehorizontalcylinder,wesawthatit couldbeusedtocomputethevolumeofahorizontaltoruswhichhasaxisofthe samelengthasthecylinder.Theexamplegavehopeforpartialgeneralizationin horizontalplanes,describedbythefollowingconjecture: Allcurvesofagivenlengthinthesamehorizontalplanewithaconstant radialfunction = R h suchthatthecurvatureoftheaxisneverexceeds 1 R h havehyperbolicvolumescomputablebytheformulaforthecylinderwhoseaxis isthesamelengthasthecurvedaxis. 43 PAGE 50 Chapter7 References [1]Anderson,JamesW., HyperbolicGeometry ,SpringerUndergraduateMathematicsSeries,1999 [2]AndrewPressley, ElementaryDierentialGeometry ,SpringerUndergraduate MathematicsSeries,2000 [3]Apostol,TomM., CalculusVolumeI:One-VariableCalculus,withanIntroductiontoLinearAlgebra ,JohnWiley&Sons,Inc. [4]Bonahan,Francis, Low-dimensionalgeometry:fromeuclideansufracestohyperbolicknots ,StudentMathematicalLibrary;v.49.IAS/ParkCitymathematicalsubseries,2009 [5]Coxeter, IntroductiontoGeometry ,JohnWiley&Sons,Inc.,ISBN0-47150458-0 [6]Beardon,AlanF., TheGeometryofDiscreteGroups ,SpringerUndergraduate MathematicsSeries,1983 44 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||