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CURVED AXIS REVOLUTIONS

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004787/00001

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Title: CURVED AXIS REVOLUTIONS
Physical Description: Book
Language: English
Creator: Huckaby, Todd
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

Subjects

Subjects / Keywords: Curved Axis Revolutions
Differential Geometry
Volumes of Rotation
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Motivated by standard solids of revolution computable by elementary calculus methods this thesis develops a construction of solids of revolution with a curved axis. We show that the volumes obtained do not depend on the curvature of the axis. In H3, we show by construction there is no straight-forward generalization. A partial generalization is conjectured to exist for curves in a horizontal plane because the natural map between the horizontal cylinder to the horizontal torus is seen to preserve volume in an example.
Statement of Responsibility: by Todd Huckaby
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Mullins, David

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Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2013 H8
System ID: NCFE004787:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004787/00001

Material Information

Title: CURVED AXIS REVOLUTIONS
Physical Description: Book
Language: English
Creator: Huckaby, Todd
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

Subjects

Subjects / Keywords: Curved Axis Revolutions
Differential Geometry
Volumes of Rotation
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Motivated by standard solids of revolution computable by elementary calculus methods this thesis develops a construction of solids of revolution with a curved axis. We show that the volumes obtained do not depend on the curvature of the axis. In H3, we show by construction there is no straight-forward generalization. A partial generalization is conjectured to exist for curves in a horizontal plane because the natural map between the horizontal cylinder to the horizontal torus is seen to preserve volume in an example.
Statement of Responsibility: by Todd Huckaby
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Mullins, David

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2013 H8
System ID: NCFE004787:00001


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CURVEDAXISREVOLUTIONS by TODDHUCKABY AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegreeof BachelorofArtsinMathematics UndertheSponsorshipof ProfessorDavidMullins,Ph.D. Sarasota,Florida May2013

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CURVEDAXISREVOLUTIONS ToddHuckaby NewCollegeofFlorida,2013 Abstract Motivatedbystandardsolidsofrevolutioncomputablebyelementary calculusmethodsthisthesisdevelopsaconstructionofsolidsofrevolutionwithacurvedaxis.Weshowthatthevolumesobtaineddonot dependonthecurvatureoftheaxis.In H 3 ,weshowbyconstruction thereisnostraight-forwardgeneralization.Apartialgeneralization isconjecturedtoexistforcurvesinahorizontalplanebecausethe naturalmapbetweenthehorizontalcylindertothehorizontaltorus isseentopreservevolumeinanexample. DavidMullins,Ph.D. DivisionofNaturalSciences

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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

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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

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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

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LISTOFFIGURES 5.3Horizontalcylinder,hyperboliccurvedaxis..............39 v

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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

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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

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uralmapbetweenthecylinderandtorusofthesameaxislengthinthesame horizontalplane z constant. 3

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Chapter2 SurfacesandSolidsofRevolution 2.1SurfacesofRevolution ThefollowingconstructionofsurfacesofrevolutionfollowsthenotationinPressley[1]. Figure2.1:SurfaceofRevolution.FigureFrom[1] 4

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Chapter3 CurvedAxisRevolutions 3.1Introduction Figure3.1:Planeviewofprolecurve = + n whichrevolvesaroundthe curvedaxis Incontrastwiththeusualsurfacesandsolidsobtainedbystraightaxisrevolutionspreviouschapter,thesurfacesandsolidspresentedinthischapterare 14

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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

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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

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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

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Thenewvariable t accountsforallradiilessthan u ,andcanbethought tollouttheinteriorofacurvedaxissurfaceofrevolutiontoobtainasolid.It isthoughtofasthesolidsobtainedbyrevolvingaregionaroundacurvedaxis ~ inFigure3.3togetthesolidontheright. Figure3.3:Planeviewofprolecurve = + n whichrevolvesaroundthe curvedaxis togettheshapeontheright 18

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Chapter5 HyperbolicRevolutionsin H 3 5.1EquidistantRegionsin H 2 Figure5.1:EquidistantRegion Thelocusofpointsdistance R h R h meaninghyperbolicradius.Thisnotation willbeusedforthehyperbolicradiusofahyperboliccircleaswellastheequidis32

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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

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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

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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

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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

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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

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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

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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

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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,
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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

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[ )]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

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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

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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


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