AnalyticProjectiveGeometry
JohnBamberg
UniversityofWesternAustralia
TimPenttila
UniversityofAdelaide
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PartIIIHigherDimensions
Preface
Projectivegeometryisthegeometryofvision.YetArthurCayleysawthatit is all geometry.ThemathematicalhistorianMorrisKlinecalleditthe‘science bornofart’,andtheveryearlyhistoryofitsdevelopmentfromthatorigin isdocumentedinthebook TheGeometryofanArt bythelatermathematicalhistorianKirstiAndersen.Someofthosedevelopments(andsomelater ones)appearinChapters 11 and 12,andwhatCayleymeantisexplained inChapters 7 through 10.FelixKleinalsoadvocatedforthecentralityof projectivegeometry,butisbetterknownforbringingoutthecentralroleof symmetryingeometryinhis ErlangenProgramme.Ourtreatmentofmost ofthetopicsinthisbookemphasisesthiscentralroleofsymmetry,through theprominentplaceweassigngroups,andwealsoexplainKlein’sviewon transformationgeometryinChapter 9.Moreover,thiswholesubjecthasa closerelationshipwithlinearalgebra,andthisunderpinsourtreatment.What JürgenRichter-Gebertcalls‘thebeautythatliesintherichinterplayofgeometricstructuresandtheiralgebraiccounterparts’isarecurringthemeofour book.Finally,wetrytoillustratesomeoftheadvantagesgainedbytakinga projectiveapproachtogeometryinChapter 10,whereweobtainconnections betweenseeminglydistant(andancient)resultsinEuclideangeometryviathe perspectivehard-wonintheearlierchapters.
Thisbookisanintroductiontoprojectivegeometry,andourcoordinatesare mostlyovertherealnumbers.However,thereisadvancedandnovelmaterial forthepractician.Chapter 6 examinesoneoftheleitmotifsofthisbook–involutions andtheirroleinprojectivegeometry.Thisistakenmuchfurther inSection 10.2,wherewebeginwithanoldresultofPappus,andexplorethe moremoderntheoremsofFerrers,Jeˇrábek,Lehmer&Daus,Gardner&Gale, Robson&Strange,andtheirastoundingsynergy.
Wefindthatanapproachthatteachesthesubjectconceptuallywhilealso sketchingitsdevelopmentresonateswithusasteachersandauthors,andalso hopethatitwillfindsympatheticvibrationsinstudentsandreaders.
FundamentalAspectsoftheRealProjective Plane
WhereasEuclideangeometrydescribesobjectsastheyare,projective geometrydescribesobjectsastheyappear.
KristenR. Schreck (2016,p.159)
Three-dimensionalEuclideanspace, R3,isperhapsthemostfamiliarandnaturalgeometrytothelayperson.Inthisintroductorysection,wewillshow howwecanbuilda‘newworldoutofnothing’(touseJánosBolyai’sasseveration)fromtheinterplaybetweenperpendicularityandparallelism,oflinesand planestogether.Thisinterplayleadstothe realprojectiveplane and duality.
1.1Parallelism
Inthree-dimensionalspace,‘parallelism’appliestotwodifferenttypesof object–tolinesandtoplanes.Alinecanbeparalleltoaplane,aplanecanbe paralleltoanotherplane,andalinecanbeparalleltoaline(seeFigure 1.1).In particular,aline isparalleltoaline iftheyare,firstly,lyinginacommon plane(i.e., coplanar)and,secondly,non-intersecting.Twonon-intersecting linesthatarenotcoplanarare skew.Twoplanesareparalleliftheyarenonintersecting.Aline andaplane π areparalleliftheyarenon-intersectingor lieswithin π
1.2Perpendicularity
Similarly,‘perpendicularity’isarelationthatworksforlinesandplanesalike. Wereallyonlyneedtounderstandperpendicularityoflinestounderstandwhat happenswhenweintroduceplanes.Twolinesareperpendiculariftheyare coplanarandperpendicularinthecommonplane.Aline isperpendicularto
Figure1.1Parallellinesandplanes.
Figure1.2Interrelationshipofperpendicularityandparallelism.
aplane π if isperpendiculartoeverylineof π thatitintersects.Twoplanes π and π areperpendiculariftheymeetinaline ,and π isperpendiculartosome lineof π . Wewillseesoonthatthemostilluminatingpropertyof R3 istheinterrelationshipofperpendicularityandparallelismarisingfromthefollowing property:
Parallel–Perpendicular Property:Let m, m belinesandlet π,π be planes.If m m , π π ,and π ⊥ m,then π ⊥ m (seeFigure 1.2).
Wecannowelevatetothenextlevelofabstraction.Foreachline ,the parallelclass [ ]of isthesetofalllinesparallelto (including itself). Similarly,write[π]fortheplanesparallelto π.Thefirstobservationwemake isthefollowing:
Property1.1
Let and π bealineandaplane(respectively).Theneither
• noelementof [ ] isparalleltoanyelementof [π],or
• everyelementof [ ] isparalleltoeveryelementof [π],andwesaythat [ ] is parallelto [π].
Soitmakessensetowrite[ ] [π].Thisrelationofparallelismbetween parallelclassesoflinesandplanesissymmetric,andcouldabstractlybean incidencerelation betweentwodifferenttypesofobjects.Withthisinmind, wemakeasecondobservation:
Property1.2
• Foranytwodifferentparallelclassesoflines [ ] and [ ],thereisaunique parallelclassofplanesthatisparalleltoboth [ ] and [ ].
• Foranytwodifferentparallelclassesofplanes [π] and [π ],thereisaunique parallelclassoflinesthatisparalleltoboth [π] and [π ]
Wecanmanufactureageometry G outoftheseparallelclassesoflinesand planes.Thegeometrywecreatewillbe planar inthesensethatwehaveonly twotypesofobject,whichwemightaswelltemporarilycall1 pistettä (singular: piste)and linjat (singular: linja).Thisnewgeometrywillconsistonlyof objectsandanincidencerelationbetweenthem—nodistance,nomidpoints, noangles,noparallelism.
PistettäParallelclassesoflinesof R3
LinjatParallelclassesofplanesof R3
IncidenceAparallelclassoflinesis‘incident’withaparallelclass ofplanesifandonlyiftheyareparallel.
Sofromwhatwehavediscussedabove,theincidencerelationhereissymmetric:twopistettälieonauniquelinjaandtwolinjathaveauniquepistein
1 Forsomereaders,theuseofthewords‘points’and‘line’willinterferewiththeirunderstanding, sotomakeitclearthatwearedefiningnewpointsandnewlines,wetemporarilyadoptanother languageforthesenewpointsandnewlines.
common.Therefore,therecannotbe‘parallel’linjatinthisgeometry G;two linjatarealwaysconcurrent.Thisgeometryisanexampleofanon-Euclidean geometry–a projectiveplane
1.3Duality
Let L bethesetofparallelclassesoflinesandlet Π bethesetofparallel classesofplanes.Thereisanaturalcorrespondencebetween L and Π:if[ ] isaparallelclassoflines,thenwetaketheset ⊥ ofallplanesthatareperpendicularto .Bytheparallel–perpendicularproperty,thissetofplanesisa parallelclassofplanesanddidnotdependontherepresentativewetookfrom .Conversely,if[π]isaparallelclassofplanes,wemaptotheset π⊥ ofall linesthatareperpendicularto π.Thuswehavethefollowingcorrespondence:
Notethatifweapply ⊥ twice,thenourobjectsareleftinvariant.Forexample, ifwetakealloftheplanesperpendiculartoaline ,andthentakeallofthelines perpendiculartothoseplanes,itwillresultintheparallelclassof .Moreover, thiscorrespondencerespectsparallelismbetweenelementsof L and Π:
]
Finally,let’sseewhattheperpendicularitycorrespondence ⊥ doesto G.We sawabovethatitpreservesincidence.Soif P isapisteof G and m isalinja, then P⊥ isincidentwith m⊥ ifandonlyif P isincidentwith m.Wealsosaw that ⊥ mapsapistetoalinja,andthenalinjatoapiste,insuchawaythatif performedtwice,itlefttheobjectsinvariant.Suchamapiscalleda polarity Thispolarityalsohasthepropertythatnopiste P isincidentwithitsimage P⊥;butwewillreturntothislateroncewehaveinvestigatedprojectiveplanes morethoroughly.
1.4TwoModelsoftheRealProjectivePlane
Wehavealreadyseenthatparallelclassesoflinesandplanesof R3 forma projectiveplane–anincidencegeometryofpointsandlinessuchthatanypair ofpointsdetermineauniqueline,andtwodistinctlinesalwaysmeetinapoint. Eachparallelclassoflineshasarepresentativepassingthroughtheorigin O of R3.Sowecanreplaceeachparallelclassbyaone-dimensionalsubspace
Table1.1 Therealprojectiveplane.
Points1-dimensionalsubspacesof R3
Lines2-dimensionalsubspacesof R3 Incidenceinclusion
of R3.Likewise,theparallelclassesofplanescanbesimulatedbytakingtwodimensionalsubspacesof R3.Formally,the realprojectiveplane PG(2, R)is theincidencestructuredefinedinTable 1.1.
Theorem1.3
(i) Anytwopointsof PG(2, R) areincidentwithauniqueline. (ii) Anytwolines PG(2, R) areincidentwithauniquepoint.
Proof Theprooffollowsfromelementarylinearalgebra.Inparticular,for(i), notethatanytwo1-dimensionalsubspacesof R3 spanaunique2-dimensional subspace.For(ii),weobservethatanytwo2-dimensionalsubspacesof R3 meetinaunique1-dimensionalsubspace.
Wedenoteapointof PG(2, R)byhomogeneouscoordinates(x, y, z), x, y, z ∈ R,notallzero.Thismeansthatwearesimplydroppingtheangledbrackets fromthesubspace (x, y, z) of R3;since
(x, y, z) = (cx, cy, cz) , forall c ∈ R\{0},
wehave(x, y, z) = (cx, cy, cz).Thisiswhatwemeanbysayingthatthecoordinatesare homogeneous,anditwillbeclearfromthecontextthatweare describingapointof PG(2, R)andnotavectorof R3.Notethat(0, 0, 0)is notthehomogeneouscoordinatesofanypointof PG(2, R).
Wedenotealinewithequation ax + by + cz = 0byhomogeneousline coordinates[a, b, c], a, b, c ∈ R,notallzero.(Again,notethat k(ax+by+cz) = 0 for k ∈ R with k 0yieldsthesameline,sothesecoordinatesareindeed homogeneous.)Apoint(x, y, z)isincidentwiththeline[a, b, c]ifandonlyif ax + by + cz = 0.Notethat[0, 0, 0]isnotthehomogeneouslinecoordinatesof anylineof PG(2, R).
AnotherwaytodefinetherealprojectiveplaneistotaketherealEuclidean planeandenlargeitalittlebit.Eachlineisequippedwithanadditionalpoint–its pointatinfinity –whichissimplytheparallelclassofthatline.Thisensures thattwoparallellinesnowbecometwointersectinglinesinthelargergeometry.Oneextralineisintroduced,anditissimplythesetofallpointsatinfinity –the lineatinfinity.Formally,the extendedEuclideanplane istheincidence structuredefinedinTable 1.2.
Table1.2 TheextendedEuclideanplane.
Pointspointsof R2
parallelclassesoflines(pointsatinfinity)
Lineslinesof R2 thelineatinfinity
Incidenceinheritedfrom R2;apointatinfinityisincidentwitheveryline ofthecorrespondingparallelclassandwiththelineatinfinity
Wehaveinsinuatedfromthebeginningthatthesemodelsofincidencegeometriesare thesame,andwehavealreadysaidthattheyaremodelsof the real projectiveplane.Tomakethismathematicallycorrect,wedefinetwoincidence geometries(ofpointsandlines)tobe isomorphic ifthereisabijectionbetween theirsetsofpointsthatrespectstheirlines.Inotherwords,thereisabijection φ mappingpointsofoneincidencegeometryontothepointsoftheother,such thatif isalineofthefirstgeometry,thentheimageofthepointsincident with (under φ)ispreciselythesetofpointsofalineofthesecondincidence geometry.
Theorem1.4
TheextendedEuclideanplaneandtherealprojectiveplaneareisomorphic.
Proof EmbedtheEuclideanplanein R3 astheplane z = 1(seeFigure 1.3).
Considertheprojectionviatheorigin O fromanon-parallelplane π notonthe originto z = 1.Theline thatistheintersectionof π and z = 0willhaveno imageandtheline m thatistheintersectionoftheplaneparallelto π with z = 1 willhavenopreimage.Thepointsof areinone-to-onecorrespondencewith parallelclassesoflinesof z = 1:namelyapoint P correspondstotheparallel classoflinesof z = 1parallelto OP.Moreover,givenapoint Q of m,each
Figure1.3TheEuclideanplaneembeddedin R3
Table1.3 FromtherealprojectiveplanetotheextendedEuclideanplane.
RealprojectiveplaneExtendedEuclideanplane
(a, b, c) , c 0 a c , b c , 1
(a, b, 0)
parallelclass {bx ay = c : c ∈ R} oftheplane z = 1.
line on Q istheimageofalineof π andtheselinesarisingfrom Q forma parallelclassin π.Moreover,theimageofalineof π (otherthan )isalineof theplane z = 1.
Now,removing π fromthepicture,wehaveaone-to-onecorrespondence betweenthe1-dimensionalsubspacesof R3 ofthepointsandparallelclasses oftheplane z = 1,andthe2-dimensionalsubspacesof R3 andthelinesand thelineatinfinityoftheplane z = 1thatpreservesincidence.Therefore,the extendedEuclideanplaneandtherealprojectiveplaneareisomorphic.Itis worthwhiledetailingthisisomorphisminTable 1.3
Composingthiswiththeisomorphism(a, b, 1) → (a, b)oftheplane z = 1 with R2,weobtain:
• (x, y, z) , z 0,intherealprojectiveplanecorrespondsto(X, Y)in R2 if andonlyif X = x z , Y = y z ;
• (1, m, 0) intherealprojectiveplanecorrespondstotheparallelclassof linesofslope m in R2;
• (0, 1, 0) intherealprojectiveplanecorrespondstotheparallelclassof verticallinesin R2
MovingbetweenCartesian(X, Y)coordinatesandhomogeneous(x, y, z) coordinatesviatheequations X = x/z, Y = y/z isdueto Hesse (1842). Thiswasadoptedby Cayley (1870)andgeneralisedto n dimensions.
Wesaythatasetofpointsis collinear iftheyareallincidentwiththesame line.Likewise,asetoflinesis concurrent iftheyareallincidentwiththe samepoint.Usingdeterminants,thereisasimpletestforwhenthreepointsare collinearorthreelinesareconcurrent.
Theorem1.5 Threepoints (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) arecollinearifandonlyif
Proof (x1, y1, z1), (x2, y2, z2), (x3, y3, z3)arecollinearifandonlyifthematrix intheabovedeterminanthasranktwo.
Theorem1.6
Threelines [a
] areconcurrentifandonlyif
Proof [a1, b1, c1], [a2, b2, c2], [a3, b3, c3]areconcurrentifandonlyifthe matrixintheabovedeterminanthasnullityone.
The dual ofastatementabouttheplaneisthestatementthatresultsafter interchanging point and line, collinear and concurrent, intersection and join andmakingthenecessarylinguisticadjustments.
The Principleof Duality (Poncelet, 1822; Gergonne, 1825/6)
Thedualofeverytheoremabout PG(2, R)isalsoatheorem.
Seealso Poncelet (1995a, 1995b).Foraproofofthe principleofduality, notethatthemaptakingapoint(a, b, c)toaline[a, b, c]preservesincidence. Foranexampleoftheprincipleofdualityatplay,notethatTheorems 1.5 and 1.6 aredual.
1.5Recap:TheRealProjectivePlaneasInvolvingPoints
andLines
WhenwethinkoftheEuclideanspace R3,wethinkoftheincidencestructure ofpoints,lines,andplanes.Nowletusbemoreabstractandinsteadthinkof theincidencestructure J whose‘points’aretheparallelclassesoflinesin Euclideanspace,andwhose‘lines’aretheparallelclassesofplanesinEuclideanspace.Theincidencerelationwouldbederivedfromthenaturalincidence relationofclassrepresentatives.
newpoints parallelclassesoflinesinEuclideanspace newlines parallelclassesofplanesinEuclideanspace
Nowdeleteaparallelclass Π ofplanesandalllinesparalleltoittoobtain anewincidencestructure A.Chooseaplane π of Π,andapoint P noton π.
Figure1.4Therealprojectiveplanemodelledonthesphere.
Considerthemap ρ thattakesaparallelclassoflinestotheintersectionofits uniquememberon P with π,andaparallelclassofplanestotheintersection ofitsuniquememberon P with π.Sincetheparallelclasseshaveparallel representativesifandonlyiftheuniquememberson P arecontainedinone another,itfollowsthattheseparallelclassesoflinesandplaneshaveparallel representativespreciselywhentheimagesunder ρ arecontainedinoneanother. Therefore, A isisomorphictothe affineplane π.(Thedeletedobjectsthus naturallygivethepointsatinfinityof π aslineson P parallelto π andalineat infinityastheplaneon P parallelto π.)
Soweseeherethattherealprojectiveplanecanberealisedasanextension oftheincidencestructure A,andthisincidencestructureisjusttheparallel classesoflinesandplanesinEuclideanspace,minusoneclassofplanes.
Exercises
1.1 Considertheunitsphere S 2 in R3.Let P bethesetofpairsofantipodal pointsof S 2,andlet L bethesetofgreatcirclesof S 2.Defineincidence betweenelementsof P and L bynaturalinclusion:apairofantipodal pointsisincidentwithagreatcircleifbothpointslieonthegreatcircle(comparewithFigure 1.4).Showthatthisincidencegeometryis isomorphicto PG(2, R).
1.2 Let(x1, y1, z1)and(x2, y2, z2)betwopointsof PG(2, R),writteninhomogeneouscoordinates.Showthattheuniquelinelyingonthesetwopoints canbecomputedusingthevectorcrossproduct.
1.3 A quadrangle isasetoffourpoints,nothreecollinear,anda quadrilateral isthedualobjectofaquadrangle.Whatisaquadrilateralwhen expressedintermsoflines?
Collineations
Thedominantroleofgrouptheoryinmathematicswaslongunsuspected;foreightyyears,theconceptofagroupwasignored.Itwas Galoiswhofirsthadaclearnotionofgroups;butitisonlysincethe workofKleinandespeciallyLiethatwehavebeguntoseethatthere isalmostnomathematicaltheorywherethisnotiondoesnotholdan importantplace.
Henri Poincaré (1921,p.137). 1
2.1TheProjectiveGeneralLinearGroup
Firstweintroducesomenotation.Ifapoint P isincidentwithaline ,thenwe write P I .If P isnotincidentwith ,thenwewrite P ✄ I .A collineation,or automorphism,of PG(2, R)isabijectiononthepointstakinglinestolines.2 Inotherwords,ifweconsidertheactionofacollineation g ontheset S of points {P : P I } incidentwithaline ,then S g = {Pg : P I } isthesetofpoints ofsomeline.Thesetofallcollineationsof PG(2, R)formsagroupunder thecompositionoperation,andwedenotethisgroupbyAut(PG(2, R)).This groupis‘large’inthesensethatitacts2-transitivelyonpoints,butitalsohas otherpropertiesthattranslatetoimportantsymmetrytraitsfortheunderlying geometry.
Givena3 × 3invertiblematrix A,themap(x, y, z) → (x, y, z)A isa collineation.Twosuchmatrices A, B definethesamecollineationifandonly if A = cB,for c ∈ R with c 0,becausethecoordinatesarehomogeneous.So thisactionofthegroup GL(3, R)ofall3×3invertiblematriceshasanon-trivial
1 ReprintedbypermissionfromSpringerNature: ActaMathematica.Rapportsurlestravauxde M.Cartan.Poincaré,H. © 1921.
2 Ifyouarefamiliarwiththepreviouschapter,thenacollineationisanisomorphismof PG(2, R) toitself.
kernel Z.Thisnormalsubgroup Z consistsofthenon-zeroscalarmatrices,and thefactorgroup GL(3, R)/Z,whichwedenoteby PGL(3, R),isthe projective generallineargroup.Wecanthinkof PGL(3, R)as3 × 3invertiblematricesthatareidentifieduptoascalar;thatis,twoelementsareequalifthe matrixrepresentationofoneofthemisascalarmultipleoftheother’smatrix representation.
Giventhewayweintroducedtherealprojectiveplaneastheextended Euclideanplane,youcouldbeforgivenforthinkingthelineatinfinitywas special.Butthisisnotthecase–alllinesoftherealprojectiveplanehavethe samestatus.
Theorem2.1
PGL(3, R) actstransitivelyonthelinesof PG(2, R)
Proof Giventwolines , m,takeabasis {v1, v2} for andextendittoabasis {v1, v2, v3} for R3,andalsotakeabasis {w1, w2} for m andextendittoabasis {w1, w2, w3} for R3.Thereisaninvertiblematrix A with vi A = wi,for i = 1, 2, 3, andthisinducesacollineationof PG(2, R)taking to m
Thisproofshowsthat PGL(3, R)acts2-transitivelyonpointsof PG(2, R). Therealaffineplane AG(2, R)isbasicallythegeometryof R2 whenwe forgetaboutdistanceandangle.Itisthe incidencegeometry of R2,withits salientpropertiesbeingtheparallelismoflinesandtheinvarianceofaffine ratio.Wecanrecovertherealaffineplane AG(2, R)fromtherealprojective planebydeletingalineandallpointsincidentwithit,reversingtheprocedure wefirstusedtoconstructtherealprojectiveplane.
Theorem2.2
Let bealineof PG(2, R).Thentheincidencestructureofpointsnoton and linesotherthan ,withtherestrictedincidence,isisomorphicto AG(2, R).
Proof ThisisanimmediateconsequenceofTheorems 1.4 and 2.1.
Theconceptof permutationalisomorphism allowsustoformallydescribe whentwogroupactionsare thesame.Essentially,twoactionsarepermutationallyisomorphicifthegroupsareisomorphicandoneoftheactionscanbeseen assimplyarelabellingofthesetacteduponintheotheraction.If G isagroup actingonaset Ω,and H isagroupactingonaset Γ,wesaythatthesetwo actionsare permutationallyisomorphic ifthereexistsagroupisomorphism ϕ : G → H andabijection β : Ω → Γ suchthat
