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Chapter5
BasicsofProject iveGeom etry
Thinkgeometri cally,provealgebraically.
- JohnTate
5.1WhyP rojectiveS paces?
Forano vic e,projectivegeometryu suallyappearstobeabitodd,and itisnot obvioustomotivatewhyi tsintroductionisinevitableand infactfruitful.Oneofthe mainmotivat ionsarisesfromalgebraicgeometr y. Themai ngoalofalgeb raicgeometry istost udythepropertiesofgeometricob- jects,suchasc urvesandsurf aces,d efinedimplicitlyintermsofalgeb raicequations.
Forins tance,theequation
x 2 +y 2 !1=0 definesacircleinR 2 equations ax 2 +by 2 +cxy+dx+ey+f=0 thesecurvesac cordingtotheirgenericg eometricshape.Thisisin dee dpossible. Exceptforso-call edsingula rcases,wegetellipses,parab olas,andhyperbolas .The samequestio ncanbeaskedforsurfaces definedbyq uadrat icequations,known areabit arti ficial.Forex ample,anellipseandahyperbo ladifferbythefact that providedthatpointsati nfinityarehandl edproperly. Anotherimportantpr oblemisthestudyofintersect ionofgeometricobjects(de- finedalgebr aically).Forexample,giventwocurvesC 1 andC 2 ofdegr eemandn, respectively,whatisthenumberofinterse ctionpointso fC 1 andC 2 ?(bydegreeof thecurv ewemeanthetotaldeg reeof thedefiningpolyn omial). 103
1045BasicsofProjectiveGeometry
Well,itdepend s!Even inthecaseoflines(whenm=n=1),there arethree possibilities:eitherthelinescoincide,orthey areparallel,orthe reisas ingleinter- sectionpoint.Ingen eral,weexpectmnintersectionpoints,butsomeofth esepoints maybemi ssingb ecausetheyareatinfi nity,becausetheycoin cide,orbecause they areimagi nary. Whatbegins totranspireisthat "point satinfinity"causetrouble.Theycaus eex- ceptionsthatinvalidategeometricth eorems(forexample,considerthemoregeneral versionsofthetheorems ofPapp usandDesa rguesfromSection2.12),andmakeit difficulttoclassifygeom etricobj ects.Projectivegeometryis design edtodealwith "pointsatinfinity"andr egu larpointsinauniformway,wi thoutma kingadistinc- tion.Pointsat infinityarenowjus tordin arypoints,andm anythingsbecomesim- pler.Forexample ,thecla ssificationofconicsandqua dricsbecomessimpler,and intersectiontheorybecomesclean er(although,tobeh onest,we needtoc onsider complexprojectivespaces ). Technically,projectivegeometrycan bedefinedaxiomatically,orby buidling uponli nearalgebra.Histori cally,theaxiomaticappr oachcamefirst (seeVeblenand Young[28,29], EmilArtin[1 ],andCoxeter [7,8,5,6]). Althoughverybeautifuland elegant,webelievethati tisahar derapproachthantheline aralg ebraicapproach.In thelinear algebraicappr oach,allnotionsareconsid ereduptoascalar.Forexample, aprojectivepointisreallyalinethroughtheorigin.Intermsof coordi nates,this correspondsto"homogenizing."Forexample, thehomogeneousequationofaconic is ax 2 +by 2 +cxy+dxz+eyz+fz 2 =0. arepoin tsofcoordinates(x,y,0)(withx,y,znotallnull,anduptoascalar).Thereis ausefulmodel(interpretation)ofplaneprojectivegeometryin termsof thecentral projectioninR 3 fromtheo riginontot heplanez=1.A notherusefulmodelis the spherical(orthehalf-spheri cal)mode l.Inthesph ericalmodel,aproject ive point correspondstoapairofantipodalpo intson the sphere. Asaf finegeometryisth estudyofpropertiesinvar iantun deraffinebijections , projectivegeometryisthestudyofprope rtiesinvariantunderbije ctiveprojective maps.Roughlys peaking,projectivemap sarelinearmapsuptoascalar.Inanalogy withourpre sentationofaffinegeometry,wewilldefineprojectivespaces,projective wede finetheprojectiv ecomplet ionofanaffinespace,andwhenwedefineduality. ofra tionalcurvesandrationals urfaces.Thegeneral ideaisthataplanerational curveisthepr ojection ofasimpler curveinalargerspace,apolynomialcurvein R 3 ,ontotheplanez=1,as wenowexpl ain. Polynomialcurvesarecurvesde finedparametricallyi ntermsofpolynomi- als.Moresp ecifically,ifEisan affines paceoffinitedimens ionn#2and (a 0 ,(e 1 ,...,e n ))isan affinef rameforE,apolynomialcurveofdegreemisam ap
F:A$Esuchthat
F(t)=a
0 +F 1 (t)e 1 +···+F n (t)e n
5.1WhyPr ojectiv eSpaces?105
forallt%A,whereF 1 (t),...,F n (t)arepol ynomialsofdegreeatmostm. Althoughmanycurvescanb edefined,iti ssomewhatembarassingtha tacircle cannotbedefinedi nsuchaw ay.Infact,manyin teresti ngcurvescann otbedefined thisway, forexample,ellip sesandhyp erbolas.Arathersimplewayt oextendthe classofcurves definedpar ametricallyisto allowrationalfunctionsinsteadof poly- nomials.Aparametricrationalcurveofdegr eemisaf unc tionF:A$Esuch that
F(t)=a
0 F 1 (t) F n+1 (t) e 1 F n (t) F n+1 (t) e n forallt%A,whereF 1 (t),...,F n (t),F n+1 (t)arepoly nomialsofdegreeatmostm.
Forexam ple,acircleinA
2 canb edefinedb ytherationalmap
F(t)=a
0 1!t 2 1+t 2 e 1 2t 1+t 2 e 2
Inth eaboveex ample,thedenomi natorF
3 (t)=1+t 2 nevertakesthev alue0 2
G(t)=a
0 t 2 t e 1 1 t e 2 ObservethatG(0)isun defined.Thecurvedefinedabove isahyperbo la,andfort closeto0,thep ointon thecurv egoestowar dinfinityinoneof thetwoa symptotic directions. inap roje ctivespace.Intuitively,thismea nsviewingarationalcurveinA n ass ome appropriateprojectionofapolyno mialcurveinA n+1 ,backontoA n GivenanaffinespaceE,foranyhyperplaneHinEandanypo inta 0 notinH,the centralprojection(o rconicprojection,orperspective projection)ofcentera 0 onto H,isthepartialmappdefinedasfollows :Foreve rypointxnotin thehyperpla ne passingthrougha 0 andparall eltoH,wedefinep(x)asth eintersectio noftheline definedbya 0 andxwiththehyp erplaneH.
Forexam ple,wecanviewGasar atio nalcurveinA
3 givenby G 1 (t)=a 0 +t 2 e 1 +e 2 +te 3
Ifwe project thiscurveG
1 (infact ,aparabolainA 3 )usingthecentralprojection (perspectiveprojection)ofcentera 0 ontothepla neofequati onx 3 =1,w egetthe previoushyperbola.For t=0,the pointG 1 (0)=a 0 +e 2 inA 3 isinth epla neof equationx 3 =0,a nditsproj ectionisundefi ned.WecanconsiderthatG 1 (0)=a 0 e 2 inA 3 isprojectedtoinfinityinthedirectionofe 2 intheplanex 3 =0.Inthesetting ofproj ectivespaces,thisdirectionc orrespondsrigorouslytoa poin tat infinity. Letusve rifyt hatthecentralproje ctionusedin theprevio usexamplehasthede- siredeffect. LetusassumethatEhasdimens ionn+1andthat(a 0 ,(e 1 ,...,e n+1 isan affinef rameforE.Wewanttodeterminethecoordinatesofthecentralprojec- tionp(x)ofapoi ntx%Eontothehype rplaneHofequa tionx n+1 =1(thecenterof
1065BasicsofProjectiveGeometry
projectionbeinga 0 ).If x=a 0 +x 1 e 1 +···+x n e n +x n+1 e n+1 assumingthatx n+1 "=0;apoi ntont helinepassing througha 0 andxhascoordi nates ofthe form(!x 1 ,...,!x n+1 );andp(x),thecentralprojectionofxontothehyper - planeHofequa tionx n+1 =1,is theinter secti onofthelinefroma 0 toxandthis hyperplaneH.Thuswemusthave!x n+1 =1,and thecoordina tesofp(x)are x 1 x n+1 x n x n+1 ,1
Notethatp(x)isun definedwhenx
n+1 =0.I nproject ivespaces,wecanmakesense ofsuc hpoints. Theabov ecalculationcon firmsthatG(t)isa cen tralprojectionofG 1 (t).Simi- larly,ifwedefin ethe curveF 1 inA 3 by F 1 (t)=a 0 +(1!t 2 )e 1 +2te 2 +(1+t 2 )e 3 thecentr alprojectionofthepolyn omialcurveF 1 (again,aparabolainA 3 )ontothe planeofequati onx 3 =1isthecircleF. Whatwejus tsketc hedisagenera lmethodtodealwithration alcurves.Wecan useour"hat const ruction"toem bedanaffinespaceEintoave ctor space
Ehaving
onemor edimension,the nconstructtheprojective spaceP E .Thisturnsoutto bethe "project ivecompletion"oftheaffinespaceE.Thenwecandefinearational curveinP E Eback ontoP E thelacko fspace,su chapre sentationisomittedfromthe maintext.However,it canbe found intheadditionalmat erialont hewebsite; seehttp://www.cis. upenn.edu/ jean/gbooks/geom2.html. Moregenera lly,theprojectivecompletionof anaffinesp aceisav er yconvenient tooltohandle "po intsatinfinity"inaclea nfashion. Thischapte rcontainsabriefprese ntationofconceptsofprojectivegeometry. Thefoll owingconceptsarepresented: projectivespaces,projectiveframes,homo- affinepatches.The projectivecompletio nofanaffine spaceispresentedusingthe "hatconst ruction."ThetheoremsofPappusandDesarguesareproved,usingthe methodinwhichpo intsar e"senttoinfinity ."Wealsodiscussthecros s-ratioand duality.Thechapterendswit havery briefexplanationofth euseofthecomplexifi- cationofaproject ivespace ino rdertodefinethenotionofangleandort hogonality inap roje ctivesetting.Wealsoincludeasho rtsectiononapplicationsofprojective anderror -correctingcodes.
5.2Proje ctiveSpaces107
5.2Proje ctiveSpaces
Asin thecas eofaffineg eometry,our presen tationofproject ivegeo metryisrather sketchyandbiasedtow ardthealg orithmicgeomet ryofcurvesandsurfaces.Fora systematictreatmentofproj ectivegeometry,werecommend Berger[3,4],Sam uel [23],Pedoe[2 1],Coxeter[7,8,5, 6],Beutel spacherandRosenbaum[2],Fres- nel[14], Sidler[24],Tis seron[26],Lehmanna ndBkouche[20],Vien ne[30], andthecl assicaltreatis ebyVeblenandYoung [28,29],which,al thoughslightly old-fashioned,isdefinitelyworthrea ding.Emi lArtin'sfamousbook[ 1]contains, amongotherthin gs,anaxiomaticp resentationofprojectivegeometry,andawealth ofgeom etricmaterialpresentedf romanalgebraicpointofview.Other" oldiesbut goodies"includethebea utifulbooksbyDarboux[ 9]andKle in[19].Foradevel- opmentofproject iv egeometryaddressingthedelicatepr oblemofor ientation, see Stolfi[25],and foranapproachg earedtowa rdscomput ergrap hics,seePennaand
Patterson[22].
First,wedefineproj ective spaces,allo wingthefieldKtobe arbitrar y(which doesnoharm,andisneededtoallowfinite andcomplexprojectivespac es).Roughly speaking,everyprojectivec onceptisalinea -algebraicconcept"uptoas calar. "For spaces,thisismad epreciseasfol lows Definition5.1.Givenavectors pac eEoverafiel dK,theprojectivespaceP(E) inducedbyEisthe set(E!{0})/&ofe quivalenceclassesofnonzerovector sinE undertheequi valence relation&definedsuchthatf orallu,v%E!{0}, u&viffv=!u,forsom e!%K!{0}. Thecanonicalprojectionp:(E!{0})$P(E)isth efunctio nassociatingthe equivalenceclass[u] modulo&tou"=0.T hedimensiondim(P(E))ofP(E)is definedasfollow s:IfEisof infinitedim ension,thendim(P(E))=dim(E),andif
Ehasfinite dimension,dim(E)=n#1thendim(P(E))=n!1.
Mathematically,aprojectivespaceP(E)isas eto fequivale nceclas sesofvectors asan"at omi c"object,forgettin gtheinternalstruct ureoftheequi valenceclass.For thisreas on,itiscustomarytocallane qui vale nceclassa=[u] apoint(theentire equivalenceclass[u] isco llapsedintoasingleobjectviewe dasap oint) .
Remarks:
(1)Ifwe vie wEasanaf fine space,thenforany nonnullvect oru%E,since [u] ={!u|!%K,!"=0}, letting
Ku={!u|!%K}
denotethesubspa ceofdime nsion1spannedbyu,themap
1085BasicsofProjectiveGeometry
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GÉOMÉTRIE AFFINE - Université Paris-Saclay