TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 189

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1 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 89 Hr lso, th ictors r orint pln surcs inicting th irction o plns in th psuo-euclin spc. s or, ctors n ictors r irnt things. This ct hs n princ lso phsicists: in th Minkowski s spc, th lctromgntic il is ictor whil th ttrpotntil is ctor. On th othr hn, th orint r is, o cours, ictor. s critrion to istinguish oth kin o mgnitus on lso uss th rrsl o coorints, which chngs th sns o ctors whil ls ictors inrint. Two ctors r si to orthogonl i thir innr prouct nishs: w w 0 So th outr prouct is th prouct th orthogonl componnt n th innr prouct is th prouct th proportionl componnt: w w w w Th prouct o two ictors ils ttrnion. Th rl n ictor prts cn sprt s th smmtric n ntismmtric prouct. Sinc: n th smmtric prouct is rl numr whos ngti lu will not hr with th smol, whrs th ntismmtric prouct, not hr with th smol, is lso ictor: w w ( w w ) w w w ( w w ) w w w ( w w ) ( w w ) ( w w ) Th outr prouct o thr ctors hs th sm prssion s or Euclin gomtr n this is nturl outcom o th tnsion thor: th prouct u w is th orint olum gnrt th surc rprsnt th ictor u whn it is trnslt prllll long th sgmnt w: u w u u u z z w w w z Finll, lt us s how th prouct o thr ctors u, n w is. Th ctor cn rsol into componnt coplnr with u n n nothr componnt prpniculr to th pln u-:

2 90 RMON GONZLEZ LVET u w u w u w Now lt us nls th prmutti proprt. In oth Euclin n hprolic plns w oun u w w u 0. In th psuo-euclin spc th prmutti proprt coms: ( u w w u) u w w u u w w u u w u w u w I tk th sm lgric hirrchis s in th ormr chptr: ll th rig proucts must oprt or th gomtric prouct, conntion qut to th ct tht in mn lgric situtions, th rig proucts must lop in sums o gomtric proucts. Th hproloi o two shts ccoring to Hilrt (Grunlgn r Gomtri, nhng V) th complt Lochsk s pln cnnot rprsnt smooth surc with constnt curtur s propos ltrmi. ut this rsult onl concrns surcs in th Euclin spc. Th surc whos points r plc t i istnc rom th origin in psuo- Euclin spc (th two-sht hproloi) is th surc srch Hilrt which rliss th Lochsk s gomtr. It is known tht it hs chrctristic istnc lik th rius o th sphr. Sinc ll th sphrs r similr, w n onl stu th unitr sphr. Likwis, ll th hproloil surcs z r r similr Figur 5. n th hproloi with unit rius (igur 5.): z whr (,, z) r rtsin lthough not Euclin coorints, suics to stu th whol Lochsk s gomtr. Th rr will in complt stu o th hproloil surc in Fr, Fountions o Euclin n Non-Euclin Gomtr, chp. VII. Th Wirstrss mol. For m, rtsin coorints r orthogonl coorints in lt spc (with null curtur), inpnntl o th Euclin or psuo-euclin ntur o th spc. Lins n plns (linr rltions twn rtsin coorints) cn lws rwn in oth spcs, lthough th psuo-euclin orthogonlit is not shown s right ngl, n th psuo-euclin istnc is not th iw lngth. Th incompltnss o rtsin coorints ws lr criticis Liniz in lttr to Hugns in 679 proposing nw gomtric clculus: «r prmirmnt j puis primr pritmnt pr c clcul tout l ntur ou éinition l igur (c qu l lgèr n it jmis, cr isnt qu q. st l qution u crcl, il

3 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 9 Th ct tht th hproloil surc owns th Lochsk s gomtr will int through th irnt projctions hr riw. I w ollow th sm orr s in th ormr chptr, w shoul irstl l with th Lochskin trigonomtr. Howr, th concpt o rc lngth on th hproloi is much lss intuiti thn on th sphr. Thror w must s th concpt n trmintion o th rc lngth or stuing th hproloil trigonomtr. Th cntrl projction (ltrmi isk) In this projction th cntr o th hproloi is th cntr o projction. Thus, w projct r point on th uppr sht o th hproloi z into nothr point on th pln z tking th origin 0, 0, z 0 s cntr o projction (igur 5.). Lt u n rtsin coorints on th pln z, which touchs th hproloi in th rt. similr tringls : u z z From whr w otin: u u u z u Th irntil o th rc lngth upon th hproloi is: s s z z ( ) u u u ( u ) ( u ) which is th ngti lu o th usul mtric o th Lochsk s surc in th ltrmi isk. Th ngti sign inicts tht th rc lngth is comprl with lngths in th - pln. Th whol uppr sht o th hproloi is projct insi th ltrmi s circl u with unit rius. Howr, i w projct th on-sht hproloi z into th pln z w in th sm mtric ut positi, wht inicts th lngths ing comprl to thos msur on th z-is. ll th uppr hl o th on-sht hproloi is projct outsi th circl o unit rius, giing ris to nw gomtr not stui up till now. ut pliqur pr l igur c qu c st t.» (Josp Mnl Prr, Gomtric lgr rsus numricl crtsinism in F. rck, R. lngh, H. Srrs, lior lgrs n Thir pplictions in Mthmticl Phsics). Now to consir tht th lgs o oth right ngl tringls r horizontl n rticl, n thror prpniculr is nough or th uction. Notwithstning, in psuo-euclin spc on must us th lgric inition o similitu: tringl with sis n is irctl similr to tringl with sis c n i n onl i c. This conition implis tht oth quotints o th moulus r qul n oth rgumnts lso.

4 9 RMON GONZLEZ LVET Lt us consir pln pssing through th origin o coorints (cntrl pln). Its intrsction with th two-sht hproloi is th hprol trmin th qutions sstm: z z, rl W srch th Frnt's trihron. irntition w in linr irntil qutions sstm : z z z whr n cn isolt: z z z z Thn th irntil o th rc lngth o this hprol is: s z z z z n its squr: s z ( ) z Now w otin th unitr ctor t tngnt to th hprol: t s s z s ( z) ( z ) ( ) Osr tht t n s h imginr componnts sinc pln onl cuts th twosht hproloi i >. nw, th gomtric ctor cn tkn with rl componnts, lthough thn its moulus is. Th riti o th tngnt ctor with rspct to th rc lngth is not onl proportionl ut qul to th norml ctor n o th hprol: t t n ( z ) κ s s cus th position ctor o r point on th two sht hproloi hs unitr moulus. Hnc th curtur κ o th hprol is constnt n qul to. ll th hprols ing intrsctions o th hproloi with plns pssing through th origin n gin point on th hproloi h th sm rius o curtur r:

5 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 9 t r s ( z ) Two consquncs r uc rom this ct: irstl thir common rius o curtur is prpniculr to th surc, n sconl, ths hprols r gosics o th hproloi. cus ll th hprols li on plns pssing through th origin, thir cntrl projctions r stright lins. In othr wors, r gosic hprol is projct into sgmnt on th ltrmi isk. Tking s qution o th lin: k u l with k n l constnt, th sustitution in s gis 4 : s u ( k ) k l k l u l u intgrtion w rri t th ollowing primiti: s u rg tgh ( k ) k k l l const log k l u k l u k k k k l l const Th lin k u l cuts th ltrmi s circl u t th points n whos coorints r th solution o th sstm o th qutions: k u l u u k l k k l u k l k k l Thus th primiti m rwrittn: s log u u u u const Figur 5. Thn, th rc lngth twn two points n on this gosic hprol is th irnc o this primiti twn oth points: s s log ( u u )( u u ) ( u u )( u u ) log ( ) 4 s si or, th rc lngth s on th two-sht hproloi hs imginr lus. I m sorr th inconninc o tht rr ccustom to tk rl lus. Howr th cohrnc o ll th gomtric lgr orc to tk ccount o th imginr unit. This lso plins wh Lochsk oun th hproloil trigonomtr choosing imginr lus or th sis in th sphricl trigonomtr.

6 94 RMON GONZLEZ LVET tht is, th hl o th logrithm o th cross rtio o th our points (igur 5.) on th ltrmi isk. Howr it is mor ntgous to writ it using cosins inst o tngnts mns o th trigonomtric intit: tgh tgh tgh tgh cosh s s s s s s tr rmoing k n l using th qution o th lin in th primiti, w rri t: cosh u u u u s s whr th prssion is rl or ll th points on th hproloi, whos projctions li on th ltrmi isk u. This prssion is quilnt to writ: ( ) ( ) ( ) ( ) cosh u u u u s s Sinc th ctor u is proportionl to th position ctor, w h: ( ) ( ) ( ) ( ) ( ) cos cosh z z z z s s s s which is th pct prssion to clcult ngls twn ctors in th psuo- Euclin spc. This rsult must commnt in mor til. Not tht r pir o ctors going rom th origin to th two-sht hproloi lws li on cntrl pln hing hprolic ntur, sinc it cuts th con 0 z o ctors with zro lngth. cus o this, th ngl msur on this pln is hprolic. hproloil tringl is th rgion o th hproloi oun thr gosic hprols. Th lngth o si o th tringl cn clcult intgrtion o th rc lngth o th hprol ccoring to th lst rsult. On th othr hn, th ngl o rt o hproloil tringl is th ngl twn th tngnt ctors o th hprols o ch si t th rt: ( )( ) ( )( ) ( )( ) cos ' ' ' ' z ' z z ' z t' t α ' ' ' '

7 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 95 Not tht r pir o tngnt ctors li on pln o Euclin ntur, pln prlll to cntrl pln not cutting th con o zro lngth. So th ngls msur on this pln r circulr. W rri t th sm conclusion proing tht th solut lu o th cosin (n lso its squr) is lssr thn th unit: ( ' ' ) ( ) ( ' ' ) < Th nomintor is positi (lthough oth squrs r ngti cus > ) n it cn pss to th right hn si o th inqulit without chnging th sns. Rmoing th nomintor n tr simpliiction: ' ' ' ' ' ' ' ' < n rrnging trms w in n quilnt inqulit: ' ' ' ' < ' ' ' ' ( ' ) ( ' ) < ( ' ' ) Just this is th conition or th istnc o th intrsction point whr two cntrl plns n th hproloi mt s I shll show now. Th plns with qutions z n ' ' z r projct into th lins u n ' u ' on th ltrmi isk. Thn th point o intrsction o oth plns n th hproloil surc is projct into th point on th ltrmi isk gin th sstm o qutions: u ' u ' whos solution is: ' u ' ' ' ' ' This point lis insi th ltrmi circl (n th point o intrsction o th hproloi n oth plns ists) i n onl i: u ( ' ) ( ' ) ( ' ' ) < which is th conition o otin. Not tht th prssion or th ngl twn gosic hprols coincis with th ngl twn th ictors o th plns contining ths hprols: cosα ( ) ( ' ' ) ' '

8 96 RMON GONZLEZ LVET It is not surprising rsult, cus it hppns likwis or th sphr: th ngl twn two mimum circls is th ngl twn th cntrl plns contining thm. Th plns pssing through th origin onl cut th two-sht hproloi i >. Ths plns lso cut th con o zro lngth z 0 so th r hprolic n th ngls msur on thm r hprolic rgumnts. On th othr hn, th plns with < o not cut th hproloi nithr th con n h Euclin ntur. In th Lochskin trigonomtr, th plns contining th sis o hproloil tringl r lws hprolic, whil th plns touching th hproloi n contining th ngls twn sis r lws Euclin, n hnc th ngls r circulr. Rsuming, rl moulus o ictor (0< ) inicts n Euclin pln n imginr moulus (0> ) hprolic pln. Now w lr h ll th ormuls to uc th hproloil trigonomtr. Hproloil (Lochskin) trigonomtr Th Lochskin trigonomtr is th stu o th trigonomtric rltions or th sis n ngls o gosic tringls on two-sht hprolois in th psuo- Euclin spc. W will tk or conninc th hproloi with unit rius z, lthough th trigonomtric intitis r qull li or hproloi with n rius. onsir thr points, n on th unitr hproloi (igur 5.). Thn. Th ngls orm ch pir o sis will not α, β n γ, n th rl lu o th sis rspctil opposit to ths ngls will smolis, n c rspctil. Figur 5. Thn is th rl lu o th rc o th gosic hprol pssing through th points n, tht is, th hprolic ngl twn ths ctors: sinh s si o, th plns intrscting th hproloi h imginr moulus. So w must ii th psuosclr imginr unit, which is lso th olum lmnt. In this qulit th hprolic sin is tkn positi n lso. α is th circulr ngl twn th sis n c o th tringl, tht is, th ngl twn th plns pssing through th origin, n, n th origin, n rspctil. Sinc th irction o pln is gin its ictor, which cn otin through th outr prouct, w h:

9 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 97 sin α ( ) ( ) whr th minus sign is u to th imginr moulus o th outr proucts o th nomintors. Now w writ th proucts o th numrtor using th gomtric prouct: sin α ( )( ) ( )( ) 8 sin α 8 W trct th ctor s common ctor t th lt, ut without writing it cus : sin α 8 ppling th prmutti proprt to th suitl pirs o proucts, w h: sin α 6 8 sinc th olum is psuosclr, which commuts with ll th lmnts o th lgr. Now th lw o sins or hproloil tringls ollows: sin α sin β sin γ sinh sinh sinh c (I) Lt us s th lw o cosins. Sinc thn: cosα ( ) ( ) ( ) ( ) sinh sinh c cosα 8 8 sinh c sinh (( )( ) ( )( ) ) ( ) Now tking into ccount tht:

10 98 RMON GONZLEZ LVET ut lso: ( )( ) 4 ( )( ) 4 n ing th n trms, w in: sinh sinh c cos α 8 ( 8 ) Etrcting common ctors n using, w m writ: sinh sinh c cos α 8 ( 8 ( ) ( ) ) sinh sinh c cos α cosh c cosh cosh cosh cosh cosh c sinh sinh c cos α (II) which is th lw o cosins or sis. Th sustitution o cosh c mns o th lw o cosins gis: cosh cosh cosh ( cosh cosh sinh sinh cos γ ) sinh sinh c cosα ( cosh ) cosh sinh sinh cos γ sinh sinh c cosα n th simpliiction o sinh : cosh sinh cosh sinh cos γ sinh c cosα Th sustitution o sinh c sinh sin γ / sin α ils: sinh sin γ cosα cosh sinh cosh sinh cos γ sin α iiing sinh : coth sinh cosh cos γ sin γ cotα (III) Strogrphic projction (Poincré isk) Figur 5.4 W projct th hproloil surc into th pln z 0 (igur 5.4) tking s cntr o projction th lowr pol (0, 0, ). Th uppr sht is projct insi th circl o unit rius whil th lowr sht is projct outsi. Lt u n th

11 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 99 rtsin coorints o th projction pln. similr tringls w h: z u z whr (,, z) r th coorints o point on th hproloi. Using its qution z, on rris t: u u rom whr th irntil o th rc lngth is otin: s z z u u 4 ( u ) ( u ) Th gosic hprols r intrsction o th hproloi with cntrl plns hing th qution: z Figur 5.5 Th sustitution th strogrphic coorints ils: ( u ) ( ) which is th qution o circl cntr t (, ) with rius r. Tht is, th gosic hprols on th hproloi r shown s circls in th strogrphic projction (igur 5.5). Th cntrl plns cutting th two-sht hproloi ulil >, so tht th rius is rl n th cntrs o ths circls r lws plc outsi th Poincré isk. Th ngl α twn two circls is otin through: r cosα r' r r' ( O O' ) ' ' ' ' Just this is th ngl twn two gosic hprols (twn th tngnt ctors t n t ) oun o. Thror, th strogrphic projction (Poincré isk) is conorml projction o th hproloil surc. On th othr hn, th gosic circls r lws orthogonl to th Poincré's circl u ( r', O' (0, 0) ) cus: r cosα O r 0

12 00 RMON GONZLEZ LVET Th irntil o r is sil otin through th moulus o th irntil ictor: ( ) ( z) ( z ) 4 u ( u ) Whn oing n inrsion o th Poincré isk cntr t point ling on th limit circl, nothr projction o th Lochsk's gomtr is otin: th Poincré s hl pln. Hr th gosics r smicircls orthogonl to th s lin o th hl pln. Sinc th inrsion is conorml trnsormtion, th uppr hl pln is lso conorml projction o th hproloi. Th igur 5.6 ispls th projction o hproloil tringl n its sis in th Poincré s uppr hl pln. Figur 5.6 zimuthl quilnt projction This projction prsrs th r n is similr to th zimuthl quilnt projction o th sphr usull us in th polr mps o th Erth. Th irntil o th r in th psuo-euclin spc is ictor whos squr is: ( ) ( z) ( z ) Th sustitution o th hproloi qution: z z z z ils: ( ) z Er zimuthl projction hs pln coorints u n proportionl to n, ing th proportionlit constnt onl unction o z: u () z () z Thn irntition o ths qulitis n sustitution o z, which is linr comintion o n w rri t: u z z '

13 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 0 Intiing oth r irntils w in th ollowing irntil qution: z z ' z I introuc th uilir unction g : g z g' z z Rwriting th qution with irntils, w rri t n ct irntil: z z z 0 g z ( g ) z g 0 ( z const) g () z () z z ( z const) z Nt to th pol n u r coincint n lso n. Thn (), which implis th intgrtion constnt n thn 5 : ( z ) () z z z u z z Wirstrss coorints n clinricl quiistnt projction Ths r st o coorints similr to th sphricl coorints, ut on th hproloi surc in th psuo-euclin spc. For hproloi with unit rius, th Wirstrss coorints r: Figur 5.7 sinhψ cosϕ sinhψ sin ϕ z coshψ Thn th irntil o rc lngth is: 5 s comprison in th cntrl projction (z) /z whil or th strogrphic projction (z) / (z ).

14 0 RMON GONZLEZ LVET s ( coshψ cosϕ ψ sinhψ sinϕ ϕ ) ( coshψ sinϕ ψ sinhψ cosϕ ϕ ) sinhψ ψ Introucing th unitr ctors n ψ ϕ s: ψ coshψ cosϕ coshψ sinϕ sinh ψ sin ϕ cosϕ ϕ th irntil o rc lngth in hproloil coorints coms: s ψ sinhψ ϕ s ( ψ sinh ψ ϕ ) ψ ϕ Not tht ψ n ϕ r orthogonl ctors, sinc thir innr prouct is zro. t ψ 0, s onl pns on ψ, so tht thr is pol. Thn ψ is th rc lngth rom th pol to th gin point on th hproloi surc (igur 5.7), whil ϕ is th rc lngth or th qutor, th prlll trmin sinhψ n ψ log( ). Th mriins (ϕ constnt) r gosics, whil th prllls (ψ constnt) incluing th qutor r not gosics cus thir plns o not pss through th origin o coorints. In th clinricl projctions, th hproloi is projct onto clinr pssing through its qutor, whos is is th z is. Th chrt is th Euclin mp otin unrolling th clinr 6. Using ϕ n ψ s th coorints u n o th chrt, th hproloi surc is projct into rctngl with π with n ininit hight. Th r on this chrt is: sinhψ ϕ ψ sinh u This projction is quiistnt or th mriins (ϕ constnt) n or th qutor (ψ log( )). linricl conorml projction I w wish to prsr th ngls twn curs, w must nlrg th mriins th sm mount s th prllls r nlrg in clinricl projction, tht is ctor /sinh ψ: ψ sinhψ ψ log tgh Th irntil o rc lngth n r r: 6 Er pln tngnt to th two-sht hproloi hs Euclin ntur. So th Lochsk s surc cn onl projct onto Euclin chrt, n th clinr o projction is not hprolic n os not long proprl to th psuo-euclin spc.

15 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 0 ( ψ sinh ψ ϕ ) sinh ( u ) 4 p( ) ( p( ) ) ( u ) s ψ 4 p( ) ( p( ) ) sinhψ ϕ ψ u linricl quilnt projction I w wish to prsr r, w must shortn th mriins in th sm mount s th prllls r nlrg in th clinricl projction, wht ils: sinhψ ψ coshψ s ( ψ sinh ψ ) ( ) u ϕ sinh ψ ϕ ψ u which clrl ispls th quilnc o th projction. Osr tht cosh ψ mns th hproloi is projct ollowing plns prpniculr to th clinr o projction (igur 5.8). Figur 5.8 onic projctions Ths projctions r m into con surc tngnt to th hproloi. Hr to tk th con z 0 is th most nturl choic, lthough s or, th con o projction hs Euclin ntur n os not long proprl to th psuo-euclin spc, whr it shoul th con o zro lngth. Th con surc unroll is circulr sctor. In conic projction prlll is shown s circl with zro istortion. Th chrctristic prmtr o conic projction is th constnt o th con n cos θ 0, ing θ 0 th Euclin ngl o inclintion o th gnrtri o th con. Th rltion with th hprolic ngl ψ 0 o th prlll touching th con (which is rprsnt without istortion in th projction) n θ 0 is: n cosθ 0 tgh ψ 0

16 04 RMON GONZLEZ LVET Sinc th grticul o th conic projctions is ril, is mor connint to us th rius r n th ngl χ : ( ψ ) ψ r χ n ϕ Th irntils o rc lngth n r or conic projction r: r sinh ψ ( ) [ ( )] s ψ sinh ψ ϕ χ ψ n sinhψ sinh ψ ϕ ψ r χ n ( ψ ) Lt us s s or th thr spcil css: quiistnt, conorml n quilnt projctions. Th irntil o r or polr coorints r, χ is r r χ. I th projction is quilnt, w must inti oth to in: sinhψ ψ n ( ψ ) ( ψ ) ( ψ ) ψ cosh n n r ψ sinh n s n n r 4 n r n r 4 r χ I th projction is quiistnt, th mriins h zro istortion so ψ r / n n: s n r sinh r n χ I th projction is conorml thn s r r χ so: sinh ψ s n r ( r r χ ) Thn w sol th irntil qution: ψ sinhψ r n r with th ounr conition tghψ 0 r0 to in th conorml projction:

17 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 05 r tghψ 0 n ψ tgh ψ 0 tgh On th congrunc o gosic tringls Two gosic tringls on th hproloi hing th sm ngls lso h th sm sis n r si to congrunt. This ollows immitl rom th rottions in th psuo-euclin spc, which r prss mns o ttrnions in th sm w s qutrnions r us in th rottions o Euclin spc. Howr, this lls out o th scop o this ook n will not trt. Th rr must prci, in spit o his Euclin s 7, tht ll th points on th hproloi r quilnt cus th curtur is lws th unit n th surc is lws prpniculr to th rius. So th pol (rt o th hproloi) is not n spcil point, n n othr point m chosn s nw pol proi tht th nw is r otin rom th ol ons through rottion. ommnt out th nms o th non-euclin gomtr Finll it is pprnt tht th irnt kins o non-euclin gomtr h n otn misnm. Th Lochskin surc ws improprl cll th hprolic pln in contriction with th ct tht it hs constnt curtur n thror is not lt. rtinl, in th Lochsk s gomtr th hprolic unctions r wil us, ut in th sm w s th circulr unctions work in th sphricl trigonomtr, s Lochsk himsl show. In th psuo-euclin pln th gomtr o th hprol is proprl rlis, th hprolic unctions ing lso in thr. Thus th psuo-euclin pln shoul nm proprl th hprolic pln, whil th Lochsk s gomtr might ltrntil cll hproloil gomtr. Th pln mols o th Lochsk s gomtr (ltrmi isk, Poincré isk n hl pln, tc.) must unrstoo s pln projctions o th hproloi in th psuo-euclin spc, which is its propr spc, n thn th Lochskin trigonomtr m cll hproloil trigonomtr. On th othr hn, th spc-tim ws th irst phsicl iscor o psuo- Euclin pln, ut on no ns to ppl to rltiit cus, s th lctor hs iw in this ook, oth signturs o th mtrics (Euclin n psuo-euclin) r inclu in th pln gomtric lgr n lso in th lgrs o highr imnsions. Erciss 5.. ccoring to th Lochsk s iom o prlllism, t lst two prlll lins pss through n trior point o gin lin. Th two lins pproching gin lin t 7 In ct, s r not Euclin ut r prct cmr whr imgs r projct on sphricl surc. Th principls o projction r likwis pplicl to th psuo-euclin spc. Our min ccustom to th orinr spc (w lrn its Euclin proprtis in th irst rs o our li) cis us whn w wish prci th psuo-euclin spc.

18 06 RMON GONZLEZ LVET th ininit r cll «prlll lins» whil thos not cutting n not pproching it r cll «ultrprlll» or «irgnt lins». Lochsk in th ngl o prlllism Π(s) s th ngl which orms th lin prlll to gin lin with its prpniculr, which is unction o th istnc s twn th intrsctions on th prpniculr. Pro tht Π(s) rctg(p(s)) using th strogrphic projction. 5.. Fin th quilnt o th Pthgorn thorm in th Lochskin gomtr tking right ngl tringl. Figur circl is in s th cur which is th intrsction o th hproloi with pln not pssing through th origin with slop lss thn th unit. Show tht: ) Th cur otin is n llips. ) Th istnc rom point tht w cll th cntr to r point o this llips is constnt, n m cll th rius o th circl. c) This llips is projct s circl in th strogrphic projction horoccl is th intrsction o th hproloi n pln with unit slop ( ). Show tht th horoccls r projct s circls touching th limit circl in th strogrphic projction. Fin th cntr o horoccl. 5.5 ) Show tht th irntil o r in th ltrmi projction is: u ( u ) / ) intgrting pro tht th r twn two lins orming n ngl ϕ n th common prlll lin (igur 5.0-) is qul to π ϕ. c) Pro tht th r o r tringl (igur 5.0-) is qul to its ngulr ct π α β γ. Figur Us th zimuthl conorml projction to show tht th r o circl (hproloil sgmnt) with rius ψ is qul to π (cosh ψ ). This rsult is nlogous to th r o sphricl sgmnt π ( cos θ ).

19 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER SOLUTIONS OF THE PROPOSE EXERISES. Th ctors n thir oprtions. Lt, th irnt sis o th prlllogrm n c, oth igonls. Thn: c c ( ) ( ) which r th sum o th squrs o th our sis o th prlllogrm.. ( ) ( ) ( ) ( ) Lt us pro tht c ( c ) : 4 c ( ) ( c c ) c c c c c c c c ( c c ) ( c c ) ( c c ) 4 ( c ) In th sm w th intit c ( c ) is pro..4 Using th intit o th prious rcis w h: c c c ( c ) ( c ) ( c ) ( c c c ) ( c c c c c c ) 0 0 whr th summns r simplii tking into ccount th prmutti proprt..5 Lt us not th componnts proportionl n prpniculr to th ctor with n. Thn using th ct tht orthogonl ctors nticommut n thos with th sm irction commut, w h:

20 08 RMON GONZLEZ LVET c ( ) ( c c ) ( ) ( c c ) ( ) ( c c ) ( c c c c ) ( c c c c ) ( c c ) ( ) c.6 Lt,, c th sis o tringl n h th ltitu corrsponing to th s. Thn th ltitu iis th tringl in two right tringls whr th Pthgorn thorm m ppli: h c h Isolting th ltitu s unction o th sis w otin: h ( c ) 4 Thn th squr o th r o th tringl is: h 4 4 ( c ) c c c 6 6 Introucing th smiprimtr s ( c ) w in: s ( s ) ( s ) ( s c ). s o ctors or th pln. rw, or mpl, two ctors o th irst qurnt n mrk u, u, n on th rtsin is. Thn clcult th r o th prlllogrm rom th rs o th rctngls: ( u ) ( u ) u u u u u. Th r o tringl is th hl o th r o th prlllogrm orm n two sis. Thror: ( 5 ) ( ) ( 9 0 ). c ( ) ( ) ( c c ) [ ( ) ] ( c c ) ( c c c c ) ( c c c c )

21 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 09 ( c c ) [ ( ) ] ( c c ) ( ) ( ) c.4 u ( ) ( 4 ) 6 7 u 5 α(u, ).570º'6".5 For this s w h: cos π/ sin π/ ppling th originl initions o th moulus o ctor on ins: u ( ) u 5 ( 4 ) u ( ) ( 4 ) u 548 u cos α u u sin α ± u α(u, ) 0.685º 4' 5" orint with th sns rom to. Whn n r not prpniculr, n hs not th mning o pur r ut th outr prouct with n r o. lso osr tht u u..6 In orr to in th componnts o or th nw s {u, u }, w must rsol th ctor into linr comintion o u n u. c u c u c u ( ) ( 5) 5 u u 6 c u ( 5) ( ) u u 7 so in th nw s (6, 7).. Th compl numrs. z t ( ) ( ) 6 ( 6 ) 8 4

22 0 RMON GONZLEZ LVET. Er compl numr z cn writtn s prouct o two ctors n, its moulus ing th prouct o th mouli o oth ctors: z z z z z*. Th qution 4 0 is sol trction o th ourth roots: 4,, 4.4 Pssing to th polr orm w h: ( ) n ( ) n π / 4 π / 4 n n n π / 4 n π/4 n n n π/4 n π/4 π Thror th rgumnts must qul cpt or k tims π: nπ 4 nπ n π π k π π k π n 4 k 4.5 Th thr cuic roots o r 8 π / 4, 8 π /, 8 9 π /..6 Using th ormul o th qution o scon gr w in: z, z..7 W suppos tht th compl nltic tnsion hs rl prt o th orm: sin K() with K(0) ppling th irst uch-rimnn conition, w in th imginr prt o : cos K( ) cos K( ) ppling th scon uch-rimnn conition: sin K' ( ) sin K( ) w rri t irntil qution or K() whos solution is th hprolic cosin: K' ( ) K( ) K(0) K() cosh Hnc th nlticl tnsion o th sin is:

23 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER sin( ) sin cosh cos sinh Th nlticl tnsions o th trigonomtric unctions m lso otin rom th ponntil unction. In th cs o th cosin, w h: ( )( cos sin ) p( ) ( cos sin ) p( z) p( z) p cos z cos z cos cosh sin sinh.8 Lt us clcult som ritis t z 0: () z ( p( z) ) log (0) log p () ( z) ' z ' () p( z) p() z p () () z '' z '' () ( p() z ) () z p() z ( p() z ) p( z) ( p() z ) 0 0 ''' ''' () IV () z p() z ( p() z ) p( z) 4 p(z) ( p() z ) ( p() z ) 6p( z) ( p() z ) 4 IV () 0 8 Hnc th Tlor sris is: 4 z z z () z log is irgnt. This is th singulrit nrst th origin. Thror th rius o conrgnc o th sris is π. Osr tht ( π ) log( 0).9 Sprting rctions: () z z z 8 6 ( z ) 6 ( z 4) n loping th scon rction:

24 RMON GONZLEZ LVET 6 z z 4 6 z z n n w in: () ( ) ( z ) z n 6 z 6 ( ) n 0 z 6 z 6... This Lurn sris is conrgnt in th nnulus 0 < z < 6, which contins th rquir nnulus < z < 4..0 Tking into ccount tht ( )..., th unction in th sris is: () z ( ) n n n 4 z 4 z 4 z ( ) Now w s tht t z /4 th unction hs pol. Thror th rius o conrgnc o this sris (cntr t z ) is /4 ( ) /4.. From th succssi ritis o th sins clcult t z 0, on otins th Tlor sris or sin z: sin z z z! 5 z... 5! n 0 ( ) n z n ( n )! so th Lurn sris or th gin unction is: sin z z n z z z! z! 5! n n ( ) 0 ( n ) Sinc th pol z 0 is th uniqu singulrit (s tht th nlticl tnsion o th rl sin in th rcis.7 hs no singulritis), th nnulus o conrgnc is 0< z <.. I (z) is nltic thn th riti t ch point is uniqu n w cn writ or two irnt irctions z n z :... z '(z) z '(z) ccoring to th rltionship twn th compl n ctoril plns, w cn multipl t th lt in orr to turn th compl irntils into ctors: z z Now lt us clcult th gomtric prouct o th ctor irntils: z '(z) z (z) z z [ (z)]* '(z) '(z) z z

25 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER Sinc '(z) is rl, oth gomtric proucts h th sm rgumnt, α(, ) α(z, z ), n hnc th trnsormtion is conorml. 4. Trnsormtions o ctors 4. I is th irction ctor o th rlction, th ctor ' rlct o with rspct to this irction is gin : ' I '' is otin rom ' rottion through n ngl α, n z is unitr compl numr with rgumnt α / thn: '' z ' z α α z cos sin Joining oth qutions: '' z z c c Tht is, on otins rlction with rspct nothr irction with ctor c z, th prouct o th irction ctor o th initil rlction n th compl numr with hl rgumnt. 4. Lt w'' th trnsorm ctor o w two conscuti rlctions with rspct irnt irctions u n : w'' w' u w u Thn w cn writ: w'' z w z z u z ing compl numr so tht it is quilnt to rottion with n ngl qul to th oul o tht orm oth irction ctors. 4. I th prouct o ch trnsorm ctor ' th initil ctor is qul to compl numr z (' n lws orm constnt ngl) thn: ' z ' z z* z z z z z ( z z ) which rprsnts n inrsion with rius z ollow rottion with n ngl qul to th oul o th rgumnt o z. s th lgr shows, oth lmntl trnsormtions commut.

26 4 RMON GONZLEZ LVET 5. Points n stright lins 5. I,, n r loct ollowing this orr on th primtr o th prlllogrm, thn : (, 5) (4, ) (, 4) (0, ) Th r is: ( 7 ) ( ) Th Eulr s thorm: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) This prouct onl nishs i, n r collinr. On th othr hn w s tht th prouct is th orint r o th tringl. 5. ) Th r o th tringl is th outr prouct o two sis: (4, 4) (, ) ( 4, ) (, ) ( ) sin 60 o 4 ) Th istnc rom to is th lngth o th ctor, tc: ( ) cos π/ 8 ( ) 4 ( ) 4 6 (, ) 8 (, ) 4 (, ) 5.4 si o th trpzoi is ctoril sum o th othr thr sis: ( ) whr th ct tht n ctors with th sm irction n contrr sns hs

27 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 5 n tkn into ccount. rrnging trms: ( ) Without loss o gnrlit w will suppos tht > : Sinc th ngl is th supplmnt o th ngl orm th ctors n, w h: ( ) cos whnc w otin th ngl : cos ( ) Th trpzoi cn onl ist or th rng < cos <, tht is: ( ) > > > ( ) 5.5 R ( p q ) O p P q Q RP ( p q ) OP q QP RP PQ ( p q ) OP PQ whnc it ollows tht: r RPQ ( p q ) r OPQ 5.6 Th irction ctor o th stright lin r is : (5, 4) (, ) (, 6) (, ) Th istnc rom th point to th lin r is: (, r) ( ) ( ) 0 0 Th ngl twn th ctors n is uc mns o th sin n cosin: cos 0 α sin α Thror, α π/. Th ngl twn two lins is lws compris rom π/ to π/

28 6 RMON GONZLEZ LVET cus rottion o π roun th intrsction point os not ltr th lins. Whn th ngl cs ths ounris, ou m or sutrct π. 5.7 ) Thr points, E, F r lign i th r linrl pnnt, tht is, i th trminnt o th coorints nishs. E F ( ) O P Q ( ) O P Q E E E ( ) O P Q F F F E F E F E F E F E F 0 whr th rcntric coorint sstm is gin th origin O n points P, Q (or mpl th rtsin sstm is trmin O (0, 0), P (, 0) n Q (0, )). Th trnsorm points ', E' n F' h th sm coorints prss or th s O', P' n Q'. Thn th trminnt is ctl th sm, so tht it nishs n th trnsorm points r lign. Thror n stright lin is trnsorm into nothr stright lin. ) Lt O', P' n Q' th trnsorm points o O, P n Q th gin init: O' ( o,o ) P' ( p, p ) Q' ( q, q ) n consir n point R with coorints (, ): R (, ) ( ) O P Q Thn R', th trnsorm point o R, is: R' ( ) O' P' Q' ( ) ( o, o ) ( p, p ) ( q, q ) ( ( p o ) (q o ) o, ( p o ) ( q - o ) o ) ( ', ' ) whr w s tht th coorints ' n ' o R' r linr unctions o th coorints o R: ' ( p o ) ( q o ) o ' ( p o ) ( q - o ) o In mtri orm: ' p ' p o o q q o o o o cus r linr (n non gnrt) mpping o coorints cn writtn in this

29 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 7 rgulr mtri orm, now w s tht it is lws n init. c) Lt us consir n thr non lign points,, n thir coorints: ( ) Q P O ( ) Q P O ( ) Q P O In mtri orm: Q P O crtin point is prss with coorints whthr or { O, P, Q } or {,, }: ( ) ( ) Q P O c c In mtri orm: ( ) ( ) c c Q P O ( ) ( ) c c which ls to th ollowing sstm o qutions: ( ) ( ) c c c c n init os not chng th coorints, o,, n, ut onl th point s - {O', P', Q'} inst o {O, P, Q}-. Thror th solution o th sstm o qutions or n c is th sm. Thn w cn writ: ( ) ' c ' ' c ' ) I th points, E, F n G r th conscuti rtics in prlllogrm thn: E GF G E F

30 8 RMON GONZLEZ LVET Th init prsrs th coorints prss in n s {, E, F}. Thn th trnsorm points orm lso prlllogrm: G' ' E' F' 'E' G'F' ) For n thr lign points, E, F th singl rtio r is: E F r E r F E ( r ) r F Th rtio r is coorint within th stright lin F n it is not chng th init: E' ( r ) ' r F' 'E' 'F' r 5.8 This rcis is th ul o th prolm. Thn I h copi n pst it chnging th wors or corrct unrstning. ) Thr lins, E, F r concurrnt i th r linrl pnnt, tht is, i th trminnt o th ul coorints nishs: E F ( ) O P Q ( ) O P Q E E E ( ) O P Q F F F E F E F E F E F E F 0 whr th ul coorint sstm is gin th lins O, P n Q. For mpl, th rtsin sstm is trmin O [0, 0] (lin 0), P [, 0] (lin 0) n Q [0, ] (lin 0). Th trnsorm lins ', E' n F' h th sm coorints prss or th s O', P' n Q'. Thn th trminnt lso nishs n th trnsorm lins r concurrnt. Thror n pncil o lins is trnsorm into nothr pncil o lins. ) Lt O', P' n Q' th trnsorm lins o O, P n Q th gin trnsormtion: O' [ o,o ] P' [ p, p ] Q' [ q, q ] n consir n lin R with ul coorints [, ]: R [, ] ( ) O P Q Thn R', th trnsorm lin o R, is: R' ( ) O' P' Q' ( ) [ o, o ] [ p, p ] [ q, q ] [ ( p o ) (q o ) o, ( p o ) ( q o ) o ] [ ', ' ]

31 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 9 whr w s tht th coorints ' n ' o R' r linr unctions o th coorints o R: ' ( p o ) ( q o ) o ' ( p o ) ( q - o ) o In mtri orm: ' ' o o o q o p o q o p cus n linr mpping (non gnrt) o ul coorints cn writtn in this rgulr mtri orm, now w s tht it is lws n init. c) Lt us consir n thr non concurrnt lins,, n thir coorints: ( ) Q P O ( ) Q P O ( ) Q P O In mtri orm: Q P O crtin lin is prss with ul coorints whthr or {O, P, Q } or {,, }: ( ) ( ) Q P O c c In mtri orm: [ ] [ ] c c Q P O [ ] [ ] c c

32 0 RMON GONZLEZ LVET which ls to th ollowing sstm o qutions: ( c) ( c) c c n init os not chng th coorints, o,, n, ut onl th lins s -{O', P', Q'} inst o {O, P, Q}-. Thror th solution o th sstm o qutions or n c is th sm. Thn w cn writ: Figur 6. ( c) ' ' c ' ' ) Th init mps prlll points into prlll points (points lign with th point (/, /), point t th ininit in th ul pln). I th lins, E, F n G r th conscuti rtics in ul prlllogrm (igur 6.) thn: E GF G E F Whr E is th ul ctor o th intrsction point o th lins n E, n GF th ul ctor o th intrsction point o G n F. Oiousl, th points EF n G r lso prlll cus rom th ormr qulit it ollows: EF G Th init prsrs th coorints prss in n s {, E, F}. Thn th trnsorm lins orm lso ul prlllogrm: G' ' E' F' 'E' G'F' ) For n thr concurrnt lins, E, F th singl ul rtio r is: E F r E r F E ( r ) r F Th rtio r is coorint within th pncil o lins F n it is not chng th init: E' ( r ) ' r F' 'E' 'F' r ) Lt us consir our concurrnt lins,, n with th ollowing ul coorints prss in th lins s {O, P, Q}: [, ] [, ] [, ] [, ]

33 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER Thn th irction ctor o th lin is: ( ) Q P O ut th irction ctors o th s ulils: 0 Q P O Thn: ( ) ( ) Q P n nlogousl: ( ) ( ) Q P ( ) ( ) Q P ( ) ( ) Q P Th outr prouct is: ( ) ( ) ( ) Q P Thn th cross rtio onl pns on th ul coorints ut not on th irction o th s ctors (s chptr 0): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Hnc it rmins inrint unr n init. Ech outr prouct cn writtn s n outr prouct o ul ctors otin sutrction o th coorints o ch lin n th ininit lin: L L,, whr L L is th ul ctor going rom th lin L t th ininit to th lin, tc. Thn w cn writ this usul ormul or th cross rtio o our lins:

34 RMON GONZLEZ LVET ( ) L L L L L L L L This rcis links with th sction Projcti cross rtio in th chptr To otin th ul coorints o th irst lin, sol th intit: ' ( ) ' c' ' ' 0 c' 0 c [0, /] For th scon lin: ' ( ) ' c' ' ' 4 c' c [4/9, /9] 9 oth lins r lign in th ul pln with th lin t th ininit (whos ul coorints r [/, /]) sinc th trminnt o th coorints nishs: / / 9 / 0 4 / 9 / / / 9 / 0 Thror th r prlll (this is lso triil rom th gnrl qutions). 5.0 Th point (, ) is th intrsction o th lins 0 n 0, whos ul coorints r: 0 [/5, /5] 0 [/, 0] Th irnc o ul coorints gis ul irction ctor or th point: [/, 0] [/5, /5] [/0, 4/0] n hnc w otin th ul continuous n gnrl qutions or th point: / c 4 4 c Th point (, ) is th intrsction o th lins 0 n 0, whos ul coorints r:

35 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 0 [4/0, /0] 0 [/4, /4] Th irnc o ul coorints gis ul irction ctor or th point: w [/4, /4] [4/0, /0] [ /0, 4/0] n hnc w otin th ul continuous n gnrl qutions or th point: / 4 / 4 c c 5 Not tht oth ul irction ctors n w r proportionl n th points r prlll in ul sns. Hnc th points r lign with th cntroi (/, /) o th coorint sstm. 6. ngls n lmntl trigonomtr 6. Lt, n c th sis o tringl tkn nticlockwis. Thn: c 0 c c ( ) c cos(π γ ) cos γ (lw o cosins) Th r s o tringl is th hl o th outr prouct o n pir o sis: s c c sin(π γ ) c sin(π α) c sin(π β ) sinγ sinα sinβ (lw o sins) c From hr on quickl otins: sin α sinβ sinα sinα sinβ sinβ iiing oth qutions n introucing th intitis or th ition n sutrction o sins w rri t:

36 4 RMON GONZLEZ LVET α β α β sin cos sinα sinβ sinα sinβ α β α β cos sin α β tg α β tg (lw o tngnts) 6. Lt us sustitut th irst cosin th hl ngl intit n conrt th ition o th two lst cosins into prouct: α β β α β γ α cos( α β ) cos( β γ ) cos( γ α ) cos cos cos Lt us trct common ctor n conrt th ition o cosins into prouct: α β cos α β α γ cos cos β α β α γ γ β cos cos cos α β β γ γ α cos cos cos Th intit or th sins is pro in similr w. 6. Using th Moir s intit: cos4α ( cosα α ) 4 sin 4α sin tr loping th right hn si w in: sin 4α 4 cos α sinα 4 cosα sin α cos 4α cos 4 α 6 cos α sin α sin 4 α n iiing oth intitis: tg 4 4 tgα 4 tg α 6 tg α tg α α Lt α, β n γ th ngls o th tringl P with rtics, n P rspctil. Th ngl γ mrcing th rc is constnt or n point P on th rc. th lw o sins w h: P P sinα sinβ P sinγ

37 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 5 Th sum o oth chors is: P P P sin α sinβ sinγ onrting th sum o sins into prouct w in: P α β α β sin cos P P P sinγ π sin γ α β cos sinγ Sinc γ is constnt, th mimum is ttin or cos(α / β /), tht is, whn th tringl P is isoscls n P is th mipoint o th rc. 6.5 From th sin thorm w h: c α β α β α β sin cos cos sinα sinβ sinγ γ γ γ sin cos sin Following th sm w, w lso h: c α β α β α β cos sin sin sinα sinβ sinγ γ γ γ sin cos cos 6.6 Tk s th s o th tringl n rw th ltitu. It iis in two sgmnts which r th projctions o th sis n c on : cosγ c sinβ n so orth. 6.7 From th oul ngl intitis w h: cosα cos α sin α sin α cos α From th lst two prssions th hl-ngl intitis ollow: α sin ± cosα α cosα cos ± Mking th quotint:

38 6 RMON GONZLEZ LVET tgα ± cosα cosα sinα cosα sinα cosα 7. Similritis n singl rtio 7. First t ll rw th tringl n s tht th homologous rtics r: P ( 0, 0 ) P' ( 4, ) Q (, 0 ) Q' (, 0 ) R ( 0, ) R' ( 5, ) PQ QR RP P'Q' Q'R' R'P' r P'Q' PQ Q'R' QR R'P' RP 5 π / 4 Th siz rtio is r 5π/4. n th ngl twn th irctions o homologous sis is 7. Lt right tringl ing th right ngl. Th ltitu cutting th s in splits in two right ngl tringls: n. In orr to simpli I introuc th ollowing nottion: c c Th tringls n r oppositl similr cus th ngl is common n th othr on is right ngl. Hnc: ( c )* c c Th tringls n r lso oppositl similr, cus th ngl is common n th othr on is right ngl. Hnc: ( c ) ( c )* c c ( c ) Summing oth rsults th Pthgorn thorm is otin: c ( c ) c c 7. First t ll w must s tht th tringls M n M r oppositl similr. Firstl, th shr th ngl M. Sconl, th ngls M n M r qul cus th mrc th sm rc. In ct, th limiting cs o th ngl whn mos to is th ngl M. Finll th ngl M is qul to th ngl M cus th sum o th ngls o tringl is π. Th opposit similrit implis: M M M M

39 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 7 M M M M Multipling oth prssions w otin: M M M M Lt Q th intrsction point o th sgmnts P n. Th tringl PQ is irctl similr to th tringl P n oppositl similr to th tringl Q: P Q P Q P P Q P P Q P P Q P P Q P P Summing oth prssions: Q Q ( P P ) P Th thr sis o th tringl r qul; thror: P P P 7.5 Lt us irstl clcult th homotht rtio k, which is th quotint o homologous sis n th similrit rtio o oth tringls: '' k Sconl w clcult th cntr o th homotht isoltion rom: O' O k O' O O ( k ) ' ( k ) O Sinc th sgmnt ' is known, this qution llows us to clcult th cntr O : O ' ( k ) ' ( '' ) 7.6 rw th lin pssing through P n th cntr o th circl. This tn imtr cuts th circl in th points R n R'. S tht th ngls R'RQ n QQ'R' r supplmntr cus th intrcpt opposit rcs o th circl. Thn th ngls PRQ n R'Q'P r qul. Thror th tringl QRP n R'Q'P r oppositl similr n w h: Figur 6. PR PQ PR' PQ' PQ PQ' PR PR' Sinc PR n PR' r trmin P n th circl, th prouct PQ PQ' (th powr o P) is constnt inpnntl o th lin PQQ'. 7.7 Th isctor o th ngl iis th tringl c in two tringls, which r

40 8 RMON GONZLEZ LVET oiousl not similr! Howr w m ppl th lw o sins to oth tringls to in: m sin sin m n sin sin n Th ngls n n m r supplmntr n sin n sin m. On th othr hn, th ngls n r qul cus o th ngl isctor. Thror it ollows tht: m n 8. Proprtis o th tringls 8. Lt, n th rtics o th gin tringl with nticlockwis position. Lt S, T n U th rtics o th thr quiltrl tringls rwn or th sis, n rspctil. Lt P, Q n R th cntrs o th tringls S, T n U rspctil. Th si U is otin rom through rottion o π/: U t with π t π / cos π sin R is / o th ltitu o th quiltrl tringls U ; thror is / o th igonl o th prlllogrm orm n U: U R ( t) Th sm rgumnt pplis to th othr quiltrl tringls: Q ( t) P ( t) From whr P, Q n R s unctions o, n r otin: P ( t) Q ( t) R ( t) Lt us clcult th ctor PQ: ( )( t) PQ Q P Introucing th cntroi: G PQ G ( t ) G G t nlogousl:

41 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 9 QR G G t RP G G z Now w ppl rottion o π/ to PQ in orr to otin QR: PQ t ( G G t ) t G t G t Sinc t is thir root o th unit ( t t 0) thn: PQ t G t G G t G t G t G t G G t GG G G t G G t QR Through th sm w w in: QR t RP RP t PQ Thror P, Q n R orm n quiltrl tringl with cntr in G, th cntroi o th tringl : P Q R ( t ) G 8. Th sustitution o th cntroi G o th tringl gis: PG ( ) P P P ( ) P P ( ) ( P) ( P) ( P) ( ) ( ) ( ) P P P P P P [ ] ( ) 8. ) Lt us clcult th r o th tringl G: G [ ( ) c ] [ ( c ) c ] ( c ) G Th proo is nlogous or th tringls G n G. ) Lt us lop PG ollowing th sm w s in th rcis 8.:

42 0 RMON GONZLEZ LVET PG [P ( c ) ] P P ( c ) ( c ) ( c ) P P P c P c c c ( P ) ( P ) c ( P ) ( ) ( ) c ( c ) c c P P c P ( c ) ( c ) c ( ) c c P P c P ( ) c ( ) c ( ) P P c P c c Th Liniz s thorm is prticulr cs o th pollonius lost thorm or c /. 8.4 Lt us consir th rtics,, n orr clockwis on th primtr. Sinc P is turn π/, Q is turn π/, tc, w h: z cos π/ sin π/ P z Q z R z S z PR QS ( P R ) ( Q S ) [ ( ) z ] [ ( ) z ] [ ( ) z ] [ ( ) z ] [ ( ) z ] [ ( ) z ] [ ( ) z ] [ ( ) z ] Th innr prouct o two ctors turn th sm ngl is qul to tht o ths ctors or th rottion. W us this ct or th irst prouct. lso w must lop th othr proucts in gomtric proucts n prmut ctors n th compl numr z: PR QS ( ) ( ) ( z z * ) Thror PR QS 0. Th sttmnt ) is pro through n nlogous w.

43 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 8.5 Th min is th sgmnt going rom rt to th mipoint o th opposit si: m I E is th intrsction point o th isctor o with th lin prlll to th isctor o, thn th ollowing qulit hols: E whr, r rl. rrnging trms w otin ctoril qulit: Th linr composition ils tr simpliiction: In th sm w, i is th intrsction o th isctor o with th lin prlll to th isctor o w h: c whr c n r rl. rrnging trms: c tr simpliiction w otin: Now w clcult th ctor E:

44 RMON GONZLEZ LVET E E Thn th irction o th lin E is gin th ctor : Whn th ctor hs th irction, th scon summn nishs, n th tringl coms isoscls. 8.7 Lt us inict th sis o th tringl with, n c in th ollowing orm: c Suppos without loss o gnrlit tht P lis on th si n Q on th si. Hnc: P k ( k ) P k k Q l ( l ) Q l l whr k n l r rl n 0 < k, l <. Now th sgmnt PQ is otin: PQ k l Sinc th r o th tringl PQ must th hl o th r o th tringl, it ollows tht: P Q ( k ) ( l ) k l Th sustitution into PQ gis: PQ k k ) I u is th ctor o th gin irction, PQ is prpniculr whn PQ u 0 which rsults in: k u 0 k whnc on otins:

45 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER k u u PQ u u u u In this cs th solutions onl ist whn u n u h irnt signs. Howr w cn lso choos th point P (or Q) ling on th nothr si c, wht gis n nlogous solution contining th innr prouct c u. Not tht i u n u h th sm sign, thn c u h th opposit sign cus: c 0 c u u u wht wrrnts thr is lws solution. ) Lt us clcult th squr o PQ: PQ k 4k quting th riti to zro, on otins th lu o k or which PQ is minimum: k Thn w otin th sgmnt PQ n its lngth: PQ PQ c) Er point m writtn s linr comintion o th thr rtics o th tringl th sum o th coicints ing qul to th unit: R ( ) R whr ll th coicints r compris twn 0 n : 0 <,, < R R Now th point R must li on th sgmnt PQ, tht is, P, Q n R must lign. Thn th trminnt o thir coorints will nish: 0 k k 0 k k t k k ( P, Q, R) l 0 l 0 0

46 4 RMON GONZLEZ LVET k k 0 ± 8 k n 4 m 8 l k 4 Thr is onl solution or positi iscriminnt: > 8 Th limiting cur is n quiltrl hprol on th pln -. Sinc th tringl m otin through n init, th limiting cur or th tringl is lso hprol lthough not quiltrl. I th trms P n Q coul mo long th prolongtions o th sis o th tringl without limittions, this woul th uniqu conition. Howr in this prolm th point P must li twn n, n Q must li twn n. It mns th itionl conition: ± 8 < k < < l < < < 4 From th irst solution (root with sign ) w h: < 8 < 4 onl isting or >/4. For /4 < < / th lt hn si is ngti so tht th uniqu rstriction is th right hn si. squring it w otin: ( 4 ) 8 < 0 < or 4 < < For >/ th right hn si is highr thn so tht th uniqu rstriction is th lt hn si: ( ) < 8 < 0 or > For th scon solution (root with sign ) w h: < 8 < 4 onl isting or < /. For /4 < < / th right hn si is positi so tht th uniqu rstriction is th lt hn si: ( ) > 8 > 0 or < < 4 For < /4 oth mmrs r ngti n tr squring w h:

47 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 5 ( ) > 8 > ( 4 ) > 0 > or ll ths conitions r plott in th igur 6.. < 4 Figur 6. It is sil pro tht th limiting lins touch th hprol t th points (/4, /) n (/, /4). For oth tringls pint with light gr thr is uniqu solution. In th rgion pint with rk gr thr r two solutions, tht is, thr r two sgmnts pssing through th point R n iiing th tringl in two prts with qul r. mns o n init th plot in th igur 6. is trnsorm into n tringl (igur 6.4). Th init is linr trnsormtion o th coorints, which conrts th limiting lins into th mins o th tringl intrscting t th cntroi n th hprol into nothr non quiltrl hprol: Figur 6.4 Osr tht th mins ii th tringl in si tringls. I point R lis outsi th show tringls, w cn lws support th trms o th sgmnt PQ pssing

48 6 RMON GONZLEZ LVET through R in nothr pir o sis. This mns tht th iision o th tringl is lws possil proi o th suitl choic o th sis. On th othr hn, i w consir ll th possiilitis, th points nighouring th cntroi mit thr sgmnts. 9. ircls 9. ( M ) ( M ) ( M ) k M M ( ) k Introucing th cntroi G ( ) / : ( ) k M M G G 9 ( M G) k ( ) ( ) ( ) 9 k 9 GM r M runs on circumrnc with rius r cntr t th cntroi o th tringl whn k is highr thn th rithmtic mn o th squrs o th thr sis: k 9. Lt n th i points n P n point on th srch gomtric locus. I th rtio o istncs rom P to n is constnt it hols tht: P k P P k P with k ing rl positi numr P P k ( P P ) ( k ) P ( k ) P k P P ( k k ) k k k W inict O th prssion: O. ing O to oth mmrs w otin: k OP k ( k ) ( ) k k

49 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 7 OP k ( ) k ( k ) ( k ) Thror, th points P orm circumrnc cntr t O, which is point o th lin, with rius: OP k k 9. Lt us rw th circumrnc circumscri to th tringl (igur 6.5) n lt n point on this circumrnc. Lt P, Q n R th orthogonl projctions o P on th sis, n rspctil. Not tht th tringls P n R r similr cus o π R. Th similrit o oth tringls is writtn s: R P Figur 6.5 Th tringls Q n R r lso similr cus o th qulit o th inscri ngls : R Q Sinc th ngls R P n R Q, th isctor o th ngl Q is lso isctor o th ngl P n R, ing its irction ctor: R Q R Q P P Lt P', Q' n R' th rlct points o P, Q n R with rspct to th isctor: R' R Multipling w h: R' R R which is n pct rsult sinc th rlction o R with rspct th isctor o th ngl R ils lws point R' lign with n. nlogousl or n w in tht th points, n Q' r lign n lso th points, n P': Q' Q R Q R R

50 8 RMON GONZLEZ LVET P' P R P R R Now w s tht th points R', Q' n P' r th trnsorm o, n unr n inrsion with cntr : R' Q' P' Sinc lis on th circumrnc pssing through, n, th inrsion trnsorms thm into lign points n th rlction prsrs this lignmnt so tht P, Q n R r lign. 9.4 I m n n c thn: ( ) ( c) ( )( c c c ) m n c c c c c c 4 c c c c c c Tking into ccount tht ( c) thn: ( c) c c c c c so w m rwrit th ormr qulit: m n c c c c c c c W ppl th prmutti proprt to som trms: m n c c c c c c c c in orr to rri t ull smmtric prssion: m n c c c Now w prss th prouct o ch pir o ctors using th ponntil unction o thir ngl: m n c c ( p( ( π γ α ) ) p( ( π β δ ) )) ing th rgumnts o th ponntil, simpliing n tking into ccount tht α β γ δ π. m n c c cos ( α γ )

51 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER Lt, n th mipoints o th sis o n tringl PQR : Q R R P P Q Th cntr o th circumrncs R, P n Q will not s, E n F rspctil. Th circumrnc P is otin rom th circumscri circumrnc PQR mns o homotht with cntr P. Thn i O is th circumcntr o th tringl PQR, th cntr o th circumrnc P is loct t hl istnc rom R: P O E PQ OR E Q O F PQ OR F PQ EF Lt Z th intrsction o E with F. Thn: ( ) E ( )F Z rrnging trms w in: E F EF Th linr composition gis: EF F E F E EF E F EF Z E F F F E F E F E P Q R O 4 which is inrint unr cclic prmuttion o th rtics P, Q n R. Thn th lins E, F n intrsct in uniqu point Z. On th othr hn, th point Z lis on th Eulr s lin (igur 9.8): Figur 9.8 G O Z W must pro tht th inrsion chngs th singl rtio o thr r clos points its conjugt lu, cus th orinttion o th ngls is chng. Lt us consir n inrsion with cntr O n rius r n lt th points ', ', ' th trnsorm o,, unr this inrsion. Thn: O' r O O' r O O' r O '' O' O' r (O O ) r O (O O) O

52 40 RMON GONZLEZ LVET r O O In th sm w: '' r O O Lt us clcult th singl rtio o th trnsorm points: ( ', ', ' ) '' '' O O O O O O ( O ) ( O ) Whn n com nr, n tn to zro n th singl rtio or th inrs points coms th conjugt lu o tht or th initil points: lim, ( ', ', ' ) O O (,, ) * 0. ross rtios n rlt trnsormtions 0. Lt quriltrl inscri in circl. W must pro th ollowing qulit: which is quilnt to: Now w inti ths quotints with cross rtios o our points,, n on circl: ( ) ( ) qulit tht lws hols cus i ( ) r thn: r r r Whn th points o not li on circl, th cross rtio is not th quotint o mouli ut compl numr: p[ ( ) ] p[ ( ) ] ccoring to th tringulr inqulit, th moulus o th sum o two compl numrs is lowr thn or qul to th sum o oth mouli:

53 TRETISE OF PLNE GEOMETRY THROUGH GEOMETRI LGER 4 0. I,, n orm hrmonic rng thir cross rtio is : In orr to isolt w must writ s ition o n : ( ) ( ) ( ) ( ) 0. Th homogrph prsrs th cross rtio: '' '' '' '' which m rwrittn s: '' '' ( '' '' ) ( '' '' ) ( ) ( ) Whn n pproch to, th ctors,, '' n '' tn to zro. So th singl rtio o thr r clos points rmins constnt: lim '' '' lim,, n thror th ngl twn tngnt ctors o curs. So th homogrph is irctl conorml trnsormtion. 0.4 I F is point on th homolog is w must pro tht: { F, } { F, ' ' ' ' } s w h sn, point ' homologous o is otin through th qution: O' O [ r F ( FO ) ] whr O is th cntr o homolog, F point on th is, th irction ctor o th is n r th homolog rtio. Thn th ctor F' is: F' FO O' FO O [ r F ( FO ) ]

54 4 RMON GONZLEZ LVET [ FO ( r F ( FO ) ) O ] [ r F ( FO ) ] [ F r FO F ( FO ) ] [ r F ( FO ) ] nlogousl: F' [ F r FO F ( FO ) ] [ r F ( FO ) ] F' [ F r FO F ( FO ) ] [ r F ( FO ) ] F' [ F r FO F ( FO ) ] [ r F ( FO ) ] Th prouct o two outr proucts is rl numr so tht w must onl p ttntion to th orr o th irst loos ctors F n FO, tc: F' F' [ F F r ( F FO F FO F F ) ( FO ) ] [ r F ( FO ) ] [ r F ( FO ) ] Using th intit o th rcis.4, w h: F' F' [ F F r F F FO ( FO ) ] [ r F ( FO ) ] [ r F ( FO ) ] F F ( r ) [ r F ( FO ) ] [ r F ( FO ) ] In th projcti cross rtio ll th ctors cpt th irst outr prouct r simplii: F' F' F' F' F' F' F' F' F F F F F F F F 0.5 ) Lt us pro tht th spcil conorml trnsormtion is iti: OP' ( OP ) OP' OP OP'' ( OP' w ) ( OP w ) ) Lt us trct th ctor OP rom OP': OP' ( OP ) OP OP' OP ( OP ) Thn w h: O' O ( O ) O' O ( O ) O' O ( O ) O' O ( O )