a beckman-quarles type theorem for linear fractional transformations of the extended double plane
Post on 06-Apr-2018
229 Views
Preview:
TRANSCRIPT
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
1/22
Vh}o+@uaknb_blovcvnlun|o
Kn|`okn|gm}Jhuvbna
]thb}hvol fw
Vh}o+@uaknb Gb}|g|u|o hi \om`bhahcw
Lotnv|kob| hi Kn|`okn|gm}
\ovvo @nu|o! GB >
nMnaygb Mhaaoco! nblvos"juaagnb"kg}Dnaukbg"mnaygb"olufMnaygb Mhaaoco! jeeogaknbDcknga"mhk
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
2/22
Vh}o+@uaknb _blovcvnlun|o Kn|`okn|gm} Jhuvbna
Yhauko >6! Bh" 6! Inaa 64>>
N Fomeknb+^unvao} |wto |`ohvok ihvagbonv ivnm|ghbna |vnb}ihvkn|ghb} hi |`o
op|oblol lhufao tanbo
Nblvos Juaagnb Kg} Jh}` Eogaknb
Nf}|vnm|" Gb |`g} tvo}ob|n|ghb! so mhb}glov |`o tvhfaok hi m`nvnm|ovgzgbc knt}
|`n| tvo}ovyo tngv} hi vgc`| `wtovfhan} hv agbo} gb |`o op|oblol lhufao tanbo s`h}o
`wtovfhagm nbcao hi gb|ov}om|ghb g} zovh" So mhb}glov |sh lg}jhgb| }tnmo} hi vgc |
`wtovfhan} nbl agbo} gb |`o op|oblol lhufao tanbo @% nbl @ nbl tvhyo |`n|fgjom|gyo knttgbc} hb |`o vo}tom|gyo }tnmo} |`n| tvo}ovyo |nbcobmw fo|soob tngv} hi
`wtovfhan} hv agbo} ku}| fo gblumol fw n agbonv ivnm|ghbna |vnb}ihvkn|ghb"
Nmebhsaolcokob|}: \`o vo}onvm tvo}ob|ol gb |`g} nv|gmao sn} }utthv|ol fw |`o Bn|ghbna
]mgobmo Ihubln|ghb ublov Cvnb| Bh" LK]+>446
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
3/22
Tnco >>9 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
> Gb|vhlum|ghb
Fomeknb nbl ^unvao} tvhyol |`n| nbw knttgbc ivhk Vb |h g|}oai! b 6! |`n| tvo}ovyo}n pol lg}|nbmo fo|soob |sh thgb|} g} bomo}}nvgaw n vgcgl kh|ghb U>[" \`g} |`ohvok sn}
nkhbc |`o v}| vo}ua|} hi s`gm` nvo bhs mnaaol m`nvnm|ovgzn|ghb} hi cohko|vgmna knttgbc}ublov kgal `wth|`o}o} U: Khfgu} |vnb}ihvkn|ghb} tvo}ovyo |`o }tnmo hi mgvmao}nbl agbo} nbl nbcao} hi gb|ov}om|ghb"
So mhb}glov |`o nbnahchu} tvhfaok ihv |`o op|oblol lhufao tanbo T ; T @! s`ovoT ;
up % we : p! w
V! e6 ; >
{nbl @ ;
u(
e'> :
V
u{{(}oo ]om|ghb 6"6'"
N} gb mhktaop nbnaw}g}! so ihubl |`n| gb |`o op|oblol lhufao tanbo! agbonv ivnm|ghbna |vnb}+ihvkn|ghb} tvo}ovyo fh|` |`o }tnmo hi yov|gmna vgc`| `wtovfhan} nbl agbo} sg|` }ahto} cvon|ov|`nb +> nbl ao}} |`nb >! lobh|ol @%! nbl |`o }tnmo hi `hvgzhb|na vgc`| `wtovfhan} nblagbo} sg|` }ahto} ao}} |`nb +> hv cvon|ov |`nb >! lobh|ol @" Iuv|`ovkhvo! |`o `wtovfhagmnbcao hi gb|ov}om|ghb `>`6 ihv nbw tngv vgc`| `wtovfhan} hv agbo} `>! `6 gb @
% hv @ g} tvo+}ovyol fw agbonv ivnm|ghbna |vnb}ihvkn|ghb}" Ihaahsgbc Ao}|ov} khloa! so tvhyo |`o ihaahsgbcFomeknb+^unvao} |wto |`ohvok"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
4/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >>3
\`ohvok 6" Ao|\ fo n fgjom|gyo knttgbc ivhk@% (vo}t" @' |h g|}oai }um` |`n|! ihv naa`>! `6 @% (vo}t" @'!
`>`6 ; 4 gi nbl hbaw gi \(`>'\(`6' ; 4"
\`ob \ g} gblumol hb@% (vo}t" @' fw n agbonv ivnm|ghbna |vnb}ihvkn|ghb hi T"
z ; p % we s ; u % ye
Igcuvo 6: Agbonv ivnm|ghbna |vnb}ihvkn|ghb} hi |`o op|oblollhufao tanbo tvo}ovyo yov|gmna nbl `hvgzhb|na vgc`| `wtovfhan}nbl agbo} nbl `wtovfhagm nbcao} hi gb|ov}om|ghb"
6 Cohko|vw gb |`o Op|oblol Lhufao TanboT
6"> \`o Lhufao Tanbo T
Lobg|ghb >" N lhufao bukfov> g} n ihvkna optvo}}ghb p % we s`ovo p! w V nble6 ; >! fu| e ( V (e g} ebhsb n} n ubgth|ob|'" Oyovw lhufao bukfov z ; p % we `n} n vonamhkthbob| Vo(z' ; p! n lhufao mhkthbob| Gk(z' ; w! nbl mhbjucn|o z ; p we"
\`o lhufao tanbo T g} |`o }o| hi naa lhufao bukfov}! s`gm` so mnaa thgb|}" \ g} knwfo |`huc`| hi n} n |sh+lgkob}ghbna yom|hv }tnmo hyov V! s`ovo onm` lhufao bukfov p % wemhvvo}thbl} |h |`o yom|hv (p! w' gb V6 sg|`
Nllg|ghb: (p>! w>' % (p6! w6' ; (p> % p6! w> % w6'5
]mnanv Kua|gtagmn|ghb: m(p>! w>' ; (mp>! mw>'5 nbl
Kua|gtagmn|ghb: (p>! w>' (p6! w6' ; (p>p6 % w>w6! p>w6 % p6w>'">\`o bnko lhufao bukfov sn} u}ol fw Wncahk" H|`ov} `nyo mnaaol |`ok tovtaop! }tag|+mhktaop! }tnmo|gko!
hv `wtovfhagm bukfov}"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
5/22
Tnco >>0 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
\`g} snw so mnb }oo |`n| lhufao bukfov} ihvk n mhkku|n|gyo nacofvn hyov V"G| g} }|vngc`|ihvsnvl |h yovgiw |`n| ihv z ; p % we!
z> loi
;
>
z ;
z
zz ;
p
we
p6 w6
g} |`o kua|gtagmn|gyo gbyov}o hi z s`oboyov p ; w" \`ovoihvo! s`gao T g} n mhkku|n|gyovgbc! g| g} bog|`ov n oal bhv oyob nb gb|ocvna lhkngb! fomnu}o oyovw bhbzovh bukfov sg|`ihvk e! V! g} n zovh+lgyg}hv"
\`o lhufao khluau} hi z ; p % we g}
qzqT loi;
qzzq ;
qp6 w6q"
\`g} g} mhb}glovol |`o lhufao lg}|nbmo hi |`o thgb| z ivhk |`o hvgcgb"
Bukfov} sg|` ihvk e nvo gb n }ob}o g}h|vhtgm! }gbmo q eqT ; 4! oyob s`ob ; 4" ]h |`ovo nvo thgb|} z>! z6 T! s`ovo z> ; z6! }um` |`n| qz> z6qT ; 4" \`ovoihvo!}|vgm|aw }tonegbc! |`o lhufao lg}|nbmo cgyob fw |`o khluau} g} bh| n ko|vgm" Fu| g| cgyo} ncohko|vw hb V6 |`n| g} ~ug|o lgfflovob| ivhk |`o Oumaglonb cohko|vw hi |`o mhktaop tanbo!s`ovo qzqM ;
p6 % w6 ; 4 gi nbl hbaw gi z ; 4"
Gb |`o mhktaop tanbo! |`o }o| hi naa thgb|} z ; p % wg M }n|g}iwgbc qzqM ; v 2 4 ihvk}n mgvmao sg|` vnlgu} v" ]gkganvaw! so `nyo |`n| |`o }o| hi naa thgb|} z T sg|` qzqT ; 2 4ihvk} n ihuv+fvnbm`ol vgc`| `wtovfhan sg|` }okg+npo} hi aobc|` ! s`h}o n}wkt|h|o} nvo |`og}h|vhtgm agbo} w ; p" N} |`o tnvnko|ov gbmvon}o}! 1 1 ! onm` fvnbm` g} lvnsbopnm|aw hbmo" ]oo Igcuvo ="
v % 4g
w
p
qzqM ; v 2 44 1 6
qzqT ; 2 44 1
4 % e
4 e
% 4e % 4e
w
p
Igcuvo =: \`o v+mgvmao nbl |`o +vgc`| `wtovfhan
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
6/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >>7
\`o lhufao nvcukob| hi z ; p % we g} nvc(z'loi; s`ovo
; |nb`> w
p: qwq 1 qpq
|nb`> pw
:
qw
q2
qp
qublobol : qwq ; qpq"
G| g} gb|ovo}|gbc |h bh|o |`n| v 2 4 nbl pol 4 1 lo|ovkgbo n ubg~uo thgb| gbM! s`ovon} 2 4 nbl V lo|ovkgbo ihuv thgb|} gb T" \`ovoihvo! |`o lhufao bukfovyov}ghb hi thanv ihvk lho} bh| naahs ihv ubg~uo votvo}ob|n|ghb hi thgb|}" Ihv khvo hb lhufaobukfov}! }oo U7! 0! >4["
6"6 @wtovfhan} gb |`o Op|oblol Lhufao Tanbo T"
\`o op|oblol lhufao tanbo T g} |`o ubghb T@! s`ovo @ ; u( e'> : Vu{{"So }hko|gko} voiov |h |`o thgb|} gb @
n} |`o thgb|} n| gbbg|w" \`g} }o| knw fo |`huc |
hi n} |sh agbo} n| gbbg|w |`n| gb|ov}om| n| (4 % 4e'>"N `wtovfhan ` gb @% hv @ g} |`o }uf}o| hiT |`n| gbmaulo} oyovw thgb| z ; p % we T
}n|g}iwgbcNzz % VoU(F % Me'z[ % L ; 4! (>'
s`ovo N!F!M!L V! " Ao|` ; UN : F : M : L[! s`ovo N!F!M!L V" \`ob(g' ` @% gi nbl hbaw gi 5
(gg' ` @ gi nbl hbaw gi "N `wtovfhan ` ; UN : F : M : L[ na}h gbmaulo} thgb|(}' n| gbbg|w" _}gbc agbonv ivnm|ghbna
|vnb}ihvkn|ghb}! so mnb yovgiw |`o thgb|(}' gb @ s`gm` ` gb|ov}om|}"
(g' (> % >e'> s`ovo > ;
NFM : F ; M : F ; M
(gg' (6 6e'> s`ovo 6 ; N
F%M: F ; M
: F ; M(Bh|gmo |`n| (4 % 4e'> ; (4 4e'>! fu| ( % e'> ; ( e'>"' GiN ; 4! |`ob `g} n yov|gmna hv `hvgzhb|na vgc`| `wtovfhan nbl gbmaulo} opnm|aw |sh thgb|} n| gbbg|w" Fu| giN ; 4! |`ob ` g} n agbo nbl hbaw gbmaulo} hbo thgb| n| gbbg|w" Khvohyov! ` g} n agbo gi nblhbaw gi ` (4 % 4e'>"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
7/22
Tnco >64 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
_}gbc }|ovohcvnt`gm tvhjom|ghb! |`o op|oblol lhufao tanbo mnb fo ygosol n} nb gbbg|o`wtovfhahgl" Mhb}glov |`o `wtovfhahgl p6 w6 % (z >'6 ; >! s`ovo |`o pw+tanbo g} |`olhufao tanbo! nbl |neo |`o thgb| (4! 4! 6' n} |`o tvhjom|ghb thgb|" Nbw agbo lvnsb |`vhuc` nthgb| hb |`o op|oblol lhufao tanbo nbl |`o tvhjom|ghb thgb| gb|ov}om|} |`o `wtovfhahgl n| n
}omhbl thgb|" @wtovfhan} gb @% mhvvo}thbl sg|` tanbnv mvh}} }om|ghb} hi |`o `wtovfhahgl|`n| |neo |`o ihvk hi oaagt}o} nbl `wtovfhan}" N }omhbl `wtovfhahgl p6 % w6 % (z >'6 ; >mhvvo}thbl} sg|` `wtovfhan} gb @"
Igcuvo " \`o }o| hi lgvom| nblgblgvom| agbonv ivnm|ghbna |vnb}ihvkn|ghb} hi |`o op|oblol lhufao bukfov tanbo ihvk n cvhut
ublov mhkth}g|ghb" So lobh|o |`g} cvhut fw AI\(T
'"Nbw lgvom| agbonv ivnm|ghbna |vnb}ihvkn|ghb g} mhkth}ol hi n| kh}| ihuv hi |`o ihaahsgbc}gktao |vnb}ihvkn|ghb}"
\vnb}an|ghb: (z' ; z % f! ihv f T Vh|n|ghb nbl Lgan|ghb: (z' ; nz! ihv n T! n ; e! s`ovo V Gbyov}ghb: (z' ; >
z
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
8/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >6>
\`o iuaa cvhut g} hf|ngbol fw gbmaulgbc mhbjucn|ghb z z" Fw iuv| ov vo}|vgm|gbc nl fm ; >! so bl |`n| AI\(T' g} `hkhkhvt`gm |h |`o cvhut ]A(6!T' sg|` |`o |sh+|h+hbomhvvo}thblobmo nz%f
mz%l! nz%f
mz%l
n fm l
" \`g} vo}|vgm|ghb g} ihv bhvknagzn|ghb tuvth}o} hbaw nbl
lho} bh| nfflom| |`o cvhut gb nbw snw"
Tvhth}g|ghb 6" Ao| UN : F : M : L[ fo n `wtovfhan gb@%! nbl ao| fo n }gktao agbonvivnm|ghbna |vnb}ihvkn|ghb"
(g' (\vnb}an|ghb' Gi (z' ; z % f! s`ovo f ; p4 % w4e ihv p4! w4 V! |`obUN : F : M : L[
UN : F 6Np4 : M% 6Nw4 : N(p64 w64' Fp4 Mw4 % L["
(gg' (Vh|n|ghb hv Lgan|ghb' Gi (z' ; nz! s`ovo n ; p4 % w4e ihv p4! w4 V fu| w4 ; p4!|`ob
UN : F : M : L[ UN : Fp4 Mw4 : Mp4 Fw4 : L(p64 w64'["
(ggg' (Gbyov}ghb' Gi (z' ;>z ! |`ob
UN : F : M : L[ UL : F : M : N["
(gy' (Mhbjucn|ghb' Gi (z' ; z! |`ob
UN : F : M : L[ UN : F : M : L["
N} kob|ghbol tvghv! oyovw AI\(T' knt} `wtovfhan} gb @% hv @ hb|h `wtovfhan}gb @% hv @" Hbo knw yovgiw |`g} fw }`hsgbc |`n| onm` hi |`o ihuv }gktao |vnb}ihvkn+|ghb} tovihvk} |`g}"
N tngv hi `wtovfhan} gb @% (vo}t" @' nvo }ngl |h fo lg}jhgb|! |nbcob| hv gb|ov}om|gbcgi |`ow }`nvo! vo}tom|gyoaw! 4! > hv 6 thgb|(}' gb T (bh|o |`n| gb|ov}om|gbc opmaulo} |nbcob|'"Fomnu}o agbonv ivnm|ghbna |vnb}ihvkn|ghb} nvo fgjom|ghb}! |`ow tvo}ovyo |`o bukfov hi gb|ov+}om|ghb thgb|}"
6"< \`o @wtovfhagm Nbcao hi Gb|ov}om|ghb gb T
Lobg|ghb =" Ao| `> ; UN : F : M : L[ nbl `6 ; UO : I : C : @[ fo lg}|gbm| `wtovfhan} gb@% (vo}t" @'" \`ob |`o `wtovfhagm nbcao hi gb|ov}om|ghb `>`6 g} lobol fw
mh}`6 `>`6 ;(6N@% 6LO% MC FI'6
(`6 g} tnv|gmuanvaw `oatiua ihv man}}giwgbc |`o voan|ghb}`gtfo|soob |sh `wtovfhan} `> nbl `6"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
9/22
Tnco >66 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
Tvhth}g|ghb =" Gi `>! `6 @% (vo}t" @'! |`ob(g' `> nbl `6 nvo lg}jhgb| gi nbl hbaw gi `>`6 g} ublobol5
(gg' `> nbl `6 nvo |nbcob| gi nbl hbaw gi `>`6 ; 45 nbl
(ggg' `> nbl `6 nvo gb|ov}om|gbc gi nbl hbaw gi `>`6 2 4"
6"? Mnbhbgmna Ihvk}
Gb huv nvcukob|! so u}o mnbhbgmna votvo}ob|n|ghb hi `wtovfhan tngv}" \`o mhbmot|ghb g} |`n|ni|ov n }ug|nfao agbonv ivnm|ghbna |vnb}ihvkn|ghb! n tngv hi `wtovfhan} knw fo n}}ukol |h`nyo n }gktagol ihvk" ]oo Igcuvo ?"
\h lokhb}|vn|o |`g}! so ao| `>! `6 fo lg}|gbm| `wtovfhan} gb @%" Bop|! so }oaom| |`voo
lg}|gbm| thgb|} t4! t>! t
`> nbl lobo 4 fw
4(z' ;(z t4'(t> t'(z t'(t> t4' "
\`g} |neo} `> |h |`o agbo U4 : 4 : > : 4[" So sg}` |h u}o n }uf}o~uob| agbonv ivnm|ghbna|vnb}ihvkn|ghb |h }gktagiw 4(`6' ; UO : I : C : @["
Bh|o |`n| |`o hbaw AI\(T' s`gm` tvo}ovyo 4(`>' ; U4 : 4 : > : 4[ nvo (z' ; nz%fmz%l !s`ovo n!f!m!l V nbl nlfm ; >" (G| g} }|vngc`|ihvsnvl |h yovgiw |`n| nbw }um` |neo} |`oagbo w ; 4 |h g|}oai5 khvohyov! n agbonv ivnm|ghbna |vnb}ihvkn|ghb |`n| tvo}ovyo} U4 : 4 : > : 4[ku}| `nyo |`g} ihvk"'
So n}}uko bop| |`n| C
4gi bomo}}nvw! kua|gtaw O!I!C! nbl @ fw +>! n} |`g} `n} bh
offlom| hb |`o `wtovfhan" N| |`g} thgb|! huv }gktagmn|ghb g} lgyglol gb|h |`voo mn}o}"Mn}o >" Gi C 1 >! |`ob gktago} |`n| I6 4(`>' ; U 4 : 4 : > : 4 [ n b l 6 > 4(`6' ; U> : 4 : : >[! s`ovo ; 6C
@
z! |`ob >
4(`>' ; U4 : 4 : > : 4[
nbl > 4(`6' ; U4 : 4 : > : >["Gi O; 4! |`ob }haygbc |`o o~un|ghb} w ; 4 nbl O(p6 w6' % I p % w % @ ; 4 ihv p wgoal}
p ; I6O" Ao|>(z' ;
>O
z % I6O!
|`ob > 4(`>' ; U4 : 4 : > : 4[ nbl > 4(`6' ; U4 : 4 : > : >["
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
10/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >6=
Ihaahs |`o}o sg|` 6(z' ; 6z%e |h co| 6>4(`>' ; U4 : 4 : > : >[ nbl 6>4(`6' ;U4 : 4 : > : >[" So mnaa |`o}o n mnbhbgmna tngv hi |nbcob| `wtovfhan} gb@%"
Mn}o =" Gi C 2 >! |`ob gktago} |`n| I6 (z' ; z %
@I
! |`ob >
4(`>' ; U4 : 4 :
> : 4[ nbl > 4(`6' ; U4 : : > : 4[! s`ovo ; IC ; 4"Gi O; 4! |`ob 4(`>' nbl 4(`6' gb|ov}om| n| z ; I
I6
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
11/22
Tnco >6< V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
Luvgbc huv vo}onvm`! so nb|gmgtn|ol |`n| \`ohvok = shual fo n thsoviua |hha ihv huvtvhhi hi \`ohvok 6 nbl n}tgvol |h nttaw |`o shve hi @nw} nbl Kg|m`oaa |h huv hsb`hsoyov!so ihubl |`n| so mhual bh| lgvom|aw nttaw |`ogv vo}ua| |h huv }|n|okob| gb \`ohvok 6" N| |`otg|` hi huv tvhhi ihv \`ohvok 6! so sgaa lobo n mov|ngb gbjom|ghb hb T |`n| mhb}o~uob|aw knt}
`wtovfhan} gb @% |h `wtovfhan} @% (}oo }om|ghb (\`o Khlgol @nw}+Kg|m`oaa \`ohvok'" Gi i : T T g} nb gbjom|gyo knttgbc|`n| }obl} `wtovfhan} gb@% |h `wtovfhan} gb@%! nblT g} n mah}ol kgllao vocghb fhublolfw n yov|gmna vgc`| `wtovfhan! |`ob |`o vo}|vgm|ghb iqT g} n agbonv ivnm|ghbna |vnb}ihvkn|ghb"
\h tvhyo Aokkn >! so sgaa nlht| |`o knjhvg|w hi |`o tvhhi ihv \`ohvok = cgyob gb U?[
nbl kneo m`nbco} gb nvon} s`gm` nvo bh| |vuo ihv huv gbjom|gyo knttgbc i" \`g} knw }ookn fg| mhbiu}gbc hb |`o }uvinmo! fu| n} |`o nvcukob| tvhmool} von}hbgbc sgaa fomhko khvomaonv" \`o tvhhi gb U?[ g} mhb}|vum|gyo! nbl gb hbo hi |`o }|ot} n `wtovfhan gb @ g} u}ol!|h s`gm` so lh bh| `nyo nmmo}}5 `hsoyov! so `nyo nb nlynb|nco! }gbmo i g} gbjom|gyo hb naaT nbl bh| ju}| hb n mah}ol kgllao vocghb! }h so knw u}o `wtovfhan} s`gm` op|oblol fowhbl|`o fhublnvw hi T! s`ovon} @nw} nbl Kg|m`oaa knw hbaw u}o `wtovfhan} s`gm` T gbmaulo}"
Gb huv }g|un|ghb! |sh khlgmn|ghb} |h |`o tvhhi gb U?[ nvo boolol" Igv}|! s`ovo @nw}nbl Kg|m`oaa u}o |`o inm| |`n| U> : 4 : 4 : : 4 : 4 : >[ (U?! ="3['! so u}o |`o tvo}ovyn|ghb hi U> : 6 : 4 : 6[ |h nvvgyon| n lgfflovob| tvo}ovyol thgb| |`n| tanw} |`o }nko vhao" ]omhbl! s`ovo @nw} nbl Kg|m oaa
ncngb u}o n `hvgzhb|na `wtovfhan |h nvcuo ihv |`o tvo}ovyn|ghb hi
> : 4 : 4 :
! `6! ihv gbjom|gyg|w gktago} |`n| n thgb|z `> `6 gi nbl hbaw gii(z' i(`>' i(`6'"
Bhs! mhb}glov `4 ; U> : 4 : 4 : >[ nbl |`o mah}ol kgllao vocghb fhublol fw g|!
T ; p % we T : p6 w6 % > 4 "
\h iuv|`ov manvgiw |`o |sh khlgmn|ghb}! so voyg}g| |`o tvhhi hi \`ohvok = n| |`o }|nco hi U?!="3[! s`ovo |`o ihaahsgbc knw fo n}}ukol hi i ni|ov `nygbc foob tvo+ nbl th}|+mhkth}olsg|` |`o }ug|nfao agbonv ivnm|ghbna |vnb}ihvkn|ghb}"
>" i tvo}ovyo} `4 ; U> : 4 : 4 : >[ nbl |`o thgb|} e nbl (4 % 4e'> (U?! ="6['"6" i knt} tnvnaaoa agbo} |h tnvnaaoa agbo} (U?! ="=['"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
12/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >6?
=" i tvo}ovyo} a> ; U4 : 4 : > : >[! a6 ; U4 : 4 : > : >[ nbl a= ; U4 : 4 : > : 4[ (U?! =" |h g|}oai! nbl mhb}o~uob|aw!
i(`4' i(`>' ; `4 `> ;
>
6%
?
6e!
>
6
?
6e
"
So sgaa }`hs |`n| i tvo}ovyo} bh| hbaw |`g} }o|! fu| onm` thgb| gb g|"Mhb}|vum| |`o agbo s`gm` tn}}o} |`vhuc` |`o thgb|} >
6%
?6
e nbl > % e!
a< ;
4 : 6
? : > :
? >
@%"
]gbmo i(a6 % ?6 e' ; >6 % ?6 e" ]ooIgcuvo 9" (Khvohyov! fomnu}o i g} gbjom|gyo! so na}h mhbmaulo |`n| i(>6
?6 e' ;
>6
?6 e"'
\`ovoihvo! so `nyo ihubl n thgb| hb `4 s`gm` g} tvo}ovyol fw i! |`u} iuaaagbc huv v}| chna"
`4 `>
a>
a6
a=
t6t>
a6
%?6
e nbl t6 ; > % e"
Bop|! so sgaa }`hs |`n|
> : 4 : 4 : : 6 : 4 : 6['" Tvhmool |h mhb}|vum| |`o
agbo |nbcob| |h `4 n|
>
6 %
?
6 e nbl |`o agbo |nbcob| |h `4 n| >
6 %
?
6 e" \`ow nvo
a? ;
4 :
>?
: > : 6?
nbl a9 ;
4 : >
?: > : 6
?
!
vo}tom|iuaaw" Fomnu}o i tvo}ovyo} |nbcobmw! knt} agbo} |h agbo}! nbl tvo}ovyo} |`o thgb|}>6 %
?6 e nbl >6 %
?6 e! g| ihaahs} |`n| i(a?' ; a? nbl i(a9' ; a9" ]gbmo a= a? 6 % 4e nbl
a= a9 6 % 4e! g| ihaahs} |`n| i(6 % 4e' ; 6 % 4e nbl i(6 % 4e' ; 6 % 4e"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
13/22
Tnco >69 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
Bop|! mhb}|vum| |sh agbo}: hbo |`vhuc` |`o thgb|} 6%4e nbl >%e nbl |`o h|`ov |`vhuc`|`o thgb|} 6%4e nbl > %e" \`o}o agbo} gb|ov}om| n| 4% 6=e" Onm` hi |`o}o agbo} g} tvo}ovyol!s`gm` gktago} |`n| i
4 % 6=e
; 4 % 6=e" ]gkganvaw! fw mhb}|vum|gbc |`o agbo |`vhuc` 6 % 4enbl >
e nbl |`o agbo |`vhuc` 6 % 4e nbl
>
e! so bl |`n| |`ogv gb|ov}om|ghb thgb| 4
6=e
g} tvo}ovyol" ]oo Igcuvo 3"
Mhb}o~uob|aw! |`o `hvgzhb|na agbo} U4 : 4 : > : 6=
[ nbl U4 : 4 : > : 6=
[ nvo tvo}ovyol fwi" \`o}o nvo |nbcob| |h U> : 4 : 4 : 6 %
?6 e
>6 ?6 e>6 ?6 e
>6 %
?6 e
> % e
> e> e
> % e
6 % 4e 6 % 4e
4 % 6=e
4 6=
e
Igcuvo 3: Agbo} nbl thgb|} tvo}ovyol fw i"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
14/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >63
< Tvhhi hi \`ohvok 6
! `6! `=! `< @% fo tngvsg}o |nbcob|" Gb mnbhbgmna ihvk! so |neo `> ; U4 :4 : > : >[ nbl `6 ; U4 : 4 : > : >[! nbl m`nvnm|ovgzo oyovw `wtovfhan UN : F : M : L[ gb@% |`n| g} |nbcob| |h fh|` `> nbl `6" Ivhk (6' gb }om|ghb 6"< nbl |`o bhvknagzn|ghb hb
UN : F : M : L[! so co| |`o ihaahsgbc }w}|ok o~un|ghb}"(6N % M'6 ; > (='
(6N % M'6 ; > ( : : 4 : > %
6
" Vomnaa |`n| fh|` `= nbl `< nvo onm` |nbcob| |h `> nbl `6 nbl |`n| |`ow nvo na}h
|nbcob| |h onm` h|`ov" Gi `= nbl `< foahbc |h |`o v}| inkgaw! |`ob `>! `6! `=! nvo `< nvo naa`hvgzhb|na agbo}! nbl |`u}
`g ; u(4 % 4e'>{! s`gm` mhb|ngb} opnm|aw hbo kokfov"
Mn}o 6" Bhs }utth}o |`n| `= nbl `< fh|` foahbc |h |`o an||ov inkgaw" So ao| `= ;> : : 4 : > %
6
: : 4 : > %
6
% 6
60 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
`6
`> p
w
`=
`p
w`= `"' @wtovfhan} `>! `6! `=! nbl `< nvo tngvsg}o|nbcob| n| opnm|aw hbo ku|una thgb| (aoi|'5 (Mn}o 6"' @wtovfhan}`= nbl `< nvo gb|ov}om|gbc (vgc`|'"
Mn}o =" Igbnaaw! }utth}o |`n| `= nbl `< onm` foahbc |h lgfflovob| inkgao}" So ao| `= ; U4 :4 : > : L[ nbl `< ;
> : : 4 : > %
6
% 6
`6
p
w`=
`
`6
p
w`=
` g} |`o hbaw ynagl cobovnagzn|ghb hi`>! `6! `= nbl `
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
16/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >67
n mhaaom|ghb hi tngvsg}o |nbcob| `wtovfhan} nbl man}}giw }tomgna inkgago} hi gbbg|oaw knbwtngvsg}o |nbcob| `wtovfhan}" Gi so tgme nbw thgb| t gb T nbl n vona bukfov k sg|` qkq 1 >!|`ob so mnb mhb}|vum| n inkgaw hi |nbcob| `wtovfhan} gb @% |`n| so lobh|o \t!k" So lgygloinkgago} gb|h ihuv |wto}"
Igv}|! s`oboyov t g} bg|o (g"o" t T'! so mnaa \t!k |`o inkgaw hi `wtovfhan} s`h}o }ahtog} k n| thgb| t"= Gb tnv|gmuanv! gi t ; p4 % w4e! |`ob
\t!kloi; uUN : k 6Np4 : > % 6Nw4 : N(p46 w46' % kp4 w4[ : N V{"
Bop|! gi t @Qu( e'>{! |`ob t> g} bg|o" S`oboyov |`g} g} |`o mn}o so lobo\t!k n} |`o inkgaw hi `wtovfhan} hf|ngbol n} gknco} hi\t>!k ublov |`o gbyov}ghb knttgbcz >
z" Gb |`g} mn}o! k lho} bh| votvo}ob| n }ahto! fu| g| lho} naahs u} |h glob|giw n }tomgm
}o| hi `wtovfhan} n| t" Gb tnv|gmuanv! gi t ; (p4 p4e'>! sg|` p4 ; ! |`ob
\t!k
loi
; uUkp4 p4 : k 6Np4 : > 6Np4 : N[ : N V{"So `nyo aoi| |h mhb}|vum| n inkgaw hi `wtovfhan} ihv t ; ( e'>" \h nmmhktag}`
|`g}! so focgb sg|` |`o thgb| (> % e'> nbl mhb}|vum| |`o }o| \(>%e'>!k! nbl |`ob u}o n}uf}o~uob| AI\(T' }h |`n| ((> % e'>' ; ( % e'>" \`o nttvhnm` g} }hkos`n|gblgvom|5 so mangk |`n| }gbmo n thgb| gb @ mhvvo}thbl} |h |`o n}wkt|h|o hi |`o `wtovfhan}mhb|ngbgbc g|! gi so ebhs s`ovo |`o n}wkt|h|o hb T cho}! |`ob so mnb glob|giw s`ovo thgb|}n| gbbg|w ch" \`o so m`hh}o |h u}o g} (z' ; z >6 " ]h s`ob t ; ( % e'>! |`ob
\t!kloi; (\(>%e'>!k'
;
k > : > 6N : > 6N : k % >! t6 T nbl ao| k>! k6 V }um` |`n| qkjq 1 >" \`ob\t>!k> ; \t6!k6 ginbl hbaw gi t> ; t6 nbl k> ; k6"
Tvhhi" Gi t> ; t6 nbl k> ; k6! |`ob hfyghu}aw \t>!k> ; \t6!k6"
=Gb |`o nvcukob| ihv |`o @ yov}ghb! so tovkg| k ; gb hvlov |h nmmhub| ihv yov|gmna agbo} nbl |`o`wtovfhan} |nbcob| |h |`ok! gb s`gm` mn}o |`o }ahto g} ublobol"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
17/22
Tnco >=4 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
Mhbyov}oaw! }utth}o so focgb sg|` \t>!k> ; \t6!k6" \`ob `>! `6 \t>!k> gktago} |`n|`>! `6 \t6!k6" Gi t> ; t6! |`ob `> nbl `6 }`nvo |sh thgb|}bnkoaw! t> nbl t6" \`g} g} nmhb|vnlgm|ghb }gbmo `> nbl `6 nvo |nbcob|" \`u} t> ; t6
}o|; t"
Fw tvo+mhkth}gbc sg|` nb nttvhtvgn|o agbonv ivnm|ghbna |vnb}ihvkn|ghb! so knw n}}uko|`n| t g} bg|o (g"o" t T'" ]gbmo naa `wtovfhan} gb \t!k> ; \t!k6 nvo |nbcob| n| hbo ku|unathgb|! |`ow ku}| }`nvo |`o }nko }ahto n| t" @obmo k> ; k6"
\`ovoihvo! so mhbmaulo |`n| \t>!k> ; \t6!k6 gi nbl hbaw gi t> ; t6 nbl k> ; k6"
w
p
t
Igcuvo >4: \`o inkgaw hi `wtovfhan} \t!k"
Aokkn ! `6! `=! `< \%t!k! |`ob \(`>'! \(`6'! \(`='! nbl \(` j
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
18/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >=>
nbl k6 sg|` qkjq 1 >! nblmhb}|vum| |`o |sh inkgago} \t!k> nbl \t!k6" Fw Aokkn
>! t
6 T nbl k>! k6
sg|` qkjq 1 > }um` |`n|
\(\t!k>' ; \t>!k> nbl \(\t!k6' ; \t6!k6"
So ku}| }`hs |`n| t>
; t6
" Huv von}hbgbc g} n} ihaahs}"Gi t> ; t6 so sgaa mhb}|vum| `wtovfhan} `> \t>!k> nbl `6 \t6!k6 |`n| nvo |nbcob| |h
hbo nbh|`ov" ]gbmo \ g} n fgjom|ghb |`n| tvo}ovyo} |nbcobmw! |`g} shual konb |`n| `wtovfhan}\>(`>' \t!k> nbl \>(`6' \t!k6 opg}| nbl nvo |nbcob| |h onm` h|`ov s`ovo k> ; k6!s`gm` g} maonvaw ina}o"
\h }gktagiw kn||ov}! so th}|+mhkth}o \ sg|` n }ug|nfao agbonv ivnm|ghbna |vnb}ihvkn|ghb gb hvlov |`n| so knw n}}uko |`n| t> ; 4! k
> ; 4 nbl t
6 g} bg|o" \`g} mnb fo svg||ob n}
n mhkth}g|ghb > 6 s`ovo 6 }obl} t> |h 4 nbl t6 |h }hko bg|o thgb|5 nbl > g} n }ug|nfaovh|n|ghb 6(z' ; nz! s`ovo n g} }hko vona bukfov"
\`o `wtovfhan `> \t>
!k>
; \4!4 mnb |`ob fo svg||ob U : 4 : > : 4[ ihv }hko V" N||`g} thgb| huv nvcukob| g} lgyglol gb|h |sh mn}o}"
Mn}o >" So n}}uko |`n| t6 ; p4%w4e s`ovo w4 ; 4" \h }gktagiw bh|n|ghb so }o| k6 ; k"So m`hh}o ; 4 }h |`n| `> ; U4 : 4 : > : 4[! nbl sgaa ahhe ihv bukfov} O!I!C !@ V }h|`n| `6 ; UO : I : C : @[ g} |nbcob| |h `> nbl foahbc} |h \t
6!k
6; \p4%w4e!k"
\`o bhvknagzn|ghb hb `6 nbl |`o |nbcobmw sg|` `> |`ob vo~ugvo
(>4'C6 ; > (>>'
Gb nllg|ghb! n} `6 mnb fo optvo}}ol gb mhhvlgbn|o} fw O(p6 w6' % I p % Cw % @ ; 4! so
`nyo |`n| `6 \p4%w4e!k vo~ugvo}
O(p64 w64' % I p4 % Cw4 % @ ; 4 (>6'O(6p4 6w4k' % I % Ck ; 4" (>='
(\`o an||ov o~un|ghb g} hf|ngbol fw gktagmg| lgfflovob|gn|ghb" Vomnaa |`n| k g} |`o }ahto hi `6n| p4 % w4e"'
Ivhk (>>'! so m`hh}o C ; > nbl }uf}|g|u|o gb|h |`o vokngbgbc o~un|ghb}" (\`o m`hgmohi %> hv > kneo} bh lgfflovobmo }gbmo kua|gtawgbc naa mhkthbob|} hi `6 fw > `n} bh offlom|
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
19/22
Tnco >=6 V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
hb `6"' Ivhk (>='! so bl I ; 6O(p4 w4k' k nbl }uf}|g|u|o gb|h |`o vokngbgbco~un|ghb}" \`ob ivhk (>6'! so na}h bl @ ; O(p64 % w
64 6p4w4k' % p4k w4 nbl }uf}|g|u|o
gb|h |`o vokngbgbc o~un|ghb}" \`u} ivhk (>4' so co| |`o ihaahsgbc o~un|ghb"
O6 >w4
O k6 k6' ; 4
\`g} g} ~unlvn|gm gb O nbl `n} vona }hau|ghb}" ]h `6 opg}|}"Mn}o 6" So n}}uko |`n| t6 ; p4 % 4e sg|` p4 ; 4 nbl so ncngb ao| k6 ; k" Fw
th}|+mhkth}gbc \ sg|` n iuv|`ov lgan|ghb so knw n}}uko |`n| p4 ; >" So m`hh}o ; >nbl sgaa ahhe ihv bukfov} O!I!C!@ V }h |`n| `6 ; UO : I : C : @[ g} |nbcob| |h `> nblfoahbc} |h \t
6!k
6; \>!k"
\`o bhvknagzn|ghb hb `6 nbl |`o |nbcobmw sg|` `> |`ob vo~ugvo |`n|
(>" (>?'
Gb nllg|ghb! n} `6 mnb fo optvo}}ol gb mhhvlgbn|o} fw O(p6 w6' % I p % Cw % @ ; 4! so
`nyo |`n| `6 \>!k vo~ugvo} |`n|
O% I % @ ; 4 (>9'
6O% I % Ck ; 4" (>3'
(\`o an||ov o~un|ghb g} hf|ngbol fw gktagmg| lgfflovob|gn|ghb"'Ivhk (>?'! so m`hh}o 6@ % C ; > nbl }uf}|g|u|o C ; > 6@ gb|h |`o vokngbgbc
o~un|ghb}" (\`o m`hgmo hi %> hv
> kneo} bh lgfflovobmo }gbmo kua|gtawgbc naa mhkthbob|} hi
`6 fw > `n} bh offlom| hb `6"' Ivhk (>9'! so na}h bl I ; (O% @' nbl }uf}|g|u|o gb|h|`o vokngbgbc o~un|ghb}" So |`ob `nyo |sh o~un|ghb}
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
20/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >==
\`g} bop| aokkn sgaa }`hs |`n| |`o thgb|sg}o knttgbc \ nm|unaaw lo|ovkgbo} |`o `wtov+fhan knttgbc \"
Aokkn 9" Gi ` @%! |`ob \(`' ; u\(t' : t `{"Tvhhi" Ao| t ` nbl ao| k V! }um` |`n| qkq 1 >! nbl mhb}|vum| \t!k" \`ob ` \t!k" FwAokkn ! }h |`n| \(\t!k' ; \t!k" \`ob\(`' \t!k ; \\(t'!k nbl \(t' \(`'" \`u} u\(t' : t `{ \(`'"
Bhs ao| t \(`' nbl }utth}o |`n| \(`' \t!k ! ihv }hko k" \`ob ihv \>! fw Aokkn! }h |`n| \(\t!k' ; \t!k " Iuv|`ovkhvo! fomnu}o \>g} fgjom|gyo! |`ovo g} n ubg~uo ` @% }um` |`n| ` ; \>(\(`'' \>(\t!k' ; \t!k! nbl|`u} t `" Fw lobg|ghb hi\! so na}h `nyo |`n| \(t' ; t! s`gm` konb} t u\(t' : t `{"@obmo \(`' u\(t' : t `{"
\`o }omhbl tnv| hi |`o tvhhi hi Aokkn 9 cgyo} u} |`n| \ g} }uvjom|gyo! }gbmo oyovwt
\(`'
`n} n tvogknco t ` ihv nbw ` @%" ]uvjom|gyg|w |oaa} u} |`n| gb|ov}om|ghb thgb|} mnbbh|fo mvon|ol fw \5 gb hvlov |h }`hs |`n| gb|ov}om|ghb thgb|} mnbbh| fo lo}|vhwol! so ku}|}`hs |`n| \ g} na}h gbjom|gyo"
Aokkn 3" \ g} gbjom|gyo"
Tvhhi" Ao| t>! t6 T nbl }utth}o |`n| \(t>' ; \(t6'" Ao| k V! qkq 1 >! nbl mhb}|vum||`o inkgago} \\(t>'!k nbl \\(t6'!k" Fw Aokkn =! \\(t>'!k ; \\(t6'!k" \`ob nttawgbc \
>!|`ovo g} nb k V sg|` qkq 1 >! }um` |`n|
\t>!k ; \>(\\(t>'!k' ; \
>(\\(t6'!k' ; \t6!k"
Fw Aokkn =! t> ; t6" @obmo \ g} gbjom|gyo"
So bhs `nyo |`n| \ g} nb gbjom|gyo knttgbc hb T s`gm` }obl} `wtovfhan} gb @% |h`wtovfhan} gb @%" \`ovoihvo! fw Aokkn > ivhk }om|ghb =! so ebhs |`n| \ g} n agbonvivnm|ghbna |vnb}ihvkn|ghb s`ob vo}|vgm|ol |h n mah}ol kgllao vocghb" So `nyo aoi| |h }`hs|`n| \ g} agbonv ivnm|ghbna hb |`o ob|gvo op|oblol lhufao bukfov tanbonbl bh| hbaw hb}hko mah}ol kgllao vocghb"
>" So sgaa }`hs |`n| > \ po}oyovw thgb| hu|}glo hi T n} soaa"
Mhktao|ghb hi |`o Tvhhi hi \`ohvok 6" ]utth}o |`n| t g} n bg|o bukfov" \`ob so mnbmhb}|vum| |sh lg}|gbm| agbo} s`gm` gb|ov}om| n| t" So mangk |`n| oyovw agbo gb @% ku}|gb|ov}om| |`o mah}ol kgllao vocghb T n| aon}| |sgmo (gb inm|! gbbg|oaw knbw |gko}'" \`ob
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
21/22
Tnco >=< V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6
T
p
w
; \qTT ; \(T'
p
w
>
> \qTT
p
w
Igcuvo >>: \`o knttgbc > \qT g} |`o glob|g|w hb |`o vocghbT"
oyovw agbo g} tvo}ovyol fw > \! nbl |`u} gb|ov}om|ghb thgb|} hi nbw tngv hi agbo} nvotvo}ovyol" @obmo >
\(t' ; t"
Bhs }utth}o t @" ]gbmo n `wtovfhan g} ubg~uoaw lo|ovkgbol fw g|} thgb|} gb T! oyovw`wtovfhan gb @% g} tvo}ovyol fw > \" \`ob g| ihaahs} |`n|
\t!k ; > \(\t!k' ; \>\(t'!k"
\`ovoihvo! fw Aokkn =! > \(t' ; t"]gbmo > \ po} oyovw thgb| hb |`o op|oblol lhufao tanbo! g| ihaahs} |`n| g| g} |`o
glob|g|w knttgbc! nbl `obmo n agbonv ivnm|ghbna |vnb}ihvkn|ghb" \`u}! fw |`o cvhut }|vum|uvo
hiAI\(T'!
\ ; > \ AI\(T'"
\`ovoihvo! fgjom|gyo knttgbc} |`n| }obl} |nbcob| `wtovfhan} gb @% |h |nbcob| `wtovfhan}gb @% nvo gblumol fw n agbonv ivnm|ghbna |vnb}ihvkn|ghb hi |`o op|oblol lhufao tanbo"
-
8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.
22/22
V@G\ _blovcvnl" Kn|`" J"! Yha" >6! Bh" 6 Tnco >=?
? Mhbmau}ghb
\`g} vo}ua| iuv|`ov} n mhbbom|ghb fo|soob |`o mhktaop bukfov}! luna bukfov}< nbl lhufaobukfov}" Iovlgbnbl} nbl Enyago }`hsol gb U=[ |`n| n fgjom|ghb hb |`o }tnmo hi tnvnfhan} |`n|
tvo}ovyo} n pol lg}|nbmo > fo|soob gb|ov}om|gbc tnvnfhan} g} gblumol fw n agbonv ivnm|ghbna|vnb}ihvkn|ghb hi |`o luna tanbo L" \`g} vo}ua| nahbc sg|` Ao}|ov} nbl huv hsb }`hs `hsgb|vgb}gm |`o agbonv ivnm|ghbna |vnb}ihvkn|ghb} nvo sg|` |`o cohko|vgmna }tnmo} gb s`gm |`ownm|" So iuv|`ov shblov gi |`ovo g} n khvo ubgol snw |h }o| ut \`ohvok 6|`n| g}! s`n|nvo |`o bomo}}nvw nbl }umgob| mhblg|ghb} ihv \ gi so |neo gb|h mhb}glovn|ghb |`o ob|gvo}tnmo hi vgc`| `wtovfhan} nbl agbo} @ ; @% @8 \`g} ~uo}|ghb mnko gb|h kgbl s`gaolo|ovkgbgbc s`o|`ov hv bh| \ g} soaa+lobol s`ob hbaw n}}ukgbc |`n| \ : @ @ g} nfgjom|ghb" G| na}h }|nbl} |h }`hs s`o|`ov hv bh| n }|vhbcov yov}ghb hi \`ohvok 6 g} |vuo fwn}}ukgbc n pol nbcao 2 4 g} tvo}ovyol"
Voiovobmo}
U>[ I" ]" Fomeknb! L" N" ^unvao}! Jv" Hb g}hko|vgo} hi Oumaglonb }tnmo}! Tvhm" Nkov"Kn|`" ]hm"! 4+0>?! >7?="
U6[ @" ]" K" Mhpo|ov" Gbyov}gyo lg}|nbmo! Nbbnag Kn|okn|gmn! 799"
U=[ \" Iovlgbnbl}! A" Enyago" N Fomeknb+^unvao} |wto |`ohvok ihv Ancuovvo |vnb}ihvkn+|ghb} gb |`o luna tanbo" Vh}o+@uaknb _blovcvnlun|o Kn|` Jhuvbna! >4(>'! 6447"
U
top related