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8/2/2019 A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane.

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> : 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

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=" 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 `> ;

>

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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%

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e nbl > % e!

a< ;

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]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

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a6

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e nbl t6 ; > % e"

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> : 4 : 4 : : 6 : 4 : 6['" Tvhmool |h mhb}|vum| |`o

agbo |nbcob| |h `4 n|

>

6 %

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nbl a9 ;

4 : >

?: > : 6

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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

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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 %

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> % e

> e> e

> % e

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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

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w

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Mn}o =" Igbnaaw! }utth}o |`n| `= nbl `< onm` foahbc |h lgfflovob| inkgao}" So ao| `= ; U4 :4 : > : L[ nbl `< ;

> : : 4 : > %

6

% 6

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p

w`=

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p

w`=

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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 >=?

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\`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|

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