PI ofthe FundamentalTheorem ofMobius Geometry

The formula

a ihi n E iprovided atthe last lecture canbe seen from the

following steps

Steph V zyzz.ES distinct extended cpxmembers

7 Mobius transformation T such that

TZ I TZE O 123 0 A

Pfofstept Teo K

Tz 5 A d

Te must be thefamz Zz

Tz p forsome complexmemberp

Then TEFL I p32 I 73

p Zits2c Z L

Hence Tz FEET

T Ustep z I a Wi

Zz O Wz

Fz co Was

jotA distinct Zyzyts distinct Wi We Ws

By step1 I T s't Tze ITZE O

Tzz A

and V sat U W F IU w 2 0

VW 5 b

T U are Miibius transformations

Then 5 2561 is a Mobius Transformation

such that Sz VIT V Tze

041 W

Similarly S7z Wa S 2 5 Ws

Hence we've proved that facing distinctzb3Zs

and distinct Wi Wa Wz I a Mobius transformation

S such that Szi Wi E 1,33

Ex Why this givesthe formula

Finally Uniqueness

If Vi Wz are Mobiustransformations at

Uh Zi Wi E 12,3 he1,2

Then U Vz

Pfof Final stepconsider Vilov then it is a Mobius transformation

suchthat VI OV Cz VI Wi Zi Eb33

Vi ov has at least 3 fixed points

as 2 i t Z are distinct

By lemma of the last lecture J

VI oo Id

V V2 XX

Corollarye All figures consisting of 3 distinctpoints are congruent in Mobius geometry

Remark This corollary Mobius geometry is not

isomorphic to Euclidean geometry and

Euclidean distance is not an invariant

InrariantsofMidbiasGeometing

Anglemeasurement

Mobiustransformations are conformal

Euclidean angle measure is an invariant

of Mobius Geometry

CrossRa

is the following function

1

Def The cross ratio

of 4 extended complex variables

Cz z z 237 FEET EFFIRemains a z E I

Zz Ze 73Z

2 If ZiZzzs are held constants thenas

a function of Z Tz ft 71,754is the unique Mobiustransformation sending

2 to 1 72 to 0 And Zz to a

Thm Let Z z Zzz be 4 distinct points on

then H S EM

5757,522,523 137522,2332

PI By the remark above

TZ 2,21,7373 is the unique Mobius

transformation such that

TZ F I TZE O TZ 3 0

Consider the composition To 5 EM

Note that To51 Sza TEI I

To St S Zz TZ 2 0

To 5 SB f Zz A

To5Cz Z Sza Sza SB ft

Therefore

TZ ToStSZ SZ SZ 572,523

HZ Zi 72,73 XX

Thin The cross ratio CFGZyZs isreal

if andonly if the 4 points lie on a

Euclidean circle a straight line

PI i CZ fifty 3 EIR

Tt TZi Tze 173 C IR V TEM

Let TEIM be the Mobius transformationsuch that Tz y TEO TZE l

ThenIR 2 7,717333 TZ l O t

TZ O l ft

Feig To

2TZHTZ

If 7,21,7323 2 then Tz

If 7 Zify B f 2 then Tze HFtF32 ftp.zzzpf

R

In any case Tz Tz Tz5fzs lie on the x axis

Therefore Z ZyzzZs lie on a Euclidean circle

or a straight line since Mobius transforms

maps linesfirdes to lines circles

Clines

Def Asubset C of the complexplane is adimeif C is a Euclidean circle a Euclidean

straight lineTim If C is a dine then TCC is a

cline t TEM

Pf Fx

Renick AH circlesandstraightlinesaryto each other in Mobius

geometry ci circle determined by 3 parts

s straight line isjust a circle passingthrough W

Ext

Symmetry

Def let C be a clone passing through 3

distinct points Zi Zz Z 3 Two points

z and are called symmetric with

respectte C if

GFzszzzsj z.se zEgMugateegaIfZyZyBare 3 distinct points on X axis

then zt za 22,73 7533Z

EEL 27 232I Is ZI Ez

E E Fo E

CE a fireEeae

ZEE suice Tze Cz Zi Zz Zz is invertible

whichis the usual mirror symmetry of t acrossthe x axis Z

I z't

Rmarke i ds In this case we see that one can take

any 3 points on the X axis to give the

symmetry wrt X axis Sanitary this is

true for any dine Cdis z z symmetric wit C

Tz Tz symmetric artTCCExt

ie1 Tz

I t

Ef If C Iz Iz a F R2 and 7543 EC

Then z z symmetric wrt C

CZYZ72,73 3ZyZzZT

EII.SI imIeer s Cz a zra zz a zsaTMobiustransfamtins

Ea Ea ZIA 75T

z.az ec s E aizRIa zFa a

E a a Ea

IIa a a a a Zz a

ata Zi Zz 23

fzt.EElaz a tz apcz a

which implies Iz aHz a1and µ z Ex

Ch6 Steinercircles

Familiesofcluies

let p g C Q the family of all lines passing

through p and g is calledthe Steinercircles

ofthefirstkindy with respect to points pand f

co

w plane

consider the transformation

W Sz ZIz q

Then p o ie Sp o

g t b lie SE n

Recall that Mobius transformations takes lines to cling

the image of the circles clines in the Steiner

circles of the 1st kind wrt p q fam the

Steiner circles of the 1stkind wrt o o

In the W plane it is easy to see that there is

another family of chutes orthogonal to the

Steiner circles ofthe 1st kind namely

thefamily of circles centered at wO

Thepull back of these circles I lw k k's inthew plane by s I form a family of Ivies on

Z plane which is called the Steinercircles

othud Court p falso called circles of Apollonius

1stkind

am

By definition the Steinercircles of 2nd kindrt p f have the equation i

t I

Z

Remarked The families of Steiner circles of Iste2nd kinds can be regarded as a generalization

of polar coordinates to Mobius geometry