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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 22 "M\366bius transformations
" }}{PARA 19 "" 0 "" {TEXT -1 12 "Mika Sepp\344l\344" }}{PARA 0 "" 0 "
" {TEXT -1 72 "M\366bius transformations are linear fractional transfo
rmations of the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "g(z)=(a*z+b)/
(c*z+d)" "6#/-%\"gG6#%\"zG*&,&*&%\"aG\"\"\"F'F,F,%\"bGF,F,,&*&%\"cGF,F
'F,F,%\"dGF,!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 5 "with " }{XPPEDIT 18 
0 "a*d-b*c <> 0;" "6#0,&*&%\"aG\"\"\"%\"dGF'F'*&%\"bGF'%\"cGF'!\"\"\"
\"!" }{TEXT -1 212 ".    Composition of mappings defines the group ope
ration in the group of M\366bius transformations.  M\366bius transform
ations mapping the upper half-plane onto itself have further the prope
rty that their coefficients " }{TEXT 257 1 "a" }{TEXT -1 2 ", " }
{TEXT 258 1 "b" }{TEXT -1 2 ", " }{TEXT 259 1 "c" }{TEXT -1 5 " and " 
}{TEXT 260 2 "d " }{TEXT -1 14 "are real and  " }{XPPEDIT 18 0 "a*d-b*
c" "6#,&*&%\"aG\"\"\"%\"dGF&F&*&%\"bGF&%\"cGF&!\"\"" }{TEXT -1 10 " > \+
0.     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
24 "Let us now assume that  " }{XPPEDIT 18 0 "g " "6#%\"gG" }{TEXT -1 
41 " maps the upper half-plane onto itself.  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We associate the matrix" 
}}{PARA 256 "" 0 "" {XPPEDIT 18 0 "G = matrix([[a/sqrt(a*d-b*c), b/sqr
t(a*d-b*c)], [c/sqrt(a*d-b*c), d/sqrt(a*d-b*c)]]);" "6#/%\"GG-%'matrix
G6#7$7$*&%\"aG\"\"\"-%%sqrtG6#,&*&F+F,%\"dGF,F,*&%\"bGF,%\"cGF,!\"\"F6
*&F4F,-F.6#,&*&F+F,F2F,F,*&F4F,F5F,F6F67$*&F5F,-F.6#,&*&F+F,F2F,F,*&F4
F,F5F,F6F6*&F2F,-F.6#,&*&F+F,F2F,F,*&F4F,F5F,F6F6" }}{PARA 0 "" 0 "" 
{TEXT -1 31 "with the M\366bius transformation " }{TEXT 261 1 "g" }
{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 47 "Clearly this matrix is
 not uniquely defined by " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 8 ",
 both  " }{TEXT 256 1 "G" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "-G" "6
#,$%\"GG!\"\"" }{TEXT -1 63 "  are matrices corresponding to a given M
\366bius transformation  " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 396 
".  Hence, to each M\366bius transformation mapping the upper half-pla
ne onto itself, corresponds a unique matrix with non-negative trace.  \+
One verifies easily that composition of M\366bius transformations corr
esponds to the multiplication of matrices.  Problems arise from the fa
ct that the trace of a product of matrices need not be positive even i
f the traces of the original matrices are positive.  " }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 5 "TOOLS" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
34 "M\366bius transformations to matrices" }}{PARA 0 "" 0 "" {TEXT -1 
43 "The following procedure computes a matrix, " }{TEXT 262 23 "with n
on-negative trace" }{TEXT -1 44 ",  corresponding to a M\366bius trans
formation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 677 "Moe2Mat := proc(g)\n    local M, nu,de,i,j;\n    n
u := collect(numer(simplify(g(z))),z): de := collect(denom(simplify(g(
z))),z):\n    if degree(nu,z)>1 then ERROR(\"Input not a Moebius trans
formation\"); fi;\n    if degree(de,z)>1 then ERROR(\"Input not a Moeb
ius transformation\"); fi; \n    M := matrix(2,2);\n    for i from 1 t
o 2 do \n        M[1,i] := coeff(nu,z,2-i):\n    od;\n    for j from 1
 to 2 do\n        M[2,j] := coeff(de,z,2-j):\n    od;\n       if linal
g[det](evalm(M)) = 0 then\n         ERROR(\"Give a non-constant mappin
g\");\n       fi;        \nif M[1,1] + M[2,2] >= 0 then map(x -> x/sqr
t(linalg[det](M)), evalm(M));\nelse  \nmap(x->-x/sqrt(linalg[det](M)),
 evalm(M)); fi;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(Moe2MatGf*
6#%\"gG6'%\"MG%#nuG%#deG%\"iG%\"jG6\"F.C+>8%-%(collectG6$-%&numerG6#-%
)simplifyG6#-9$6#%\"zGF>>8&-F36$-%&denomGF7F>@$2\"\"\"-%'degreeG6$F1F>
-%&ERRORG6#QCInput~not~a~Moebius~transformationF.@$2FG-FI6$F@F>FK>8$-%
'matrixG6$\"\"#FX?(8'FGFGFX%%trueG>&FT6$FGFZ-%&coeffG6%F1F>,&FXFGFZ!\"
\"?(8(FGFGFXFen>&FT6$FXF_o-Fjn6%F@F>,&FXFGF_oF]o@$/-&%'linalgG6#%$detG
6#-%&evalmG6#FT\"\"!-FL6#Q<Give~a~non-constant~mappingF.@%1Fap,&&FT6$F
GFGFG&FTFWFG-%$mapG6$f*6#%\"xGF.6$%)operatorG%&arrowGF.*&F<FG-%%sqrtG6
#-Fio6#T#F]oF.F.6$F)FTF^p-F\\q6$f*F_qF.FaqF.,$FdqF]oF.F.F[rF^pF.F.F." 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 270 "The above procedure takes as i
ts input a linear fractional transformation defined as a function, and
 returns the corresponding matrix normalized so that it has a non-nega
tive trace. The procedure gives some error messages in cases where the
 input is not of desired type." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
34 "Matrices to M\366bius transformations" }}{PARA 0 "" 0 "" {TEXT -1 
104 "To transform a matrix to a M\366bius transformation is straightfo
rward.  The following procedure does that." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 173 "Mat2Moe := proc(M, matrix)\n     if linalg[det](M)
=0 then ERROR(\"Give a matrix with non-zero determinant.\"); fi;\n    \+
 unapply((M[1,1]*z + M[1,2])/(M[2,1]*z + M[2,2]),z);\nend;" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%(Mat2MoeGf*6$%\"MG%'matrixG6\"F)F)C$@$/-&%
'linalgG6#%$detG6#9$\"\"!-%&ERRORG6#QIGive~a~matrix~with~non-zero~dete
rminant.F)-%(unapplyG6$*&,&*&&F36$\"\"\"FAFA%\"zGFAFA&F36$FA\"\"#FAFA,
&*&&F36$FEFAFAFBFAFA&F36$FEFEFA!\"\"FBF)F)F)" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 11 "Cross Ratio" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }
{TEXT 263 12 "cross-ratio " }{TEXT -1 71 "of four  distinct finite num
bers a, b, c and  d  is defined as follows\n" }}{PARA 256 "" 0 "" 
{TEXT -1 1 "(" }{XPPEDIT 18 0 "a,b,c,d" "6&%\"aG%\"bG%\"cG%\"dG" }
{TEXT -1 4 ") = " }{XPPEDIT 18 0 "(a-c)*(b-d)/(a-d)/(b-c);" "6#**,&%\"
aG\"\"\"%\"cG!\"\"F&,&%\"bGF&%\"dGF(F&,&F%F&F+F(F(,&F*F&F'F(F(" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "This can be seen as the \+
image of  " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 45 "  under the M
\366biust ransformation that takes " }{XPPEDIT 18 0 "b" "6#%\"bG" }
{TEXT -1 7 " to 1, " }{XPPEDIT 18 0 "c" "6#%\"cG" }{TEXT -1 10 " to 0 \+
and " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "
infinity" "6#%)infinityG" }{TEXT -1 18 ".  We assume that " }{XPPEDIT 
18 0 "b" "6#%\"bG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c" "6#%\"cG" }
{TEXT -1 6 " and  " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 75 "  are p
airwise different.  The following procedure defines the cross-ratio." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 548 "CR := proc(a,b,c,d)\n      if b=c or c=d or b=d then \n        \+
  ERROR(\"Cross ratio not defined.\") \n      fi;\n      if a<> infini
ty then \n      if d = infinity then \n         (a-c)/(b-c);\n      el
if c = infinity then \n         (b-d)/(a-d);\n      elif b = infinity \+
then \n         (a-c)/(a-d);\n      else \n         (a-c)*(b-d)/(a-d)/
(b-c); \n      fi;\n      else\n      if d = infinity then\n          \+
infinity;\n      elif c = infinity then\n          0;\n      elif b = \+
infinity then \n         1;\n      else\n         (b-d)/(b-c);\n      \+
fi;\n      fi; \nend; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#CRGf*6&%
\"aG%\"bG%\"cG%\"dG6\"F+F+C$@$55/9%9&/F29'/F1F4-%&ERRORG6#Q9Cross~rati
o~not~defined.F+@%09$%)infinityG@)/F4F=*&,&F<\"\"\"F2!\"\"FB,&F1FBF2FC
FC/F2F=*&,&F1FBF4FCFB,&F<FBF4FCFC/F1F=*&FAFBFHFC**FAFBFGFBFHFCFDFC@)F?
F=FE\"\"!FIFB*&FGFBFDFCF+F+F+" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
31 "Types of M\366bius transformations" }}{PARA 0 "" 0 "" {TEXT -1 
225 "M\366bius transformations mapping the upper half-plane onto itsel
f  have either one or two fixed points (one of which may be the infini
ty).  If a M\366bius transformation has only one fixed-point, then the
 transformation is called " }{TEXT 264 9 "parabolic" }{TEXT -1 428 ". \+
 M\366bius transformations mapping the upper half-plane onto itself an
d having two fixed-points are either hyperbolic or elliptic or the ide
ntity.  If the fixed points are real (or the infinity), the M\366bius \+
transformation in question is hyperbolic.  If the fixed points are not
 real, then they are pairwise complex conjugate, and the elliptic tran
sformation is a rotation about the fixed-points.  Such a transformatio
n is called " }{TEXT 265 8 "elliptic" }{TEXT -1 114 ".  The following \+
procedure finds out the type of a M\366biust ransformation mapping the
 upper half-plane onto itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 993 "WhatTypeM := proc(g)\n     \+
local x,y, TYPE;\n     if Im(g(I))<=0 then \n       ERROR(\"Tansformat
ion does not map the upper half-plane onto itself.\"); \n     fi;  \n \+
    if diff(g(z),z)=1 then\n           TYPE := 'identity'; \n     elif
 abs(limit(g(z),z=infinity))=infinity then\n         if nops([solve(g(
z)=z,z)])=0 then TYPE := 'parabolic';  \n         elif nops([solve(g(z
)=z,z)])=1 and type(solve(g(z)=z,z),numeric) then \n           TYPE :=
 'hyperbolic';\n         elif nops([solve(g(z)=z,z)])=1 and not(type(s
olve(g(z)=z,z),numeric)) then\n           TYPE := 'parabolic'\n       \+
  else ERROR(\"Wrong type of input.\") \n     fi;\n     fi;\n     if a
bs(limit(g(z),z=infinity))<>infinity \n     then if  \n         nops([
solve(g(z)=z,z)])=2 and Im(solve(g(z)=z,z)[1])=0 then TYPE := 'hyperbo
lic';              elif \n         nops([solve(g(z)=z,z)])=2 and Im(so
lve(g(z)=z,z)[1])<>0 then TYPE := 'elliptic'; \n     elif\n           \+
nops([solve(g(z)=z,z)])=1 then TYPE := 'parabolic'; fi;\n     fi;\n   \+
  TYPE;\nend;\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*WhatTypeMGf*6#%
\"gG6%%\"xG%\"yG%%TYPEG6\"F,C&@$1-%#ImG6#-9$6#^#\"\"\"\"\"!-%&ERRORG6#
QgnTansformation~does~not~map~the~upper~half-plane~onto~itself.F,@&/-%
%diffG6$-F46#%\"zGFDF7>8&.%)identityG/-%$absG6#-%&limitG6$FB/FD%)infin
ityGFQ@)/-%%nopsG6#7#-%&solveG6$/FBFDFDF8>FF.%*parabolicG3/FTF7-%%type
G6$FX%(numericG>FF.%+hyperbolicG3Fjn4F[o>FFFgn-F:6#Q5Wrong~type~of~inp
ut.F,@$0FJFQ@(3/FT\"\"#/-F16#&FX6#F7F8>FFF`o3F\\p0F_pF8>FF.%)ellipticG
Fjn>FFFgnFFF,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 353 "The type of
 a M\366bius transformation mapping the upper half-plane onto itself c
an also be deduced from the corresponding matrix.  If the absolute val
ue of the trace of the matrix is >2, then the transformation is hyperb
olic, if it =2, then the transformation is parabolic and otherwise it \+
is elliptic.  The following procedure uses this characterization." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 299 "WhatType := proc(g)\n      local M;\n      M := Moe2Mat(g);\n  \+
    if diff(g(z),z)=1 then\n            identity;        \n      elif \+
evalf(abs(linalg[trace](M)))>2 then\n          hyperbolic;\n      elif
 evalf(abs(linalg[trace](M)))=2 then\n          parabolic;\n      else
\n          elliptic;\n      fi;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%)WhatTypeGf*6#%\"gG6#%\"MG6\"F*C$>8$-%(Moe2MatG6#9$@)/-%%diffG
6$-F16#%\"zGF9\"\"\"%)identityG2\"\"#-%&evalfG6#-%$absG6#-&%'linalgG6#
%&traceG6#F-%+hyperbolicG/F>F=%*parabolicG%)ellipticGF*F*F*" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 49 "Fixed-points of hyperbolic M\366b
ius transformations" }}{PARA 0 "" 0 "" {TEXT -1 16 "Next we compute " 
}{TEXT 266 79 "the attracting and repelling fixed points of a hyperbol
ic M\366bius transformation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 566 "HyperMoeFixPoints := proc(g)\n      local x,y,a,r;\n
      if WhatType(g) <> hyperbolic then \n         ERROR(\"Not a hyper
bolic Moebius transformation.\");  \n      fi;\n      if abs(limit(g(z
),z=infinity))=infinity then \n         x:= infinity;\n         y := s
olve(g(z)=z,z);\n       if abs(g(y+I)-y)<1 then \n         a,r := y,x;
 \n       else \n         a,r := x,y;\n       fi;\n     else\n     x :
= solve(g(z)=z,z)[1]; y := solve(g(z)=z,z)[2];\n     if abs(g((x+y)/2)
-x) < abs((y-x)/2) then \n         a,r := x,y;\n     else \n         a
,r := y,x;\n     fi;\n     fi;\n     a,r; \nend;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%2HyperMoeFixPointsGf*6#%\"gG6&%\"xG%\"yG%\"aG%\"rG6\"
F-C%@$0-%)WhatTypeG6#9$%+hyperbolicG-%&ERRORG6#QINot~a~hyperbolic~Moeb
ius~transformation.F-@%/-%$absG6#-%&limitG6$-F46#%\"zG/FD%)infinityGFF
C%>8$FF>8%-%&solveG6$/FBFDFD@%2-F=6#,&-F46#,&FK\"\"\"^#FXFXFXFK!\"\"FX
>6$8&8'6$FKFI>Ffn6$FIFKC%>FI&FL6#FX>FK&FL6#\"\"#@%2-F=6#,&-F46#,&*&#FX
FcoFXFIFXFX*&F]pFXFKFXFXFXFIFZ-F=6#,&*&F]pFXFKFXFX*&#FXFcoFXFIFXFZ>Ffn
F[o>FfnFinFfnF-F-F-" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 48 "Multipli
er of a hyperbolic M\366bius transformation" }}{PARA 0 "" 0 "" {TEXT 
-1 44 "The cross-ration can be used to compute the " }{TEXT 267 10 "mu
ltiplier" }{TEXT -1 39 " of a hyperbolic M\366bius transformation." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
238 "Multiplier := proc(g)\n     local a,r;\n     a,r := HyperMoeFixPo
ints(g);     \n     if a = infinity then \n     CR(g(r+1),r+1,r,a);\n \+
    elif r = infinity then\n     CR(g(a+1),a+1,r,a);\n     else\n     \+
CR(g((a+r)/2),(a+r)/2,r,a);\n     fi;\nend;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%+MultiplierGf*6#%\"gG6$%\"aG%\"rG6\"F+C$>6$8$8%-%2Hyp
erMoeFixPointsG6#9$@'/F/%)infinityG-%#CRG6&-F46#,&F0\"\"\"F>F>F=F0F//F
0F7-F96&-F46#,&F/F>F>F>FDF0F/-F96&-F46#,&*&#F>\"\"#F>F/F>F>*&FKF>F0F>F
>FIF0F/F+F+F+" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 42 "Creating hyper
bolic M\366bius transformations" }}{PARA 0 "" 0 "" {TEXT -1 99 "A hype
rbolic M\366bius transformation is defined by knowing its attracting a
nd repelling fixed-points " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 5 "
 and " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 20 " and its multiplier \+
" }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 79 ".    If these parameters \+
are finite, the corresponding M\366bius transformation is" }}{PARA 0 "
" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 
18 0 "g(z)=((k*a-y)*z-x*y*(k-1))/((k-1)*z + x - k*y)" "6#/-%\"gG6#%\"z
G*&,&*&,&*&%\"kG\"\"\"%\"aGF.F.%\"yG!\"\"F.F'F.F.*(%\"xGF.F0F.,&F-F.F.
F1F.F1F.,(*&,&F-F.F.F1F.F'F.F.F3F.*&F-F.F0F.F1F1" }}{PARA 0 "" 0 "" 
{TEXT -1 87 "with obvious generalizations to the case that one of the \+
fixed points is the infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 47 "The following procedure uses this formula
 for  " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 47 ".  Observe that, by
 definition, the multiplier " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 
28 " of a M\366bius transformation " }{XPPEDIT 18 0 "g" "6#%\"gG" }
{TEXT -1 11 " satisfies " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 46 " \+
> 1.  The following works, however, also for " }{XPPEDIT 18 0 "k=1" "6
#/%\"kG\"\"\"" }{TEXT -1 61 ", in which case it creates the identity m
apping, and for 0 < " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 50 " < 1.
  In fact, using the notation defined below,\n" }}{PARA 256 "> " 0 "" 
{MPLTEXT 1 0 21 "CreateHypMoe(a,r,1/k)" }{TEXT -1 3 " = " }{MPLTEXT 1 
0 19 "CreateHypMoe(r,a,k)" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 526 "CreateHypMoe := proc(a,r,k)\n  if a=r then\n     ERR
OR(\"Fix points (the first two argument) must be distinct.\");     fi;
\n  if k=infinity then \n     ERROR(\"The last argument (multiplier) m
ust be a positive number.\"); \n  fi;  \n  if not(k>0) then \n     ERR
OR(\"The last argument (multiplier) must be a positive number.\");\n  \+
fi;\n  if a <> infinity and r <> infinity then\n    unapply(((k*a-r)*z
-a*r*(k-1))/((k-1)*z + a - k*r),z);\n  elif a = infinity then\n    una
pply(k*z - r*(k-1),z);\n  else\n    unapply(z/k + a*(1-1/k),z);\n  fi;
\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-CreateHypMoeGf*6%%\"aG%\"
rG%\"kG6\"F*F*C&@$/9$9%-%&ERRORG6#QVFix~points~(the~first~two~argument
)~must~be~distinct.F*@$/9&%)infinityG-F16#QZThe~last~argument~(multipl
ier)~must~be~a~positive~number.F*@$42\"\"!F6F8@'30F.F70F/F7-%(unapplyG
6$*&,&*&,&*&F6\"\"\"F.FKFKF/!\"\"FK%\"zGFKFK*(F.FKF/FK,&F6FKFKFLFKFLFK
,(*&FOFKFMFKFKF.FK*&F6FKF/FKFLFLFM/F.F7-FD6$,&*&F6FKFMFKFK*&F/FKFOFKFL
FM-FD6$,&*&FMFKF6FLFK*&F.FK,&FKFK*&FKFKF6FLFLFKFKFMF*F*F*" }}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 39 "Creating elliptic M\366bius transformat
ion" }}{PARA 0 "" 0 "" {TEXT -1 192 "Elliptic M\366bius transformation
s can be created by the same formula as what we used for hyperbolic M
\366bius transformations.  In this case the multiplier is, however, a \+
complex number of the form" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "k = exp
(theta*I);" "6#/%\"kG-%$expG6#*&%&thetaG\"\"\"%\"IGF*" }{TEXT -1 1 ",
" }}{PARA 0 "" 0 "" {TEXT -1 7 "where  " }{XPPEDIT 18 0 "theta" "6#%&t
hetaG" }{TEXT -1 119 "  is the rotation angle.  So we can copy the abo
ve code to form a procedure to create ellipitic M\366bius transformati
ons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 668 "CreateEllMoe := proc(a,r,theta)\n  if a=r then\n    \+
 ERROR(\"Fix points (the first two argument) must be distinct.\");    \+
 fi;\n  if theta=infinity then \n     ERROR(\"The last argument (rotat
ion angle) must be between 0 and 2*Pi \"); \n  fi;  \n#  if not(theta>
=0) or not(theta>=evalf(2*Pi)) then \n#     ERROR(\"The last argument \+
(rotation angle) must be between 0 and #2*Pi.\");\n#  fi;\n  if a <> i
nfinity and r <> infinity then\n    unapply(((exp(I*theta)*a-r)*z-a*r*
(exp(I*theta)-1))/((exp(I*theta)-1)*z + a - exp(I*theta)*r),z);\n  eli
f a = infinity then\n    unapply(exp(I*theta)*z - r*(exp(I*theta)-1),z
);\n  else\n    unapply(z/exp(I*theta) + a*(1-1/exp(I*theta)),z);\n  f
i;\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-CreateEllMoeGf*6%%\"aG%
\"rG%&thetaG6\"F*F*C%@$/9$9%-%&ERRORG6#QVFix~points~(the~first~two~arg
ument)~must~be~distinct.F*@$/9&%)infinityG-F16#QinThe~last~argument~(r
otation~angle)~must~be~between~0~and~2*Pi~F*@'30F.F70F/F7-%(unapplyG6$
*&,&*&,&*&-%$expG6#*&F6\"\"\"^#FKFKFKF.FKFKF/!\"\"FK%\"zGFKFK*(F.FKF/F
K,&FGFKFKFMFKFMFK,(*&FPFKFNFKFKF.FK*&FGFKF/FKFMFMFN/F.F7-F@6$,&*&FGFKF
NFKFK*&F/FKFPFKFMFN-F@6$,&*&FNFKFGFMFK*&F.FK,&FKFK*&FKFKFGFMFMFKFKFNF*
F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 8 "Example." }{TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "e:=CreateEllMoe(I,-I,Pi/3
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6#%\"zG6\"6$%)operatorG%
&arrowGF(*&,(*&,&*&,&#\"\"\"\"\"#F3*&^#F2F3\"\"$F2F3F3^#F3F3F3F8F3F39$
F3F3F2F3*&^##!\"\"F4F3F7F2F3F3,(*&,&#F3F4F=F5F3F3F9F3F3F8F3F0F3F=F(F(F
(" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Experiments" }}{PARA 0 "" 
0 "" {TEXT -1 99 "Consider hyperbolic M\366bius transformations  g  an
d  h  with intersecting axes.  Here is an example." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "g:=CreateHypMoe(0,2,5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gGf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*(\"\"#\"\"
\"9$F/,&*&\"\"%F/F0F/F/\"#5!\"\"F5F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "h:=CreateHypMoe(1,3,7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"hGf*6#%\"zG6\"6$%)operatorG%&arrowGF(*&,&*&\"\"%\"
\"\"9$F0F0\"#=!\"\"F0,&*&\"\"'F0F1F0F0\"#?F3F3F(F(F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "L
et us next compute the corresponding inverse M\366bius transformations
 G  and  H:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "G:=CreateHyp
Moe(2,0,5); H:= CreateHypMoe(3,1,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"GGf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*(\"#5\"\"\"9$F/,&*&\"\"%
F/F0F/F/\"\"#F/!\"\"F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG
f*6#%\"zG6\"6$%)operatorG%&arrowGF(*&,&*&\"#?\"\"\"9$F0F0\"#=!\"\"F0,&
*&\"\"'F0F1F0F0\"\"%F3F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Now form the matrices,
 with positive traces, corresponding to the above transformations." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "m:= Moe2Mat(g);n:=Moe2Mat(h
); M := Moe2Mat(G); N:= Moe2Mat(H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"mG-%'matrixG6#7$7$,$*&\"\"&!\"\"F,#\"\"\"\"\"#F/\"\"!7$,$*(F0F/F,
F-F,F.F-*$F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG-%'matrixG6#7$
7$,$*(\"\"#\"\"\"\"\"(!\"\"F.#F-F,F/,$*(\"\"*F-F.F/F.F0F-7$,$*(\"\"$F-
F.F/F.F0F/,$*(\"#5F-F.F/F.F0F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
MG-%'matrixG6#7$7$*$\"\"&#\"\"\"\"\"#\"\"!7$,$*(F.F-F+!\"\"F+F,F-,$*&F
+F3F+F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'matrixG6#7$7$,$*
(\"#5\"\"\"\"\"(!\"\"F.#F-\"\"#F-,$*(\"\"*F-F.F/F.F0F/7$,$*(\"\"$F-F.F
/F.F0F-,$*(F1F-F.F/F.F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The commutator of  " }
{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "h" "
6#%\"hG" }{TEXT -1 7 "  is   " }{XPPEDIT 18 0 "c = g@(H@(G@h))" "6#/%
\"cG-%\"@G6$%\"gG-F&6$%\"HG-F&6$%\"GG%\"hG" }{TEXT -1 1 "." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Let us compute th
e commutator:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "c := unappl
y(simplify(g(H(G(h(z))))),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"c
Gf*6#%\"zG6\"6$%)operatorG%&arrowGF(*&,&*&\"#P\"\"\"9$F0F0\"$)>!\"\"F0
,&*&\"\"'F0F1F0F0F0F0F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The corresponding matrix
 is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MatrixOfCommutator =
 Moe2Mat(c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3MatrixOfCommutatorG
-%'matrixG6#7$7$#\"#P\"#N#!$)>F,7$#\"\"'F,#\"\"\"F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "T
he commutator of the corresponding matrices is:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "CommutatorOfMatrices = evalm(m&*N&*M&*n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5CommutatorOfMatricesG-%'matrixG6#7$
7$#!#P\"#N#\"$)>F,7$#!\"'F,#!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
269 12 "Observation." }{TEXT -1 60 "  Trace of the commutator of the m
atrices corresponding to  " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 7 "
  and  " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 81 "  above is negativ
e.  This is, in fact, a particular case of a general theorem.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 270 10 "Example.  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "g := unapply(-1/z,z);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"gGf*6#%\"zG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.9$!\"\"F0F(
F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The transformation  " }
{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 42 "  is elliptic with fixed-poi
nts I and -I. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "WhatType(
g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)ellipticG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Fur
thermore,  g  is an involution:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "g(g(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"zG" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 30 "Hence the group generated by  " }
{XPPEDIT 18 0 "