(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.0'

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Notebook[{
Cell["The Poincare Model", "Title"],

Cell["W. Goldman, 11 July 2004", "Subtitle"],

Cell["\<\
This graphics package draws objects in the hyperbolic
plane using the Poincare model. The pictures are
drawn in the unit disc, but the calculations occur in the
upper half plane, where the isometry group is PSL(2,R).\
\>", "Subsubtitle"],

Cell["Calculations in the hyperbolic plane", "Section"],

Cell[BoxData[{
    \(Off[General::"\<spell1\>"]\ \), "\[IndentingNewLine]", 
    \(Off[Power::"\<infy\>"]\), "\[IndentingNewLine]", 
    \(Off[General::"\<spell\>"]\)}], "Input"],

Cell["Draw in the unit disc.", "Subsection"],

Cell[BoxData[{
    \(\(UnitCircle = Circle[{0, 0}, 1];\)\), "\n", 
    \(ShowP[l_] := 
      Show[Graphics[Prepend[l, UnitCircle]], 
        PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-1.1\), 1.1}}, 
        AspectRatio \[Rule] 1]\)}], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(ShowP[{}];\)\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
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\>"], "Graphics",
  ImageSize->{288, 288},
  ImageMargins->{{43, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCacheValid->False]
}, Open  ]],

Cell["Compute in the upper half plane.", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[{\(ToCircle[
        r_] := {2 
           r\ /\((\ r^2\  + 1)\), \((\ r^2\  - 1)\)/\((\ 
            r^2 + 1)\)}\), "\n", \(ToCircle[ComplexInfinity] := {0, 
        1}\), "\[IndentingNewLine]", 
    RowBox[{\(ToCircle::usage\), "=", 
      "\"\<ToCircle[r] gives the point on the unit circle corresponding to r \
\!\(\*
StyleBox[\"\[Element]\",\nFontSize->16]\)\!\(\*
StyleBox[\" \",\nFontSize->16]\)\[DoubleStruckCapitalR] \[Union] \
{\[Infinity]}.\nHere 0 is at the bottom, \[Infinity] on top, -1 to to left \
and 1 to right.\>\""}]}], "Input"],

Cell[BoxData["\<\"ToCircle[r] gives the point on the unit circle \
corresponding to r \\!\\(\\* \
StyleBox[\\\"\[Element]\\\",\\nFontSize->16]\\)\\!\\(\\* StyleBox[\\\" \
\\\",\\nFontSize->16]\\)\[DoubleStruckCapitalR] \[Union] {\[Infinity]}. Here \
0 is at the bottom, \[Infinity] on top, -1 to to left and 1 to right.\"\>"], \
"Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(ToUnitDisc[ComplexInfinity] = I;\)\), "\n", 
    \(\(ToUnitDisc[\(-I\)] = ComplexInfinity;\)\), "\n", 
    \(ToUnitDisc[
        z_] := \ \((z - I)\)/\((\(-I\)\ z\  + \ 
            1)\)\), "\[IndentingNewLine]", 
    \(ToUnitDisc::usage = "\<ToUnitDisc[z] maps the point z in the upper half \
plane to the corresponding point w in the unit disc. When z is real, this \
agrees with ToCircle[z].\>"\)}], "Input"],

Cell[BoxData[
    \("ToUnitDisc[z] maps the point z in the upper half plane to the \
corresponding point w in the unit disc. When z is real, this agrees with \
ToCircle[z]."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(ToR2[z_] := {Re[z], Im[z]}\), "\[IndentingNewLine]", 
    \(ToR2::usage = \*"\"\<ToR2[w] gives the point in \!\(\
\[DoubleStruckCapitalR]\^2\) whose\ncoordinates are the real and imaginary \
parts of w.\>\""\)}], "Input"],

Cell[BoxData[
    \("ToR2[w] gives the point in \!\(\[DoubleStruckCapitalR]\^2\) whose \
coordinates are the real and imaginary parts of w."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(NormSquared[{x_, y_}] := x^2\  + \ y^2\), "\n", 
    \(InvertInUnitCircle[v_] := \ 
      1/NormSquared[v]\ v\), "\[IndentingNewLine]", 
    \(NormSquared::usage = \*"\"\<NormSquared[v] gives the square of the \
Euclidean norm of a vector in \!\(\[DoubleStruckCapitalR]\^2\).\>\""\)}], \
"Input"],

Cell[BoxData[
    \("NormSquared[v] gives the square of the Euclidean norm of a vector in \
\!\(\[DoubleStruckCapitalR]\^2\)."\)], "Output"],

Cell["Drawing Geodesics", "Section"],

Cell["\<\
When the geodesic from z1 to z2 passes through i,
the corresponding geodesic in the unit disc passes through 0 and is \
represented by a Euclidean line segment,  a diameter of the unit disc.\
\>", \
"Subsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(LinearDependentQ[z1_, 
        z2_] := \((Chop[Im[z1\ \ Conjugate[z2]]] \[Equal] 
          0)\)\), "\[IndentingNewLine]", 
    \(LinearDependentQ::usage = "\<LinearDependentQ[z1,z2] tests
whether complex numbers z1 and z2 are linearly dependent
over \[DoubleStruckCapitalR].\>"\)}], "Input"],

Cell[BoxData[
    \("LinearDependentQ[z1,z2] tests whether complex numbers z1 and z2 are \
linearly dependent over \[DoubleStruckCapitalR]."\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\(PGSegment[{z1_, z2_}] := \ 
      Module[{w1 = ToUnitDisc[z1], w2 = ToUnitDisc[z2]}, 
        Line[{ToR2[w1], ToR2[w2]}] /; 
          LinearDependentQ[w1, w2]]\)\(\[IndentingNewLine]\)
    \)\)], "Input"],

Cell["\<\
The general case is more complicated.  First we find the
endpoints of the Poincare geodesic between two points
z1, z2 in the upper half plane.\
\>", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(MidpointEndpointsPGeodesic[z1_, z2_] := 
      Re[\((z1\  + \ z2)\)/2\  - \ 
          I/2\ \((Im[z1 + z2]/
                  Re[z1 - z2]\ \((z1 - z2)\))\)]\[IndentingNewLine]
    MidpointEndpointsPGeodesic::usage = "\<MidpointEndpointsPGeodesic[z1,z2] \
gives the midpoint
of the pair of endpoints of the Poincare geodesic containing points z1 and z2 \
in the upper half plane.\>"\)\(\ \)\)\)], "Input"],

Cell[BoxData[
    \("MidpointEndpointsPGeodesic[z1,z2] gives the midpoint of the pair of \
endpoints of the Poincare geodesic containing points z1 and z2 in the upper \
half plane."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(EndpointsPGeodesic[z1_, z2_] := 
      Module[{distance, 
          midpoint = 
            MidpointEndpointsPGeodesic[z1, 
              z2]}, \[IndentingNewLine]distance = 
          Abs[z1 - midpoint]; \[IndentingNewLine]{midpoint - distance, 
          midpoint + distance}]\), "\n", 
    \(EndpointsPGeodesic[z1_, 
        z2_] := {Re[z1], 
          ComplexInfinity} /; \((Re[z1] == 
            Re[z2])\)\), "\[IndentingNewLine]", 
    \(EndpointsPGeodesic[z_, ComplexInfinity] := {Re[z], 
        ComplexInfinity}\), "\[IndentingNewLine]", 
    \(EndpointsPGeodesic[ComplexInfinity, z_] := {Re[z], 
        ComplexInfinity}\), "\[IndentingNewLine]", 
    \(EndpointsPGeodesic::usage = "\<EndpointsPGeodesic[z1,z2] gives the \
endpoints of the Poincare geodesic containing z1 and z2 in the upper half \
plane.\>"\)}], "Input"],

Cell[BoxData[
    \("EndpointsPGeodesic[z1,z2] gives the endpoints of the Poincare geodesic \
containing z1 and z2 in the upper half plane."\)], "Output"]
}, Open  ]],

Cell["Now we transform to the unit disc.", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(xxParam[r1_, r2_] := \((ToCircle[r1] + ToCircle[r2])\)/
        2\), "\[IndentingNewLine]", 
    \(xxParam::usage = "\<xxParam[r1,r2] gives the midpoint of the Euclidean \
line segment joining the points on the unit circle corresponding to r1,r2 \
\[Element] \[DoubleStruckCapitalR] \[Union] {\[Infinity]}.\>"\), "\n", 
    \(LengthxxParam[r1_, r2_] := Sqrt[NormSquared[xxParam[r1, r2]]]\), "\n", 
    \(LengthxxParam::usage = "\<LengthxxParam[r1,r2] gives the length of the \
vector
to the midpoint of the Euclidean line segment joining the points on the unit \
circle corresponding to r1,r2 \[Element] \[DoubleStruckCapitalR] \[Union] {\
\[Infinity]}.\>"\)}], "Input"],

Cell[BoxData[
    \("xxParam[r1,r2] gives the midpoint of the Euclidean line segment \
joining the points on the unit circle corresponding to r1,r2 \[Element] \
\[DoubleStruckCapitalR] \[Union] {\[Infinity]}."\)], "Output"],

Cell[BoxData[
    \("LengthxxParam[r1,r2] gives the length of the vector to the midpoint of \
the Euclidean line segment joining the points on the unit circle \
corresponding to r1,r2 \[Element] \[DoubleStruckCapitalR] \[Union] {\
\[Infinity]}."\)], "Output"]
}, Open  ]],

Cell["\<\
Now we can determine the center and radius of the circle containing \
the Poincare geodesic passing through the points w1 and w2 in the unit \
disc.\
\>", "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(RadiusPGeodesic[r1_, r2_] := 
      Module[{x = LengthxxParam[r1, r2]}, Re[Sqrt[1 - x^2]/Abs[x]]]\), "\n", 
    \(CenterPGeodesic[r1_, r2_] := 
      Module[{v = xxParam[r1, r2]}, InvertInUnitCircle[v]]\), "\n", 
    \(InclinationAngle[v_] := Apply[ArcTan, v]\), "\[IndentingNewLine]", 
    \(RadiusPGeodesic::usage = "\<RadiusPGeodesic[r1,r2] gives the radius of \
the circle
representing the Poincare geodesic with endpoints corresponding to r1,r2 \
\[Element] \[DoubleStruckCapitalR] \[Union] {\[Infinity]}.\>"\), "\
\[IndentingNewLine]", 
    \(CenterPGeodesic::usage = "\<CenterPGeodesic[r1,r2] gives the center of \
the circle
representing the Poincare geodesic with endpoints corresponding to r1,r2 \
\[Element] \[DoubleStruckCapitalR] \[Union] {\[Infinity]}.\>"\)}], "Input"],

Cell[BoxData[
    \("RadiusPGeodesic[r1,r2] gives the radius of the circle representing the \
Poincare geodesic with endpoints corresponding to r1,r2 \[Element] \
\[DoubleStruckCapitalR] \[Union] {\[Infinity]}."\)], "Output"],

Cell[BoxData[
    \("CenterPGeodesic[r1,r2] gives the center of the circle representing the \
Poincare geodesic with endpoints corresponding to r1,r2 \[Element] \
\[DoubleStruckCapitalR] \[Union] {\[Infinity]}."\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "The angular interval needed for ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s graphics object Circle must be specified in numeric order. Furthermore \
the angular interval corresponding to a Poincare geodesic must be < \[Pi]. \
The output of  ",
  StyleBox["Mathematica's ",
    FontSlant->"Italic"],
  " ArcTan",
  StyleBox[" ",
    FontSlant->"Italic"],
  "function, lies in the interval ",
  StyleBox["(-\[Pi], \[Pi]] .\n",
    FontWeight->"Plain"],
  "For this reason, we need a routine to prepare the input\nto use Circle to \
draw a circular arc."
}], "Subsection"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(ArcSort[l_] := \ 
      Module[{sorted = Sort[l]}, \[IndentingNewLine]If[
          sorted[\([2]\)] - sorted[\([1]\)] > \ 
            Pi, \[IndentingNewLine]{sorted[\([2]\)], 
            2\ Pi\  + \ 
              sorted[\([1]\)]}, \[IndentingNewLine]sorted]\ \ \
\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]\), "\
\[IndentingNewLine]", 
    \(ArcSort::usage = "\<ArcSort[l] takes a list l of two angles (residue \
classes modulo 2\[Pi]) and puts them in increasing order, such that their \
difference is < \[Pi].\>"\)}], "Input"],

Cell[BoxData[
    \("ArcSort[l] takes a list l of two angles (residue classes modulo \
2\[Pi]) and puts them in increasing order, such that their difference is < \
\[Pi]."\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(PGSegment[{z1_, z2_}] := \ 
      Module[{w1 = ToUnitDisc[z1], 
          w2 = ToUnitDisc[z2], \[IndentingNewLine]endpoints, r1, r2}, 
        endpoints = EndpointsPGeodesic[z1, z2]; r1 = endpoints[\([1]\)]; 
        r2 = endpoints[\([2]\)]; \[IndentingNewLine]center = 
          CenterPGeodesic[r1, r2]; 
        radius = 
          RadiusPGeodesic[r1, 
            r2]; \[IndentingNewLine]Circle[\(ToR2[center]\)[\([1]\)], radius, 
          ArcSort[\[IndentingNewLine]Map[
              Apply[ArcTan, ToR2[#] - center] &, {w1, 
                w2}]\[IndentingNewLine]]\ \ \ \ \ \ \ \ \ \ \ ]\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ ]\)], "Input"],

Cell["Test for roundoff error:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(PGSegment[{E^\((I\ N[Pi]/5)\), E^\((I\ 4\ N[Pi]/5)\)}] // 
        Chop;\)\), "\n", 
    \(\(PGSegment[{E^\((I\ N[Pi]/5)\),  .000000001\  + 
              E^\((I\ 4\ N[Pi]/5)\)}] // Chop;\)\)}], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    StyleBox[\(Affine[l_] := Drop[l, \(-1\)]/Last[l]\),
      FontWeight->"Bold"], "\[IndentingNewLine]", 
    StyleBox[\(Affine::usage = "\<Affine[l] gives the inhomogeneous \
coordinates of the list of homogeneous coordinates of a point in projective \
space,\>"\),
      FontWeight->"Bold"]}], "Input",
  FontWeight->"Plain"],

Cell[BoxData[
    \("Affine[l] gives the inhomogeneous coordinates of the list of \
homogeneous coordinates of a point in projective space,"\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(IdealFixedPoints[M_] := 
      Map[\(Affine[#]\)[\([1]\)] &, 
        Eigenvectors[M]]\), "\[IndentingNewLine]", 
    \(IdealFixedPoints::usage = "\<IdealFixedPoints[M} gives extended real \
numbers, representing
the attracting fixed point, then the repelling fixed point of the linear \
fractional
transformation represented by the 2x2 matrix M.\>"\)}], "Input"],

Cell[BoxData[
    \("IdealFixedPoints[M} gives extended real numbers, representing the \
attracting fixed point, then the repelling fixed point of the linear \
fractional transformation represented by the 2x2 matrix M."\)], "Output"]
}, Open  ]],

Cell["Generalized Triangle Groups", "Section"],

Cell[TextData[{
  "We use Fricke's parametrization of ",
  StyleBox["SL(2) ",
    FontWeight->"Bold"],
  " of a rank two free group by traces.\nHere the group has a presentation  \
",
  StyleBox["< A, B, C  |  A B C = 1 >",
    FontWeight->"Bold"],
  " and the Fricke\nparameters are the traces ",
  StyleBox["a, b, c ",
    FontWeight->"Bold"],
  "of  ",
  StyleBox["A, B, C",
    FontWeight->"Bold"],
  " respectively."
}], "Text"],

Cell["Generators of the free group", "Subsection",
  FontWeight->"Plain"],

Cell[BoxData[{
    \(AMatrix[a_, b_, c_] := \ {{a, \(-1\)}, {1, 0}}\), "\n", 
    \(BMatrix[a_, b_, c_] := \ {{0, cc1[c]}, {\(-cc2[c]\), b}}\)}], "Input"],

Cell["\<\
Going from traces to representations (the slice) involves one \
quadratic extension:\
\>", "Text"],

Cell[BoxData[{
    \(\(cc1[c_] := \ \((c\  - Sqrt[c^2\  - \ 4])\)/2;\)\), "\n", 
    \(\(cc2[c_] := \ \((c\  + \ Sqrt[c^2\  - 4])\)/2;\)\), "\n", 
    \(CMatrix[a_, b_, 
        c_] := \ {{cc1[c], \(-a\)\ cc1[c] + b}, {0, cc2[c]}}\)}], "Input"],

Cell[BoxData[
    \(ABCMatrices[a_, b_, c_] := {AMatrix[a, b, c], BMatrix[a, b, c], 
        CMatrix[a, b, c]}\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Map[MatrixForm, ABCMatrices[a, b, c]] // Simplify\)], "Input"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{
        TagBox[
          RowBox[{"(", "\[NoBreak]", GridBox[{
                {"a", \(-1\)},
                {"1", "0"}
                }], "\[NoBreak]", ")"}],
          Function[ BoxForm`e$, 
            MatrixForm[ BoxForm`e$]]], ",", 
        TagBox[
          RowBox[{"(", "\[NoBreak]", GridBox[{
                {"0", \(1\/2\ \((c - \@\(\(-4\) + c\^2\))\)\)},
                {\(1\/2\ \((\(-c\) - \@\(\(-4\) + c\^2\))\)\), "b"}
                }], "\[NoBreak]", ")"}],
          Function[ BoxForm`e$, 
            MatrixForm[ BoxForm`e$]]], ",", 
        TagBox[
          RowBox[{"(", "\[NoBreak]", GridBox[{
                {\(1\/2\ \((c - \@\(\(-4\) + c\^2\))\)\), \(b + 
                    1\/2\ a\ \((\(-c\) + \@\(\(-4\) + c\^2\))\)\)},
                {"0", \(1\/2\ \((c + \@\(\(-4\) + c\^2\))\)\)}
                }], "\[NoBreak]", ")"}],
          Function[ BoxForm`e$, 
            MatrixForm[ BoxForm`e$]]]}], "}"}]], "Output"]
}, Open  ]],

Cell["Involutions", "Section"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(InvolutionFixing[z1_, z2_] := 
      1/\((z1\  - \ z2)\)\ {\ {z1\  + \ 
              z2, \ \(-2\)\ z1\ z2}, {2, \ \(-\((z1\  + \ 
                  z2)\)\)}}\), "\[IndentingNewLine]", 
    \(InvolutionFixing[z_, 
        ComplexInfinity] := {{\(-1\), 2\ z}, {0, 
          1}}\), "\[IndentingNewLine]", 
    \(InvolutionFixing[ComplexInfinity, 
        z_] := {{\(-1\), 2\ z}, {0, 1}}\), "\[IndentingNewLine]", 
    \(InvolutionFixing::usage = "\<InvolutionFixing[z1,z2]
gives the 2x2 matrix representing an involution fixing
complex numbers z1, z2. It is normalized to have determinant -1.\>"\)}], \
"Input"],

Cell[BoxData[
    \("InvolutionFixing[z1,z2] gives the 2x2 matrix representing an \
involution fixing complex numbers z1, z2. It is normalized to have \
determinant -1."\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(Clear[M, a, b, c, d]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(InvolutionFixing[3, ComplexInfinity]\)], "Input"],

Cell[BoxData[
    \({{\(-1\), 6}, {0, 1}}\)], "Output"]
}, Open  ]],

Cell[BoxData[{
    \(Projective[z_] := {z, 1}\), "\[IndentingNewLine]", 
    \(Projective[ComplexInfinity] := {1, 0}\)}], "Input"],

Cell[BoxData[{
    \(ApplyLinearFracTransf[M_, 
        r_] := \ \(Affine[
          M . Projective[r]]\)[\([1]\)]\), "\[IndentingNewLine]", 
    \(\(ApplyLinearFracTransf::usage = "\<ApplyLinearFracTransf[M,r] applies \
the linear
fractional transformation defined by the 2x2 matrix M
to the extended real number r.\>";\)\)}], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ApplyLinearFracTransf[InvolutionFixing[3, ComplexInfinity], 
      ComplexInfinity]\)], "Input"],

Cell[BoxData[
    \(ComplexInfinity\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Traceless[M_]\  := \ 
      M\  - \ Tr[M]/2\ IdentityMatrix[2]\), "\[IndentingNewLine]", 
    \(Traceless::usage = "\<Traceless[M] gives the orthogonal projection of a \
2x2 matrix M to the Lie algebra of traceless matrices.\>"\)}], "Input"],

Cell[BoxData[
    \("Traceless[M] gives the orthogonal projection of a 2x2 matrix M to the \
Lie algebra of traceless matrices."\)], "Output"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(Lie[A_, B_]\  := \ A . B\  - \ B . A\), "\[IndentingNewLine]", 
    \(Lie::usage = "\<Lie[A,B] gives the Lie product (additive commutator) of \
matrices A, B.\>"\)}], "Input"],

Cell[BoxData[
    \("Lie[A,B] gives the Lie product (additive commutator) of matrices A, \
B."\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[BoxData[
    \(SLNormalize[M_] := M/Sqrt[Det[M]]\)], "Input"],

Cell[BoxData[
    \(UpperHalfPlaneQ[z_] := \((Im[z] > 0)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(EllipticQ[
        M_] := \ \((Det[M] \[NotEqual] 0)\) && \ \((Tr[M]^2/Det[M] < 
            4)\)\), "\[IndentingNewLine]", 
    \(EllipticQ::usage = "\<EllipticQ[M] tests whether M represents an \
elliptic transformation.\>"\)}], "Input"],

Cell[BoxData[
    \("EllipticQ[M] tests whether M represents an elliptic \
transformation."\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(FixedPointOfElliptic[
        M_] := \(Select[Flatten[Map[Affine, Eigenvectors[SLNormalize[M]]]]\ , 
            UpperHalfPlaneQ]\)[\([1]\)] /; 
        EllipticQ[M]\), "\[IndentingNewLine]", 
    \(FixedPointOfElliptic::usage = "\<FixedPointOfElliptic[M] gives the \
fixed point (in the upper half plane) of the isometry determined
by M, which is assumed to be elliptic.\>"\)}], "Input"],

Cell[BoxData[
    \("FixedPointOfElliptic[M] gives the fixed point (in the upper half \
plane) of the isometry determined by M, which is assumed to be elliptic."\)], \
"Output"]
}, Open  ]],

Cell[BoxData[
    \(EllipticQ[
        M_] := \ \((Det[M] \[NotEqual] 0)\) && \ \((Tr[M]^2/Det[M] < 
            4)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(CommonOrthogonalSegment[A1_, A2_] := 
      Module[{lie12 = Lie[A1, A2]}, 
        Map[FixedPointOfElliptic[lie12 . Traceless[#]] &, {A1, 
            A2}]\ ]\), "\[IndentingNewLine]", 
    \(CommonOrthogonalSegment::usage = "\<CommonOrthogonalSegment[A1,A2] \
gives the shortest segment from the invariant axis of A1 to the invariant \
axis of A2.\>"\)}], "Input"],

Cell[BoxData[
    \("CommonOrthogonalSegment[A1,A2] gives the shortest segment from the \
invariant axis of A1 to the invariant axis of A2."\)], "Output"]
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