(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 1908352, 45093]*) (*NotebookOutlinePosition[ 1909851, 45137]*) (* CellTagsIndexPosition[ 1909667, 45129]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[ \(\((uu = Sqrt[2]/2\ \ {{1 - I, 0}, {0, 1 + I}})\) // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(\(1 - \[ImaginaryI]\)\/\@2\), "0"}, {"0", \(\(1 + \[ImaginaryI]\)\/\@2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((ww = 1/2\ {{1 - I, 1 - I}, {\(-1\) - I, 1 + I}})\) // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(1\/2 - \[ImaginaryI]\/2\), \(1\/2 - \[ImaginaryI]\/2\)}, {\(\(-\(1\/2\)\) - \[ImaginaryI]\/2\), \(1\/2 + \ \[ImaginaryI]\/2\)} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", FontSize->24] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(uu . uu\)], "Input"], Cell[BoxData[ \({{\(-\[ImaginaryI]\), 0}, {0, \[ImaginaryI]}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(uu . uu . uu . uu\)], "Input"], Cell[BoxData[ \({{\(-1\), 0}, {0, \(-1\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ww . ww . ww\)], "Input"], Cell[BoxData[ \({{\(-1\), 0}, {0, \(-1\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((uu . ww)\) . \((uu . ww)\)\)], "Input"], Cell[BoxData[ \({{\(-1\), 0}, {0, \(-1\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pp = ww . 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Conjugate[qq]\)], "Input"], Cell[BoxData[ \({{1, 0}, {0, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Det[ww]\)], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\(\(**\)\(**\)\)*\(\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(**\)\(\ **\)\(**\)\)*ford . m\n routines\ for\ computing\ isometric\ spheres\ of\ elements\ of\ SL \ \((2, C)\)\n\ *****************************************************************************)\ \)\(\[IndentingNewLine]\)\( (*Needs["\"]*) \)\(\n\)\(\ \[IndentingNewLine]\)\(Normalize::usage = "\"\[IndentingNewLine]\n IsometricCircle::usage = "\"\ \[IndentingNewLine]\n Lft::usage = "\"\[IndentingNewLine]\n SetWidth::usage = "\"\[IndentingNewLine]\n DrawCircles::usage = "\"\[IndentingNewLine]\n Trc::usage = "\"\ \[IndentingNewLine]\n AbTr::usage = "\"\[IndentingNewLine]\n ComplexToR2Vector::usage = "\"\[IndentingNewLine]\n CircumCircle::usage = "\"\[IndentingNewLine]\n CenterCircumCircle::usage = "\"\[IndentingNewLine]\n BTetr::usage = "\"\[IndentingNewLine]\n PoincareGeodesic::usage = "\"\[IndentingNewLine]\n \ (*****************************************************************************\ ) \[IndentingNewLine]\[IndentingNewLine] Lft[a_, b_, c_] := DiagonalMatrix[\((b - a)\)/\((b - c)\), 1] . {{1, \(-a\)}, {1, \(-c\)}}\[IndentingNewLine]\n IsometricCircle[m_] := If[m[\([2, 1]\)] \[Equal] 0. , Return[Null], Circle[\(-m[\([2, 2]\)]\)/m[\([2, 1]\)], 1/Abs[m[\([2, 1]\)]]]]\[IndentingNewLine]\n ComplexToR2Vector[z_] := {Re[z], Im[z]}\n Unprotect[Circle]\n Circle[center_?NumberQ, opts___] := Circle[ComplexToR2Vector[center], opts]\n Protect[Circle]\[IndentingNewLine]\n ComplexNormSquared[z_] := z\ Conjugate[z]\n CircumCircle[z1_, z2_, z3_] := Block[{w2, w3, wcenter}, w2 = z2 - z1; w3 = \((z3 - z1)\)/w2; \[IndentingNewLine]If[Im[w3] \[Equal] 0, Line[{ComplexToR2Vector[z1], ComplexToR2Vector[z2]}], wcenter = CenterCircumCircle[w3]; \[IndentingNewLine]Circle[ z1 + w2\ wcenter, Abs[w2\ wcenter]]]]\n CenterCircumCircle[z_] := 1/2 + I\ \((ComplexNormSquared[z] - Re[z])\)/\((2\ Im[z])\)\n Normalize[m_] := Block[{det}, det = Det[m]; \[IndentingNewLine]If[det \[Equal] 0. , Return[Null], m/Sqrt[det]]]\[IndentingNewLine]\n Powers[m_, n_] := Block[{list}, list = {m}; \[IndentingNewLine]Do[ list = Append[list, m . Last[list]], {i, n}]; \[IndentingNewLine]list]\[IndentingNewLine]\n ListOfIsometricCircles[m_, n_] := Map[IsometricCircle, Join[Powers[m, n], Powers[Inverse[m], n]]]\[IndentingNewLine]\n PosListOfIsometricCircles[m_, n_] := Map[IsometricCircle, Powers[m, n]]\[IndentingNewLine]\n Width = 1\ (*default\ value*) \n SetWidth[Width_] := SetOptions[Graphics, PlotRange \[Rule] {{\(-Width\), Width}, {\(-Width\), Width}}, AspectRatio \[Rule] 1]\n SetWidth[Width]\n DrawCircles[mm_, n_] := \((m = Normalize[mm]; \[IndentingNewLine]picture = Graphics[ ListOfIsometricCircles[m, n - 1], {PlotLabel \[Rule] StringJoin["\", ToString[2\ n], "\< Ford Circles for \>", ToString[mm]]}]; \[IndentingNewLine]Show[ picture])\)\n (*Display["\", picture]*) \[IndentingNewLine]\[IndentingNewLine] DrawPosCircles[mm_, n_] := \((m = Normalize[mm]; \[IndentingNewLine]picture = Graphics[ PosListOfIsometricCircles[m, n - 1], {PlotLabel \[Rule] StringJoin["\", ToString[2\ n], "\< Ford Circles for \>", ToString[mm]]}]; \[IndentingNewLine]Show[ picture])\)\n (*Display["\", picture]*) \[IndentingNewLine]\[IndentingNewLine] Test[trace_, n_] := DrawCircles[{{trace, \(-1. \)/3. }, {3. , 0. }}, n]\[IndentingNewLine]\n PosTest[trace_, n_] := DrawPosCircles[{{trace, \(-1. \)/3. }, {3. , 0. }}, n]\[IndentingNewLine]\n Trc[m_] := m[\([1, 1]\)] + m[\([2, 2]\)]\n AbTr[m_] := Abs[Trc[Normalize[m]]]\[IndentingNewLine]\n Test[u_] := Show[Graphics[{u, Circle[{0, 0}, 1]}, UnitCircleOpts]]\[IndentingNewLine]\n (*Examples : Test[2.0 + I, 5]; \[IndentingNewLine]Test[ .9\ I, 5];*) \[IndentingNewLine]\[IndentingNewLine] \)\)\)], "Input"], Cell[BoxData[ \("Normalize[m] returns the unimodular matrix corresponding to the 2x2 \ complex matrix m."\)], "Output"], Cell[BoxData[ \("IsometricCircle[m] returns the Ford isometric circle of the linear \ fractional transformation corresponding to element m of SL(2,C). If m is \ upper triangular, Null is returned."\)], "Output"], Cell[BoxData[ \("Lft[a,b,c] returns the linear fractional transformation taking 0 to a, \ 1 to b, infinity to c."\)], "Output"], Cell[BoxData[ \("SetWidth[w] sets PlotRange to square of width 2w centered at 0."\)], \ "Output"], Cell[BoxData[ \("DrawCircles[m,n] draws the Ford isometric circles for the first n \ powers (both positive and negative) of the Mobius transformation m."\)], \ "Output"], Cell[BoxData[ \("Trc[m] returns the trace of the matrix m."\)], "Output"], Cell[BoxData[ \("AbTr[m] returns the absolute value of the trace of m after it has been \ normalized."\)], "Output"], Cell[BoxData[ \("ComplexToR2Vector[z] converts complex number z to real 2-vector."\)], \ "Output"], Cell[BoxData[ \("CircumCircle[z1,z2,z3] returns the circle passing through the three \ complex numbers z1,z2,z3."\)], "Output"], Cell[BoxData[ \("CenterCircumCircle[z] returns the center of the circle passing through \ the complex numbers 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