(************** 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 1664380, 28833]*) (*NotebookOutlinePosition[ 1665320, 28862]*) (* CellTagsIndexPosition[ 1665276, 28858]*) (*WindowFrame->Normal*) Notebook[{ Cell["The Poincare Model (version 7)", "Title", FontSize->24], Cell["\<\ W. 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When z is real, this agrees with \ ToCircle[z]."\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(ToR2[z_] := {Re[z], Im[z]}\), "\[IndentingNewLine]", \(ToR2::usage = \*"\"\\""\)}], "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]", \(NormSquared::usage = \*"\"\\""\)}], \ "Input"], Cell[BoxData[ \("NormSquared[v] gives the square of the Euclidean norm of a vector in \ \!\(\[DoubleStruckCapitalR]\^2\)."\)], "Output"] }, Open ]], Cell["Drawing Geodesics", "Subtitle"], Cell[CellGroupData[{ 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.\ \>", \ 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\(PGSegment[{ComplexInfinity, 1}]\)], "Input"], Cell[BoxData[ \(PGSegment[{ComplexInfinity, 1}]\)], "Output"] }, Open ]] }, Open ]], Cell[TextData[{ "In this case the Poincare geodesic is a circular arc (in ", StyleBox["Mathematica", FontSlant->"Italic"], " notation ", Cell[BoxData[ \(Circle[{1, 1}, 1, {\[Pi], \(3\ \[Pi]\)\/2}]\)]], ",\nfor which we presently develop the code." }], "Text"], 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 = "\"\)\(\ \)\)\)], "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"], 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]", \(EndpointsPGeodesic::usage = "\"\)}], "Input"], Cell[BoxData[ \("EndpointsPGeodesic[z1,z2] gives the endpoints of the Poincare geodesic \ containing z1 and z2 in the upper half plane."\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Now we transform to the unit disc.", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[{ \(xxParam[r1_, r2_] := \((ToCircle[r1] + ToCircle[r2])\)/ 2\), "\[IndentingNewLine]", \(xxParam::usage = "\"\), "\n", \(LengthxxParam[r1_, r2_] := Sqrt[NormSquared[xxParam[r1, r2]]]\), "\n", \(LengthxxParam::usage = "\"\)}], "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 ]] }, Open ]], Cell[CellGroupData[{ 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]", \(RadiusPGeodesic::usage = "\"\), "\ \[IndentingNewLine]", \(CenterPGeodesic::usage = "\"\)}], "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 ]] }, Open ]], Cell[CellGroupData[{ 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]", \(ArcSort::usage = "\"\)}], "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[CellGroupData[{ Cell[BoxData[{ \(RepairCircularArc[ Circle[{a_, b_}, r_, {theta1_, theta2_}]] := \[IndentingNewLine]Circle[{a, b}, r, {Min[theta2, theta1], Max[theta2, theta1]}]\), "\[IndentingNewLine]", \(RepairCircularArc::usage = \ "\"\)}], "Input"], Cell[BoxData[ \("RepairCircularArc[Circle[center,radius,{t1,t2}]] ensures that t2 is \ not less than t1."\)], "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]RepairCircularArc[ 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[BoxData[ \(Line[{{0.5095254494944288`, 0}, {\(-0.5095254494944287`\), 0}}]\)], "Output"], Cell[BoxData[ \(Circle[{0, \(-2.4721353651577`*^9\)}, 2.4721353651577005`*^9, {1.5707963265887892`, 1.5707963270010041`}]\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(ShowPList[l_] := \ ShowP[Map[PGSegment, l]]\), "\n", \(ShowPList[l_, viewingTransformation_] := ShowPList[Map[ApplyLFT[viewingTransformation, #] &, l]]\), "\n", \(ShowPList::usage = "\"\)}], "Input"], Cell[BoxData[ \("ShowPList[l] draws segments in the Poincare unit disc in a list l of \ pairs of extended complex numbers in the upper half-plane. When called with \ two arguments, it applies a viewing transformation (a 2x2 real matrix) before \ drawing."\)], "Output"] }, Open ]], Cell["Here are some examples:", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(ShowP[{PGSegment[{ComplexInfinity, 0}]}];\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.454545 0.5 0.454545 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath 0 g .5 Mabswid [ ] 0 setdash newpath .5 .5 .45455 0 365.73 arc s .5 .95455 m .5 .04545 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 288}, ImageMargins->{{43, 0}, {0, 0}}, 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