(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.1' 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[ 10108, 358]*) (*NotebookOutlinePosition[ 10774, 381]*) (* CellTagsIndexPosition[ 10730, 377]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Translation Invariant Helicoid", "Title"], Cell[TextData[StyleBox["Matthias Weber\nIndiana University\n\ http://www.indiana.edu/~minimal", "Commentary"]], "Author"], Cell[BoxData[ \(<< Own`Mesh`\)], "Input"], Cell[CellGroupData[{ Cell["Weierstrass Data", "Section"], Cell[BoxData[ \(\(\[Alpha]0 = 70.7083362972048057`;\)\)], "Input"], Cell[BoxData[ \(\(\[Tau]0 = Exp[\[Alpha]0\ Degree\ I];\)\)], "Input"], Cell[BoxData[ \(\(\[Theta][z_, \[Tau]_] := EllipticTheta[1, Pi\ z, Exp[\ Pi\ I\ \ \[Tau]]];\)\)], "Input"], Cell[BoxData[ \(G[z_, b_, \[Tau]_] := \((E\^\(I\ \[Pi]\ \((b - 2\ z + 2\ \[Tau] + b\ \ \[Tau])\)\)\ \[Theta][ z + 1\/2\ \((\(-2\) + b)\)\ \((1 + \[Tau])\), \[Tau]]\ \[Theta][ z - 1\/2\ \((1 + b)\)\ \((1 + \[Tau])\), \[Tau]])\)/\((\[Theta][ z + 1\/2\ \((\(-1\) + b)\)\ \((1 + \[Tau])\), \[Tau]]\ \[Theta][ z - 1\/2\ b\ \((1 + \[Tau])\), \[Tau]])\)\)], "Input"], Cell[BoxData[ \(dh[z_, b_, \[Tau]_] := \(\[Theta][z + 1\/2\ \((\(-2\) + b)\)\ \((1 + \[Tau])\ \), \[Tau]]\ \[Theta][z - 1\/2\ b\ \((1 + \[Tau])\), \[Tau]]\)\/\(\[Theta][z \ + 1\/2\ \((\(-1\) + b)\)\ \((1 + \[Tau])\), \[Tau]]\ \[Theta][z - 1\/2\ \((1 \ + b)\)\ \((1 + \[Tau])\), \[Tau]]\)\)], "Input"], Cell[BoxData[ \(\(dhper = 0.386191090012370175`\[InvisibleSpace] - 0.169838749468014027`\ I;\)\)], "Input"], Cell[BoxData[ \(\(\[Omega]3[z_] := dh[z, b0, \[Tau]0]/dhper;\)\)], "Input"], Cell[BoxData[ \(\(\[Rho]1 = 108.369522264594063`\[InvisibleSpace] - 62.8417365006266681`\ I;\)\)], "Input"], Cell[BoxData[ \(\(b0 = 0.629065098323904514`;\)\)], "Input"], Cell[BoxData[ \(G[z_] := \[Rho]1\ G[z, b0, \[Tau]0]\)], "Input"], Cell[BoxData[{ \(\(\[Omega]2[z_]\ := I \((\ G[z]\ \[Omega]3[z] + 1/G[z]\ \[Omega]3[z]\ )\)/2;\)\), "\n", \(\(\(\[Omega]1[ z_]\ := \(-\((\ G[z]\ \[Omega]3[z] - 1/G[z]\ \[Omega]3[z])\)\)/ 2;\)\(\n\) \)\)}], "Input"], Cell["Weierstra\[SZ] integrals:", "Text"], Cell[BoxData[ \(ww2[z_] := Re[NIntegrate[{\[Omega]1[w], \[Omega]2[w], \[Omega]3[w]}, {w, 1, z}]]\)], "Input"], Cell[BoxData[ \(\(w0 = ww2[\[Tau]0/2];\)\)], "Input"], Cell[BoxData[ \(\(f[z_] := Re[NIntegrate[{\[Omega]1[w], \[Omega]2[w], \[Omega]3[w]}, {w, \[Tau]0/ 2, z}]] + w0;\)\)], "Input"], Cell[BoxData[ \(n[z_] := StereographicProjection[G[z]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Defining the Parameterization", "Section"], Cell[BoxData[ \(\(\[Epsilon] = 10. ^\(-7\);\)\)], "Input"], Cell["\<\ The strategy is to parametrize half of the rhombus. This is a \ rectangular domain with two punctures at the ends. This rectangle is \ parametrized by the upper half plane,using elliptic integrals (via \ Schwarz-Christoffel maps). For the upper half plane, polar coordinates are \ used so that r=0 and r=\[Infinity] correspond to the ends. Special care is \ taken for the radii which go to the rectangle corners. Also, so that \ fundamental pieces fit well together, the radii have to lie symmetric. \ \>", \ "Text"], Cell["Polar coordinates for upper half plane:", "Text"], Cell[BoxData[ \(HalfStripToUpperHalfPlane[z_] := Exp[z]\)], "Input"], Cell["\<\ Rearrange punctures in upper half plane so that 0 and \[Infinity] \ become ends:\ \>", "Text"], Cell[BoxData[ \(tr[z_, r_, a_] := \((\(-a\) - r\ z)\)/\((\(-1\) + a\ z)\)\)], "Input"], Cell["\<\ Map from upper half plane to rectangle so that: \t\t-1\[Rule]0, 1\[Rule]1,-r\[Rule]i b, r\[Rule]1+i b where b is the height of the rectangle. r i sthe modulus of the \ rectangle:\ \>", "Text"], Cell[BoxData[ \(\(tst[z_, r_] := EllipticF[ArcSin[z], 1/r^2]/\((2 EllipticF[\[Pi]\/2, 1/r^2])\) + .5;\)\)], "Input"], Cell["Determine modulus from height:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindRoot[ Im[\((1 + \[Tau]0)\)/2 \((1 + tst[\(-r\), r])\) - \[Tau]0], {r, 1.1, 3}]\)], "Input"], Cell[BoxData[ \({r \[Rule] 2.4305051649092477`}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(r0 = 2.43050611112724901`;\)\)], "Input"], Cell["\<\ Redefine tst as tst[z_]:=tst[z,r0]; For efficiency use \ instead:\ \>", "Text"], Cell[BoxData[ \(\(quot0 = \((2 EllipticF[\[Pi]\/2, 1/r0^2])\);\)\)], "Input"], Cell[BoxData[ \(\(tst[z_] := EllipticF[ArcSin[z], 1/r0^2]/quot0 + .5;\)\)], "Input"], Cell["\<\ Find the parameter a which determines the \ M\[ODoubleDot]biustransformation which sends the ends to 0 and \[Infinity]:\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindRoot[tst[a] - \((1 - b0)\), {a, \(-1\), 1}]\)], "Input"], Cell[BoxData[ \({2.1995197724666147` \[Rule] \(-0.40995577890340235`\)}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(a0 = \(-0.409955776251214221`\);\)\)], "Input"], Cell["Redefine tr:", "Text"], Cell[BoxData[ \(tr[z_] := tr[z, r0, a0]\)], "Input"], Cell[BoxData[ \(par[z_] := tst[tr[HalfStripToUpperHalfPlane[z]]]\)], "Input"], Cell["Finally, the actual parametrization:", "Text"], Cell[BoxData[ \(\(g[w_] := par[w] \((1 + \[Tau]0)\)/2;\)\)], "Input"], Cell["\<\ Determine the special points corresponding to the reactangle \ corners:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[tr[HalfStripToUpperHalfPlane[z]] == 1, z]\)], "Input"], Cell[BoxData[ \({{z \[Rule] \(-0.35981157777783046`\)}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[tr[HalfStripToUpperHalfPlane[z]] == \(-1\), z]\)], "Input"], Cell[BoxData[ \({{z \[Rule] \(-1.571524464735095`\) + 3.141592653589793`\ \[ImaginaryI]}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[tr[HalfStripToUpperHalfPlane[z]] == r0, z]\)], "Input"], Cell[BoxData[ \({{z \[Rule] 0.683424952885058356`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[tr[HalfStripToUpperHalfPlane[z]] == \(-r0\), z]\)], "Input"], Cell[BoxData[ \({{z \[Rule] \(-0.528287934072206422`\) + 3.14159265358979311`\ I}}\)], "Output"] }, Open ]], Cell["Symmetry position:", "Text"], Cell[BoxData[ \(\(sym = \(-0.359811577777830482`\) - \ 0.528287934072206422`;\)\)], "Input"], Cell["Mesh resolution parameters:", "Text"], Cell[BoxData[{ \(\(nr1 = 3;\)\), "\[IndentingNewLine]", \(\(nr2 = 5;\)\), "\[IndentingNewLine]", \(\(nr3 = 2;\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(nt = 15;\)\)}], "Input"], Cell["r1 controls the size of the helicoidal end:", "Text"], Cell[BoxData[ \(\(r1 = \(-2.5\);\)\)], "Input"], Cell[BoxData[ \(\(xlist0 = NRange[r1, \(-1.57152446473509517`\), nr1] \[Union] NRange[\(-1.57152446473509517`\), \(-0.528287934072206422`\), nr2] \[Union] NRange[\(-0.528287934072206422`\), \(-0.44404975592501844`\), nr3];\)\)], "Input"], Cell[BoxData[ \(\(xlist = Union[xlist0, \((sym - xlist0)\), SameTest \[Rule] AlmostEqual];\)\)], "Input"], Cell[BoxData[ \(\(mesh1 = RectangularDomain[xlist, NRange[\[Epsilon], Pi - \[Epsilon], 10]];\)\)], "Input"], Cell[BoxData[ \(\(mesh2 = MeshApply[g, mesh1];\)\)], "Input"], Cell[BoxData[ \(\(g2 = DomainToGraphics[mesh2];\)\)], "Input"], Cell["This shows the parameter domain with its coordinate lines:", "Text"], Cell[BoxData[ \(\(Show[g2, AspectRatio \[Rule] Automatic];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Drawing a surface patch", "Section"], Cell[BoxData[ \(\(mp1 = MeshPlot3D[{f[x + I\ y], n[x + I\ y]}, {x, y}, mesh2];\)\)], "Input"], Cell[BoxData[ \(\(Show[gl = Mesh3DToGraphics3D[mp1], PlotRange \[Rule] All, Axes \[Rule] False];\)\)], "Input"], Cell[BoxData[ \(\(mp2 = MeshJoin[mp1, MeshRotate[mp1, StraightLine[{0, 0, 0}, {0, 0, 1}]]];\)\)], "Input"], Cell[BoxData[ \(\(mp3 = MeshJoin[mp2, MeshTranslate[mp2, {0, 0, 2}], MeshTranslate[mp2, {0, 0, \(-2\)}]];\)\)], "Input"], Cell[BoxData[ \(\(Show[gl = Mesh3DToGraphics3D[mp3], PlotRange \[Rule] All, Axes \[Rule] False];\)\)], "Input"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.1 for Macintosh", ScreenRectangle->{{57, 1680}, {0, 1028}}, AutoGeneratedPackage->None, WindowSize->{957, 805}, WindowMargins->{{Automatic, 0}, {Automatic, 21}} ] (******************************************************************* Cached data follows. 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