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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[ 48042, 1540]*) (*NotebookOutlinePosition[ 49192, 1579]*) (* CellTagsIndexPosition[ 49103, 1573]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontFamily->"Times", FontSize->18, FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[0, 0.500008, 0]], StyleBox[" Graphics in the Internet:\nAdditional Lighting and Clipping in \ LiveGraphics3D ", FontFamily->"Times", FontSize->18, FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0]], StyleBox["\n", FontFamily->"Times", FontSize->18, FontWeight->"Bold"], StyleBox["\n", FontFamily->"Times", FontSize->12, FontWeight->"Bold"], StyleBox[ButtonBox["Ralf Schaper", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/"], None}, ButtonStyle->"Hyperlink"], FontFamily->"Times New Roman"], StyleBox["\nUniversit\[ADoubleDot]t Gesamthochschule Kassel", FontFamily->"Times", FontSize->12, FontWeight->"Bold"], StyleBox["\n", FontFamily->"Times", FontSize->18, FontWeight->"Bold"], StyleBox["\n", FontFamily->"Times New Roman", FontSize->18, FontWeight->"Bold"], StyleBox["Juli 2001 (revised September 2001)", FontFamily->"Arial", FontSize->12] }], "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell[TextData[{ "Since 1997 ", ButtonBox["Martin Kraus", ButtonData:>{ URL[ "http://wwwvis.informatik.uni-stuttgart.de/~kraus/index.html"], None}, ButtonStyle->"Hyperlink"], " from Stuttgart develops ", StyleBox[ButtonBox["LiveGraphics3D", ButtonData:>{ URL[ "http://wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGraphics3D/index.\ html"], None}, ButtonStyle->"Hyperlink"], FontSlant->"Italic", FontVariations->{"Underline"->True}], ". This is a ", StyleBox[ButtonBox["Java 1.1", ButtonData:>{ URL[ "http://java.sun.com/"], None}, ButtonStyle->"Hyperlink"], FontVariations->{"Underline"->True}], "-Applet to display and to rotate three-dimensional graphics in HTML pages \ produced by ", StyleBox["Mathematica", FontSlant->"Italic"], ". The project is supported by ", StyleBox[ButtonBox["Wolfram Research", ButtonData:>{ URL[ "http://www.wolfram.com/"], None}, ButtonStyle->"Hyperlink"], FontVariations->{"Underline"->True}], ", especially in showing ", ButtonBox["polytopes", ButtonData:>{ URL[ "http://polyhedra.wolfram.com/"], None}, ButtonStyle->"Hyperlink"], ". There is also a short ", StyleBox[ButtonBox["user manual", ButtonData:>{ URL[ "http://polyhedra.wolfram.com/live.html"], None}, ButtonStyle->"Hyperlink"], FontVariations->{"Underline"->True}], ". \nIn his well written ", StyleBox[ButtonBox["documentation", ButtonData:>{ URL[ "http://wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGraphics3D/\ documentation.html"], None}, ButtonStyle->"Hyperlink"], FontVariations->{"Underline"->True}], " Martin Kraus mentions two ", StyleBox[ButtonBox["restrictions:\n", ButtonData:>{ URL[ "http://wwwvis.informatik.uni-stuttgart.de/~kraus/LiveGraphics3D/\ documentation.html#subsection: Append-Limitations"], None}, ButtonStyle->"Hyperlink"], FontVariations->{"Underline"->True}], StyleBox["\"", Background->None, FontVariations->{"CompatibilityType"->0}], StyleBox["LiveGraphics3D", FontSlant->"Italic", Background->None, FontVariations->{"CompatibilityType"->0}], StyleBox[" does not clip primitives which are outside of the PlotRange. \ \nIn some situations the color of \ the wrong face of a polygon is used. The reason is the simple (but fast) \ algorithm being used to decide which face is painted.\" \nTwo years ago I \ worked with ", Background->None, FontVariations->{"CompatibilityType"->0}], StyleBox["LiveGraphics3D", FontSlant->"Italic", Background->None, FontVariations->{"CompatibilityType"->0}], StyleBox[" and I found some solutions. Now at last I will give a \ description.", Background->None, FontVariations->{"CompatibilityType"->0}] }], "Text", TextJustification->1] }, Open ]], Cell[CellGroupData[{ Cell["Initialization", "Subsection"], Cell[TextData[{ "Don`t be alarmed that seven packages will be loaded now. Further on \ working with this notebook will become easier with these packages. Some hints \ about installation are given below. Only ", StyleBox["ExtendGraphics`Geometry3D`", FontFamily->"Courier"], StyleBox[" and ", FontFamily->"Times New Roman"], StyleBox["LiveGraphics3D`", FontFamily->"Courier"], " are not part of the standard ", StyleBox["Mathematica", FontSlant->"Italic"], " packages." }], "Text", TextJustification->1], Cell[BoxData[{ \(\(Off[\ General\ ::\ spell\ ];\)\), "\[IndentingNewLine]", \(\(\(Off[\ General\ ::\ spell1\ ];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\(Needs[\ "\"\ ];\)\), \ "\[IndentingNewLine]", \(\(Needs[\ "\"\ ];\)\), "\[IndentingNewLine]", \(\(Needs[\ "\"\ ];\)\), "\[IndentingNewLine]", \(\(Needs[\ "\"\ ];\)\), \ "\[IndentingNewLine]", \(\(Needs[\ "\"\ ];\)\), "\[IndentingNewLine]", \(\(Needs[\ "\"\ ];\)\), "\[IndentingNewLine]", \(\(\(Needs[\ "\"\ ];\)\(\[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(\($DefaultFont\ = \ {"\", \ 12};\)\)}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 1: Lighting", "Subsection"], Cell[TextData[{ "Take an icosahedron to get an impression of the problem of lighting \ \"backward\" faces of polygons and surfaces with ", StyleBox["LiveGraphics3D", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], ". When you have loaded the package ", StyleBox["Graphics`Polyhedra`", FontFamily->"Courier"], " you get an image of an icosahedron with the following expression:" }], "Text", TextJustification->1], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(gr1\ = \ Show[\ Polyhedron[\ Icosahedron\ ], Boxed\ \[Rule] \ False\ ];\)\)\)], "Input"], Cell[TextData[{ StyleBox["WriteLiveForm", FontFamily->"Courier"], " of ", StyleBox["LiveGraphics3D", FontSlant->"Italic"], " will produce the HTML-file. But you will see \"shadows\" on the backward \ faces." }], "Text", TextJustification->1], Cell[BoxData[ \(\(WriteLiveForm[\ "\", \ gr1\ ];\)\)], "Input"], Cell[TextData[{ "Now you can look ", ButtonBox["at the result", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu1.html"], None}, ButtonStyle->"Hyperlink"], ". " }], "Text"], Cell[TextData[{ "You can look around the icosahedron from all directions with ", StyleBox["SpinShow", FontFamily->"Courier"], " from the package ", StyleBox["Graphics`Animation`:", FontFamily->"Courier"], " Also the backward faces are colored:" }], "Text", TextJustification->1], Cell[BoxData[ \(animatgr1\ = \ SpinShow[\ gr1, \ SpinRange\ \[Rule] \ {\ 0\ Degree, \ 360\ Degree\ }, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ SpinDistance\ \[Rule] \ 2, SpinOrigin\ \[Rule] \ {0, \ 0, \ 0}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Frames\ \[Rule] \ 24\ ]\)], "Input"] }, Open ]], Cell[TextData[{ "You may think that the wrong colors of the backward faces of the \ icosahedron may be due to the closed surface. But you get the same phenomenon \ with surfaces like the hyperbolic paraboloid with the simple equation ", StyleBox["z", FontSlant->"Italic"], " = ", StyleBox["x\[CenterDot]y", FontSlant->"Italic"], " ." }], "Text", TextJustification->1], Cell[BoxData[ \(\(gr2\ = \ Plot3D[\ x\ y, \ {x, \(-2\), 2}, {y, \(-2\), 2}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes\ \[Rule] \ None, PlotPoints\ \[Rule] \ 7\ ];\)\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", \ gr2\ ];\)\)], "Input"], Cell[TextData[{ "Now the ", ButtonBox["LiveGraphics3D-version", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu2.html"], None}, ButtonStyle->"Hyperlink"], ". " }], "Text"], Cell["The faces from \"below\" are colored:", "Text"], Cell[BoxData[ \(\(Show[\ gr2, \ ViewPoint\ \[Rule] \ {\(-0.5\), 0, \ \(-2\)}\ ];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Problem 2: Clipping", "Subsection"], Cell[TextData[{ "There are several possibilities in ", StyleBox["Mathematica", FontSlant->"Italic"], " to restrict values of functions or parts of the domain of definition, \ e.g. in the image ", StyleBox["gr2", FontFamily->"Courier", FontWeight->"Bold"], ":" }], "Text", TextJustification->1], Cell[BoxData[{ \(\(\[Alpha]\ = \ 1.5;\)\), "\[IndentingNewLine]", \(\(gr3\ = \ Plot3D[\ x\ y, {x, \(-2\), 2}, {y, \(-2\), 2}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints\ \[Rule] \ 7, \ PlotRange\ \[Rule] \ {\(-\[Alpha]\), \[Alpha]}\ ];\)\)}], "Input"], Cell["\<\ A little change and a restriction of the plot domain produce a more agreeable \ image: \ \>", "Text"], Cell[BoxData[ \(\(Plot3D[\ x\ y, \ {x, \(-2\), 2}, {y, \(-2\), 2}, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ PlotPoints\ \ \[Rule] \ 7, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ PlotRange\ \[Rule] \ \ {{\(-\[Alpha]\), \[Alpha]}, \ {\(-\[Alpha]\), \[Alpha]}, \ {\(-\[Alpha]\), \ \[Alpha]}}\ ];\)\)], "Input"], Cell[TextData[{ StyleBox["Axes \[Rule] None", FontFamily->"Courier"], " will suppress the ticks at the box. They are not needed for the \ transformation with ", StyleBox["LiveGraphics3D", FontSlant->"Italic", Background->None, FontVariations->{"CompatibilityType"->0}], " ." }], "Text"], Cell[BoxData[ \(\(gr4\ = \ Plot3D[\ x\ y, \ {x, \(-2\), 2}, {y, \(-2\), 2}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints\ \[Rule] \ 7, \ Axes\ \[Rule] \ None, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotRange\ \[Rule] \ {{\(-\[Alpha]\), \[Alpha]}, \ {\(-\[Alpha]\), \ \[Alpha]}, \ {\(-\[Alpha]\), \[Alpha]}}\ ];\)\)], "Input"], Cell[BoxData[{ \(\(WriteLiveForm[\ "\", \ gr3\ ];\)\), "\[IndentingNewLine]", \(\(WriteLiveForm[\ "\", \ gr4\ ];\)\)}], "Input"], Cell[TextData[{ "In every case ", StyleBox["LiveGraphics3D", FontSlant->"Italic"], " does not produce the desired result:\n", ButtonBox["klafu3.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu3.html"], None}, ButtonStyle->"Hyperlink"], "\n", ButtonBox["klafu4.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu4.html"], None}, ButtonStyle->"Hyperlink"] }], "Text"], Cell[TextData[{ "I think that the following ", StyleBox["Mathematica", FontSlant->"Italic"], " figure will be adequate:" }], "Text"], Cell[BoxData[ \(\(gr5\ = \ Show[\ Graphics3D[\ gr2\ ], \ PlotRange\ \[Rule] \ {\(-\[Alpha]\), \[Alpha]}\ ];\)\)], "Input"], Cell[TextData[{ "Also transforming this figure with ", StyleBox["LiveGraphics3D", FontSlant->"Italic", Background->None, FontVariations->{"CompatibilityType"->0}], " gives not the desired result:" }], "Text"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", \ gr5\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu5.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu5.html"], None}, ButtonStyle->"Hyperlink"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["A solution of problem 1 : Lighting", "Subsection"], Cell[TextData[{ "To switch on some \"additional lights\" gives an obvious solution. You \ will find some information about the option ", StyleBox[ButtonBox["LightSources", ButtonStyle->"RefGuideLink"], FontVariations->{"Underline"->True}], " in the ", StyleBox["Mathematica", FontSlant->"Italic"], " Help-Browser. This option will be used in the next expressions. More \ information can be found in [Smith, Blachman, p. 254], [Wickham-Jones, p. \ 213] and [Schaper, p. 189] ." }], "Text", TextJustification->1], Cell[TextData[{ "You can get information on the default ", StyleBox["LightSources", FontFamily->"Courier"], ":" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Options[\ Plot3D, \ LightSources\ ]\)], "Input"], Cell[BoxData[ \({LightSources \[Rule] {{{1.`, 0.`, 1.`}, RGBColor[1, 0, 0]}, {{1.`, 1.`, 1.`}, RGBColor[0, 1, 0]}, {{0.`, 1.`, 1.`}, RGBColor[0, 0, 1]}}}\)], "Output"] }, Open ]], Cell[TextData[{ StyleBox["LightSources", "MR"], " ", StyleBox["->", "MR"], " ", StyleBox["{", FontFamily->"Courier"], StyleBox["direction", "TI"], StyleBox[",", "MR"], " ", StyleBox["color", "TI"], StyleBox["}", FontFamily->"Courier"], " produces parallel light with the direction ", StyleBox["direction", FontSlant->"Italic"], " and with the color ", StyleBox["color", FontSlant->"Italic"], ".\nNow additional lightsources are used:" }], "Text"], Cell[BoxData[ \(\(lights\ = \ {{{\ \ 1, \ \ 0, \ \ \ \ 1}, Red\ \ }, {{\ \ 1, \ \ \ 1, \ \ 1}, Green}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 0, \ \ \ 1, \ \ \ 1}, Blue}\ , \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (*\ \(additional\)\(:\)\ *) \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\(-1\), \ \ 0, \(-1\)}, Red\ \ }, {{\(-1\), \(-1\), \(-1\)}, Green}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 0, \(-1\), \(-1\)}, Blue}\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ };\)\)], "Input"], Cell[BoxData[ \(\(SetOptions[\ Plot3D, \ LightSources\ \[Rule] \ lights\ ];\)\)], "Input"], Cell["\<\ As above the hyperbolic paraboloid and the icosahedron are used to give some \ explanations.\ \>", "Text"], Cell[BoxData[ \(\(\(gr6 = \ Plot3D[\ x\ \ y, \ {x, \(-2\), 2}, {y, \(-2\), 2}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes\ \[Rule] \ None, Boxed\ \[Rule] \ True, \ \ \ \ \ \ \ \ \ \ \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints\ \[Rule] \ 7\ ];\)\(\[IndentingNewLine]\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)], "Input"], Cell[TextData[{ "Using an adequate choice of ", ButtonBox["ViewPoint", ButtonStyle->"RefGuideLink"], " you can look at the surface from \"below\". " }], "Text"], Cell[BoxData[ \(\(Show[\ gr6, \ ViewPoint\ \[Rule] \ {\(-2\), 0, \ \(-3\)}\ ];\)\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr6\ ];\)\)], "Input"], Cell[TextData[{ ButtonBox["Here", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu6.html"], None}, ButtonStyle->"Hyperlink"], " you can look at the HTML-version. And all polygons are colored!\n\nThe \ icosahedron will be treated in an analogous manner :" }], "Text"], Cell[BoxData[ \(\(gr7\ = \ Show[\ Polyhedron[\ Icosahedron\ ], \ LightSources\ \[Rule] \ lights, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Boxed\ \[Rule] \ False\ ];\)\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr7\ ];\)\)], "Input"], Cell[TextData[{ "You can see the ", ButtonBox["positive effect", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu7.html"], None}, ButtonStyle->"Hyperlink"], " . Compare this example with the ", ButtonBox["old one", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu1.html"], None}, ButtonStyle->"Hyperlink"], " ." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["A solution of problem 2 : Clipping", "Subsection"], Cell[TextData[{ "Again we will begin with the hyperbolic paraboloid. The image ", StyleBox["gr3", FontFamily->"Courier", FontWeight->"Bold"], " is reproduced as ", StyleBox["gr8\[Alpha]", FontFamily->"Courier", FontWeight->"Bold"], " :" }], "Text"], Cell[BoxData[ \(\(gr8\[Alpha]\ = \ Graphics3D[\ Plot3D[\ x\ y, {x, \(-2\), 2}, {y, \(-2\), 2}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Axes\ \[Rule] \ True, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints\ \[Rule] \ 7, \ PlotRange\ \[Rule] \ {\(-\[Alpha]\), \[Alpha]}\ ]]\ ;\)\)], \ "Input"], Cell[TextData[{ "Now the surface is clipped by ", StyleBox["Clip3D", FontFamily->"Courier"], " from the package ", StyleBox["ExtendGraphics`Geometry3D`", FontFamily->"Courier"], " of Tom Wickham-Jones:" }], "Text"], Cell[BoxData[{ \(\(gr8\[Beta] = \ Fold[\ \ Clip3D[#1, #2] &, gr8\[Alpha], \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\ Plane[{0, 0, \ \ \[Alpha]}, {0, \ 0, \(-1\)}], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plane[{0, 0, \(-\[Alpha]\)}, {0, \ 0, \ \ 1}]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }];\)\ \ \), "\[IndentingNewLine]", \(\(gr8\[Gamma]\ = Show[\ gr8\[Beta]\ , \ Axes\ \[Rule] \ True\ ];\)\)}], "Input"], Cell[TextData[{ "The option ", StyleBox["Axes", FontFamily->"Courier"], " will get the value ", StyleBox["None", FontFamily->"Courier"], ":" }], "Text"], Cell[BoxData[ \(\(gr8\ = Show[\ gr8\[Gamma]\ , \ Axes\ \[Rule] \ None\ ];\)\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr8\ ];\)\)], "Input"], Cell[TextData[{ "As ", ButtonBox["you can see", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu8.html"], None}, ButtonStyle->"Hyperlink"], " ", StyleBox["LiveGraphics3D ", FontSlant->"Italic", Background->None, FontVariations->{"CompatibilityType"->0}], "and the clipping work together. Since above the option ", StyleBox["LightSources", FontFamily->"Courier"], " of ", StyleBox["Plot3D", FontFamily->"Courier"], " has been changed also the backward faces are colored. You get the same \ effect with the icosahedron. Now different colors for the lightsourses are \ chosen. " }], "Text", TextJustification->1], Cell[BoxData[ \(\(gr9\ = \ Show[\ Polyhedron[\ Icosahedron\ ], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ LightSources\ \[Rule] {{{1, \ \ \ 0, \ \ 1}, \ Red}, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ 1, \ \ \ 1, \ \ \ 1}, Green}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ 0, \ \ \ 1, \ \ \ 1}, Blue}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\(-1\), \ \ \ 0, \(-1\)}, Tomato}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ 0, \(-1\ \), \(-1\)}, SkyBlueDeep}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\(-1\)\ , \(-1\), \(-1\)}, GreenDark}\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Boxed\ \[Rule] \ False\ ];\)\)], "Input"], Cell[BoxData[{ \(\(\[Alpha]\ = \ 3/4;\)\), "\[IndentingNewLine]", \(\(gr10 = \ Fold[\ Clip3D[#1, #2] &, gr9, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Plane[{0, 0, \ \ \[Alpha]}, {0, \ 0, \(-1\)}], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plane[{0, 0, \(-\[Alpha]\)}, {0, 0, \ \ 1}]}];\)\), "\[IndentingNewLine]", \(\(Show[\ gr10\ ];\)\)}], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr10\ ];\)\)], "Input"], Cell[TextData[{ "Here you get the ", ButtonBox["WWW images", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu10.html"], None}, ButtonStyle->"Hyperlink"], " ." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Applications", "Subsection"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Klein Bottle", FontWeight->"Bold"]], "Text"], Cell[TextData[{ "At the beginning three different forms of the Klein bottle are clipped. In \ the ", StyleBox["Mathematica", FontSlant->"Italic"], "-Book [1999] you will find the first form on page 995 resp. in the Help \ Browser at ", StyleBox[ButtonBox["Light Source Variations.", ButtonData:>{"G.2.46", None}, ButtonStyle->"MainBookLink"], FontVariations->{"Underline"->True}], " [Schaper, S. 257, ff.] gives parametrisations of the first and second \ form. You can get the parametrisations of the second and third form in from \ this ", ButtonBox["page", ButtonData:>{ URL[ "http://www.uib.no/people/nfytn/surfaces.htm"], None}, ButtonStyle->"Hyperlink"], " . This ", StyleBox[ButtonBox["page", ButtonData:>"Klein Bottle Formula", ButtonStyle->"GettingStartedLink"], FontVariations->{"Underline"->True}], " of the Help Browser of ", StyleBox["Mathematica", FontSlant->"Italic"], " gives supplementary information on the second form. Also see [Gray, p. \ 327].\n", "(Inserted in September 2001: The parametrisation of the second form can be \ found in Dieudonn\[EAcute] on page 192 resp. Seite 194. See also: The ", StyleBox["Mathematica", FontSlant->"Italic"], " Journal, 1, (3), 1991, p. 65.)" }], "Text", TextJustification->1], Cell[CellGroupData[{ Cell["First form:", "Subsubsection"], Cell["\<\ bx = 6 Cos[u] (1 + Sin[u]); by = 16 Sin[u]; rad = 4 - 2 Cos[u]; X = If[ Pi < u <= 2 Pi, bx + rad Cos[v + Pi], bx + rad Cos[u] Cos[v] ]; Y = If[ Pi < u <= 2 Pi, by, by + rad Sin[u] Cos[v] ]; Z = rad Sin[v];\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}, FontFamily->"Courier New", Background->GrayLevel[1]], Cell[BoxData[ \(\(SetOptions[\ ParametricPlot3D, \ LightSources\ \[Rule] \ lights];\)\)], "Input"], Cell["\<\ gr11 = ParametricPlot3D[ {X, Y, Z}, {u, 0, 2\[Pi]}, {v, 0, 2\[Pi]}, PlotPoints -> {48, 12}, Axes -> False, Boxed -> False, ViewPoint-> {0,-2,-2}];\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}, FontFamily->"Courier New", Background->GrayLevel[1]], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr11\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu11.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu11.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell[TextData[{ "If ", StyleBox["v", FontSlant->"Italic"], " \[Element] [0, \[Pi]] you get this \"insight\":" }], "Text"], Cell[BoxData[ \(\(gr12\ = ParametricPlot3D[\ {X, Y, Z}, {u, 0, 2 \[Pi]}, {v, 0, \[Pi]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ PlotPoints \ \[Rule] {48, 12}, Axes \[Rule] False, Boxed \[Rule] False, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ ViewPoint \[Rule] {0, \ \(-2\), \(-2\)}];\)\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr12\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu12.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu12.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell["Now another clipping:", "Text"], Cell[BoxData[{ \(\(gr13\ = \ Fold[\ Clip3D[#1, #2] &, gr11, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Plane[{0, \(-10\), \(-1\)}, {4, 0, \(-1\)}]}];\)\), "\[IndentingNewLine]", \(\(Show[\ gr13, \ ViewPoint\ \[Rule] \ {\(-1.5\), 0, 0}\ ];\)\)}], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr13\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu13.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu13.html"], None}, ButtonStyle->"Hyperlink"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Second form:", "Subsubsection"], Cell[BoxData[{ \(\(X = \((2 + Cos[u/2]\ Sin[v] - Sin[u/2]\ Sin[2 v])\)\ Cos[ u];\)\), "\n", \(\(Y = \((2 + Cos[u/2]\ Sin[v] - Sin[u/2]\ Sin[2 v])\)\ Sin[ u];\)\), "\n", \(\(Z = Sin[u/2]\ Sin[v] + Cos[u/2]\ Sin[2 v];\)\)}], "Input"], Cell[BoxData[ \(\(gr14 = ParametricPlot3D[\ {X, Y, Z}, {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ PlotPoints \[Rule] 51, Boxed \[Rule] False, Axes \[Rule] None\ ];\)\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr14\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu14.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu14.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell["Now the clipping with two planes:", "Text"], Cell[BoxData[{ \(\(gr15\[Alpha]\ = Fold[\ Clip3D[#1, #2] &, gr14, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Plane[{0.5, 0, 0.4}, {\(-0.5\), 0.25, \(-1\)}], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plane[{\ \ \ \ 0, \ 0, 0.8}, {\ \ \ \ \ \ \ 0, \ \ \ \ \ \ \ 0, \(-1\)}]}];\)\), "\ \[IndentingNewLine]", \(\(gr15\ = Show[\ gr15\[Alpha]\ ];\)\)}], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr15\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu15.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu15.html"], None}, ButtonStyle->"Hyperlink"]], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Third form:", "Subsubsection"], Cell[TextData[{ "A polynomial representation of the Klein bottle due to Ian Stewart can be \ found on this ", ButtonBox["web page", ButtonData:>{ URL[ "http://www.uib.no/people/nfytn/surfaces.htm"], None}, ButtonStyle->"Hyperlink"], " . The equation looks odd\":" }], "Text", TextJustification->1], Cell["\<\ (x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+16*x*z*(x^2+y^2+z^2-2*y-1) \ == 0\ \>", "Text"], Cell[TextData[{ "You can get the following part of the Klein bottle with the package ", StyleBox["ImplicitPlot3D", FontFamily->"Courier"], " and the following ", StyleBox["Mathematica", FontSlant->"Italic"], " expression. ", StyleBox["Attention please:", FontColor->RGBColor[1, 0, 0]], " The computation needs a lot of time!" }], "Text", TextJustification->1], Cell[BoxData[ \( (*\[IndentingNewLine]h\ = \ 7; p\ = \ 51; \[IndentingNewLine]gr16\ = \ \[IndentingNewLine]ImplicitPlot3D[\ \[IndentingNewLine]\ \ \((x^2 + y^2 + z^2 + 2*y - 1)\)*\((\((x^2 + y^2 + z^2 - 2*y - 1)\)^2 - 8*z^2)\) + 16*x*z*\((x^2 + y^2 + z^2 - 2*y - 1)\) == 0, \[IndentingNewLine]\ \ \ \ {x, \(-2\), h}, {y, \(-2\), h}, {z, \(-2\), h}, \[IndentingNewLine]\ \ \ \ \ PlotPoints \[Rule] {p, p, p}, \ ViewPoint\ \[Rule] \ {\(-3\), 0, 1}, \[IndentingNewLine]\ \ Boxed\ \[Rule] \ False, \ LightSources\ \[Rule] \ lights]\ ;\[IndentingNewLine]*) \)], "Input"], Cell[BoxData[ \( (*\ \(WriteLiveForm[\ "\", gr16\ ];\)\ *) \)], "Input"], Cell[TextData[ButtonBox["klafu16.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu16.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell[BoxData[ \( (*\[IndentingNewLine]gr17\ = \ Fold[\ Clip3D[#1, #2] &, gr16, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ {Plane[{\(-1\), \ \(-1.5\), 3}, {3, \(-2\), \(-2\)}]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ }]; \[IndentingNewLine]gr18\ = \ Show[\ gr17\ , Boxed\ \[Rule] \ False\ ];\[IndentingNewLine]*) \)], "Input"], Cell[BoxData[ \( (*\ \(WriteLiveForm[\ "\", gr18\ ];\)\ *) \)], "Input"], Cell[TextData[ButtonBox["klafu18.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu18.html"], None}, ButtonStyle->"Hyperlink"]], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Fermat", "Subsubsection"], Cell[TextData[{ "At ", ButtonBox["MathSource", ButtonData:>{ URL[ "http://www.mathsource.com"], None}, ButtonStyle->"Hyperlink"], " there is a notebook of Andrew J. Hanson: ", ButtonBox["FermatSolution.nb", ButtonData:>{ URL[ "http://www.mathsource.com/Content/Applications/Mathematics/0206-288"], None}, ButtonStyle->"Hyperlink"], " .", "\n\"Solution of Fermat's Equation ", Cell[BoxData[ \(TraditionalForm\`z\_1\^n\)]], " + ", Cell[BoxData[ \(TraditionalForm\`z\_2\^n\)]], " = 1\nThe notebook shows a projection from four-dimensional space of the \ so-called projective variety that represents all possible solutions of the \ equation ", Cell[BoxData[ \(TraditionalForm\`x\^n\)]], " + ", Cell[BoxData[ \(TraditionalForm\`y\^n\)]], " == ", Cell[BoxData[ \(TraditionalForm\`z\^n\)]], " for varying ", StyleBox["n", FontSlant->"Italic"], ". What Fermat's Last Theorem states is that none of these solutions can \ correspond to integer values of ", StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["y", FontSlant->"Italic"], " and ", StyleBox["z", FontSlant->"Italic"], ". \"" }], "Text", TextJustification->1], Cell[TextData[{ "Look also at the figures on page 39 of the second edition of the ", StyleBox["Mathematica", FontSlant->"Italic"], "-Book. Having done some calculations not incoorperated in this notebook, \ it is possible to work with the following expressions:" }], "Text", TextJustification->1], Cell["\<\ (* gr19 = Show[ Graphics3D[ Table[surface[k1, k2], {k1, 1, n}, {k2, 1, n}]], \t\tBoxed -> False, \t\tLightSources\[Rule] lights ]; *) \ \>", "Input", AspectRatioFixed->True], Cell[BoxData[ \( (*\ \(WriteLiveForm["\", gr19];\)\ *) \)], "Input"], Cell[TextData[ButtonBox["klafu19.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu19.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell["You will get more \"insight\" after clipping:", "Text"], Cell[BoxData[ \( (*\[IndentingNewLine]\[Alpha]\ = \ 0.2; \[IndentingNewLine]gr20\ = \ Fold[\ Clip3D[#1, #2] &, gr19, \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ {Plane[{\[Alpha], \ \[Alpha], \[Alpha]}, {\(-1\), 1, \(-1\)}]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ }]; \ \ \ \[IndentingNewLine]gr21\ = \ Show[\ gr20, \ PlotRange\ \[Rule] \ All\ ];\[IndentingNewLine]*) \)], "Input"], Cell[BoxData[ \( (*\ \(WriteLiveForm[\ "\", gr21\ ];\)\ *) \)], "Input"], Cell[TextData[ButtonBox["klafu21.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu21.html"], None}, ButtonStyle->"Hyperlink"]], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Outlook", "Subsection"], Cell["\<\ Some years ago I explored the following surface. At the first glance it seems \ to be boaring. But there are \"inner values\" which can be seen when the \ graphic is rendered or after an adequate clipping. The changed options of \ lightsources are used.\ \>", "Text", TextJustification->1], Cell[BoxData[ \(\(SetOptions[\ Graphics3D, Boxed \[Rule] False\ ];\)\)], "Input"], Cell["\<\ gr22\[Alpha] = ParametricPlot3D[ {u Cos[v] Sin[u], u Cos[u] Cos[v],-u Sin[v] }, {u, 0, 3Pi, Pi/18}, {v, 0, 2Pi, Pi/18}, Boxed -> False, Axes -> False ];\ \>", "Input", PageWidth->Infinity, ImageRegion->{{0, 1}, {0, 1}}, FontFamily->"Courier New", Background->GrayLevel[1]], Cell[TextData[{ "Remembering abraded shells on the shore I took the following choice of ", StyleBox["PlotRange", FontFamily->"Courier"], ":" }], "Text"], Cell[BoxData[ \(\(gr22\[Beta]\ = Show[\ gr22\[Alpha], PlotRange \[Rule] {All, {\(-0.5\), 10}, All}\ ];\)\)], "Input"], Cell[BoxData[ \(gr22\[Gamma]\ = \ Fold[\ Clip3D[#1, #2] &, gr22\[Alpha], \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Plane[{0, \(-0.5\), 0}, {0, 1, 0}], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plane[{0, 1.5\ \[Pi], 0}, {0, \(-1\), 0}]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }]; gr22\[Delta]\ = Show[\ gr22\[Gamma]\ ];\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr22\[Gamma]\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu20.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu20.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell[BoxData[ \(gr22\[Delta]\ = \ Fold[\ Clip3D[#1, #2] &, gr22\[Alpha], \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Plane[{0, \(-0.7\), 0.5}, {\(-0.2\), 1, \(-1.5\)}], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plane[{0, \(-0.5\), \ \ \ \ 0}, {\ \ \ \ \ \ \ \ 0, 1, \ \ \ \ \ \ 0}]\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }]; gr22\ = Show[\ gr22\[Delta]\ ];\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr22\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu22.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu22.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell[TextData[{ "I think that Michael Trott is the author of the following ", StyleBox["Mathematica", FontSlant->"Italic"], "-Oneliner. \n", StyleBox["SetOptions[ Graphics3D, Boxed \[Rule] False ]", FontFamily->"Courier"], " is needed:" }], "Text"], Cell[BoxData[ \(\(gr23\[Alpha]\ = With[{stellDode = Stellate@ Delete[Dodecahedron[], 6]}, \ \[IndentingNewLine]\ \ \ \ \ \ \ Show@\ \ \(Graphics3D@{EdgeForm@\(Thickness@0.001\), Stellate[ OpenTruncate@Geodesate[stellDode, 3, {0, 0, 0}, 1.6], 1.05], \ \[IndentingNewLine]\ \ \ \ Stellate@\(OpenTruncate@ Geodesate[stellDode, 3]\)}\)];\)\)], "Input"], Cell["Clipping and lighting is applied to this object:", "Text"], Cell[BoxData[{ \(\(gr23\[Beta]\ = \ Fold[\ Clip3D[#1, #2] &, gr23\[Alpha], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Plane[{0, \ \ 0.15, 0}, {0, \(-1\), \ 0}], \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Plane[{0, \(-0.15\), 0}, {0, \ \ 1, 0}]}];\)\), "\n", \(\(gr23\ = \ Show[\ gr23\[Beta], \ Boxed\ \[Rule] \ False, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ PlotRange\ \ \[Rule] \ All, \ ViewPoint\ \[Rule] {0, \(-3\), 0}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ LightSources \ \[Rule] \ lights\ ];\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \)}], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr23\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu23.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu23.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell["Now the hole is filled:", "Text"], Cell[BoxData[{ \(\(Needs[\ "\"\ ];\)\), "\n", \(\ \(gr24\[Alpha]\ = Graphics3D[\ Sphere[0.98, 32, 16\ ]\ ];\)\), "\n", \(\(Show[\ gr24\[Alpha]\ ];\)\)}], "Input"], Cell["\<\ There is a remark on \"placing several lightsources at the same point\" in [ \ Smith, Blachman, p.260]. That will produce \"more intense illumination\".\ \>", "Text"], Cell[BoxData[ \(\(gr25\ = \ Show[{gr23, gr24\[Alpha]}, \ Boxed\ \[Rule] \ False, \[IndentingNewLine]ViewPoint\ \[Rule] \ {0, \(-3\), 0}, \ \[IndentingNewLine]\ LightSources \[Rule] \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{{\ \ \ 1, \ \ 0, \ \ \ 1}, Gold}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 0, \ \ 1, \ \ \ 1}, Gold}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 1, \ \ 1, \ \ \ 1}, Yellow}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 1, \ \ 1, \ \ \ 1}, Red}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 1, \ \ 0, \ \ \ 1}, Gold}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 0, \ \ 1, \ \ \ 1}, Gold}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 1, \ \ 1, \ \ \ 1}, Gold}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \(-1\), \ \ 0, \(-1\)}, Yellow}, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \ \ 0, \(-1\), \(-1\)}, Yellow}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \(-1\), \(-1\), \(-1\)}, Yellow}, \ \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\ \(-1\), \(-1\), \(-1\)}, Red}\ \ \ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }];\)\)], "Input"], Cell[BoxData[ \(\(WriteLiveForm[\ "\", gr25\ ];\)\)], "Input"], Cell[TextData[ButtonBox["klafu25.html", ButtonData:>{ URL[ "http://www.mathematik.uni-kassel.de/~rascha/Live3D/klafu25.html"], None}, ButtonStyle->"Hyperlink"]], "Text"], Cell["Let the sun shine bright over the conferences at Klagenfurt.", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Hints for the Installation", "Subsection"], Cell[TextData[{ "You have to set the right path in the package ", StyleBox["LiveGraphics3D`", FontFamily->"Courier"], " to save the HTML-files." }], "Text"], Cell[TextData[{ "In his book Tom Wickham-Jones describes a ", StyleBox["Mathematica", FontSlant->"Italic"], "-package for additional clipping routines of 3D-graphics. \nYou can get \ the packages on ", StyleBox["MathSource", FontSlant->"Italic"], " with this URL:\n", ButtonBox["http://www.mathsource.com/Content/Enhancements/Graphics/3D/0208-\ 976", ButtonData:>{ URL[ "http://www.mathsource.com/Content/Enhancements/Graphics/3D/0208-976"], None}, ButtonStyle->"Hyperlink"], "\n\nOur notebook needs only the packages ", StyleBox["Geometry.m", FontFamily->"Courier"], ", ", StyleBox["Geometry3D.m", FontFamily->"Courier"], ", ", StyleBox["NonConvexTriangulate.m", FontFamily->"Courier"], " and ", StyleBox["SimpleHull.m", FontFamily->"Courier"], " . Work will be more easy if the mentioned files are loaded in a directory \ also named ", StyleBox["ExtendGraphics", FontFamily->"Times New Roman"], " in the directory Autoload in AddOns:\nE.g.: ", StyleBox["Mathematica/4.1/AddOns/Autoload/ExtendGraphics", FontFamily->"Courier"], ".\nYou can find the following example in the book of Wickham-Jones ( p. \ 440 ). 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