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Cell[CellGroupData[{ Cell["Eine 3D-Grafik: Ein Viereck im Raum ", "Subsection"], Cell["\<\ Eine 3D-Grafik: Wie ein Viereck mit zuf\[ADoubleDot]llig \ gew\[ADoubleDot]hlten Eckpunkten im Raum dargestellt wird. 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"1"}], "}"}], ",", RowBox[{"{", RowBox[{"4", ",", "1", ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "1", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"5", ",", "3", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"3", ",", "1", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"6", ",", "3", ",", "2"}], "}"}]}], "}"}], "]"}]}], "]"}]], "Output"] }, Open ]], Cell[TextData[{ "Beide sind ", StyleBox["GraphicsComplex", FontWeight->"Bold"], "-Konstrukte, aus denen sich sofort ein erstes Bild erzeugen \ l\[ADoubleDot]sst." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"g", "=", RowBox[{"Graphics3D", "[", RowBox[{ RowBox[{"{", RowBox[{"cube", ",", "octahedron"}], "}"}], ",", RowBox[{"Boxed", "\[Rule]", "False"}]}], "]"}]}]], "Input"], Cell[BoxData[ Graphics3DBox[{ GraphicsComplex3DBox[ NCache[{{Rational[-1, 2], Rational[-1, 2], Rational[-1, 2]}, { Rational[-1, 2], Rational[-1, 2], Rational[1, 2]}, { Rational[-1, 2], Rational[1, 2], Rational[-1, 2]}, { Rational[-1, 2], 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Open ]], Cell["\<\ Das Oktaeder ragt aus dem W\[UDoubleDot]rfel heraus, muss also noch skaliert \ werden. Komisch, dass es nur an zwei Seiten aus dem W\[UDoubleDot]rfel \ herausragt. Ein Drahtgitterbild bringt den Grund ans Licht: Das Oktaeder liegt \"falsch \ herum\" im W\[UDoubleDot]rfel und muss auch noch gedreht werden.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Graphics3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"FaceForm", "[", "]"}], ",", "cube", ",", "octahedron"}], "}"}], ",", RowBox[{"Boxed", "\[Rule]", "False"}]}], "]"}]], "Input"], Cell[BoxData[ Graphics3DBox[ {FaceForm[None, None], GraphicsComplex3DBox[ NCache[{{Rational[-1, 2], Rational[-1, 2], Rational[-1, 2]}, { Rational[-1, 2], Rational[-1, 2], Rational[1, 2]}, { Rational[-1, 2], Rational[1, 2], Rational[-1, 2]}, { Rational[-1, 2], Rational[1, 2], Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2], Rational[-1, 2]}, { Rational[1, 2], Rational[-1, 2], Rational[1, 2]}, { Rational[1, 2], Rational[1, 2], Rational[-1, 2]}, { Rational[1, 2], Rational[1, 2], Rational[ 1, 2]}}, {{-0.5, -0.5, -0.5}, {-0.5, -0.5, 0.5}, {-0.5, 0.5, -0.5}, {-0.5, 0.5, 0.5}, {0.5, -0.5, -0.5}, {0.5, -0.5, 0.5}, {0.5, 0.5, -0.5}, {0.5, 0.5, 0.5}}], Polygon3DBox[{{8, 4, 2, 6}, {8, 6, 5, 7}, {8, 7, 3, 4}, {4, 3, 1, 2}, {1, 3, 7, 5}, {2, 1, 5, 6}}]], GraphicsComplex3DBox[ NCache[{{Rational[-1, 2], Rational[-1, 2], 0}, { Rational[-1, 2], Rational[1, 2], 0}, {0, 0, -2^Rational[-1, 2]}, { 0, 0, 2^Rational[-1, 2]}, {Rational[1, 2], Rational[-1, 2], 0}, { Rational[1, 2], Rational[1, 2], 0}}, {{-0.5, -0.5, 0}, {-0.5, 0.5, 0}, { 0, 0, -0.7071067811865476}, {0, 0, 0.7071067811865476}, { 0.5, -0.5, 0}, {0.5, 0.5, 0}}], Polygon3DBox[{{4, 5, 6}, {4, 6, 2}, {4, 2, 1}, {4, 1, 5}, {5, 1, 3}, {5, 3, 6}, {3, 1, 2}, {6, 3, 2}}]]}, Boxed->False]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["cube", FontWeight->"Bold"], " und ", StyleBox["octahedron", FontWeight->"Bold"], " sind ", StyleBox["GraphicsComplex", FontWeight->"Bold"], "-Primitive, auf denen sich geometrische Transformationen leicht ausf\ \[UDoubleDot]hren lassen. Das erste Argument eines ", StyleBox["GraphicsComplex", FontWeight->"Bold"], "-Primitivs ist eine Liste von Punkt-Koordinaten, das zweite eine \ Beschreibung der Linien- und Fl\[ADoubleDot]chenelemente des Objekts relativ \ zu diesen Punkten. Wir m\[UDoubleDot]ssen also nur die Koordinaten der Punkte \ mit der entsprechenden Transformationsmatrix multiplizieren und daraus das \ neue Grafikprimitiv nach denselben Anweisungen aufbauen.\nHier ist schon mal \ eine Funktionsdefinition, die das erledigt:" }], "SmallText"], Cell[BoxData[ RowBox[{ RowBox[{"gcTransform", "[", RowBox[{"r_", ",", "g_GraphicsComplex"}], "]"}], ":=", RowBox[{"GraphicsComplex", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"r", ".", "#"}], "&"}], ")"}], "/@", RowBox[{"g", "\[LeftDoubleBracket]", "1", "\[RightDoubleBracket]"}]}], ",", RowBox[{"g", "\[LeftDoubleBracket]", "2", "\[RightDoubleBracket]"}]}], "]"}]}]], "Input"], Cell["\<\ Die erforderliche Transformation des Oktaeders k\[ODoubleDot]nnen wir aus \ zwei Schritten zusammensetzen. Zun\[ADoubleDot]chst die Drehung in z-Richtung \ um 90\[Degree].\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"r", "=", RowBox[{"RotationMatrix", "[", RowBox[{ RowBox[{"\[Pi]", "/", "4"}], ",", RowBox[{"{", RowBox[{"0", ",", "0", ",", "1"}], "}"}]}], "]"}]}], ")"}], "//", "MatrixForm"}]], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { FractionBox["1", SqrtBox["2"]], RowBox[{"-", FractionBox["1", SqrtBox["2"]]}], "0"}, { FractionBox["1", SqrtBox["2"]], FractionBox["1", SqrtBox["2"]], "0"}, {"0", "0", "1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"p", "=", RowBox[{"Graphics3D", "[", RowBox[{ RowBox[{"{", RowBox[{"cube", ",", RowBox[{"gcTransform", "[", RowBox[{"r", ",", "octahedron"}], "]"}]}], "}"}], ",", RowBox[{"Boxed", "\[Rule]", "False"}]}], "]"}]}]], "Input"], Cell[BoxData[ Graphics3DBox[{ GraphicsComplex3DBox[ NCache[{{Rational[-1, 2], Rational[-1, 2], Rational[-1, 2]}, { Rational[-1, 2], Rational[-1, 2], Rational[1, 2]}, { Rational[-1, 2], Rational[1, 2], Rational[-1, 2]}, { Rational[-1, 2], Rational[1, 2], Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2], Rational[-1, 2]}, { Rational[1, 2], Rational[-1, 2], Rational[1, 2]}, { Rational[1, 2], Rational[1, 2], Rational[-1, 2]}, { Rational[1, 2], Rational[1, 2], Rational[ 1, 2]}}, {{-0.5, -0.5, -0.5}, {-0.5, -0.5, 0.5}, {-0.5, 0.5, -0.5}, {-0.5, 0.5, 0.5}, {0.5, -0.5, -0.5}, {0.5, -0.5, 0.5}, {0.5, 0.5, -0.5}, {0.5, 0.5, 0.5}}], Polygon3DBox[{{8, 4, 2, 6}, {8, 6, 5, 7}, {8, 7, 3, 4}, {4, 3, 1, 2}, {1, 3, 7, 5}, {2, 1, 5, 6}}]], GraphicsComplex3DBox[ NCache[{{0, -2^Rational[-1, 2], 0}, {-2^Rational[-1, 2], 0, 0}, { 0, 0, -2^Rational[-1, 2]}, {0, 0, 2^Rational[-1, 2]}, { 2^Rational[-1, 2], 0, 0}, {0, 2^Rational[-1, 2], 0}}, {{ 0, -0.7071067811865476, 0}, {-0.7071067811865476, 0, 0}, { 0, 0, -0.7071067811865476}, {0, 0, 0.7071067811865476}, { 0.7071067811865476, 0, 0}, {0, 0.7071067811865476, 0}}], Polygon3DBox[{{4, 5, 6}, {4, 6, 2}, {4, 2, 1}, {4, 1, 5}, {5, 1, 3}, {5, 3, 6}, {3, 1, 2}, {6, 3, 2}}]]}, Boxed->False]], "Output"] }, Open ]], Cell[TextData[{ "Rund um die Grafik ist noch extrem viel Platz. Das hat damit zu tun, dass \ die 3D-Begrenzungsbox durch die Mitten des Oktaeders bestimmt wird, also um \ den Faktor ", Cell[BoxData[ FormBox[ SqrtBox["2"], TraditionalForm]]], ", wie wir gleich sehen werden, gr\[ODoubleDot]\[SZ]er ist als der W\ \[UDoubleDot]rfel. Diese Begrenzungsbox muss ihrerseits \ standardm\[ADoubleDot]\[SZ]ig vollst\[ADoubleDot]ndig ins 2D-Bildfenster \ passen. Mit der Option ", StyleBox["ViewAngle", FontWeight->"Bold"], " k\[ODoubleDot]nnen Sie die Gr\[ODoubleDot]\[SZ]e der Szene beeinflussen." }], "SmallText"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Show", "[", RowBox[{"p", ",", RowBox[{"ViewAngle", "->", RowBox[{"20", "\[Degree]"}]}]}], "]"}]], "Input"], Cell[BoxData[ Graphics3DBox[{ GraphicsComplex3DBox[ NCache[{{Rational[-1, 2], Rational[-1, 2], Rational[-1, 2]}, { Rational[-1, 2], Rational[-1, 2], Rational[1, 2]}, { Rational[-1, 2], Rational[1, 2], Rational[-1, 2]}, { Rational[-1, 2], Rational[1, 2], Rational[1, 2]}, { Rational[1, 2], Rational[-1, 2], Rational[-1, 2]}, { Rational[1, 2], Rational[-1, 2], Rational[1, 2]}, { Rational[1, 2], Rational[1, 2], Rational[-1, 2]}, { Rational[1, 2], Rational[1, 2], Rational[ 1, 2]}}, {{-0.5, -0.5, -0.5}, {-0.5, -0.5, 0.5}, {-0.5, 0.5, -0.5}, {-0.5, 0.5, 0.5}, {0.5, -0.5, -0.5}, {0.5, -0.5, 0.5}, {0.5, 0.5, -0.5}, {0.5, 0.5, 0.5}}], Polygon3DBox[{{8, 4, 2, 6}, {8, 6, 5, 7}, {8, 7, 3, 4}, {4, 3, 1, 2}, {1, 3, 7, 5}, {2, 1, 5, 6}}]], GraphicsComplex3DBox[ NCache[{{0, -2^Rational[-1, 2], 0}, {-2^Rational[-1, 2], 0, 0}, { 0, 0, -2^Rational[-1, 2]}, {0, 0, 2^Rational[-1, 2]}, { 2^Rational[-1, 2], 0, 0}, {0, 2^Rational[-1, 2], 0}}, {{ 0, -0.7071067811865476, 0}, {-0.7071067811865476, 0, 0}, { 0, 0, -0.7071067811865476}, {0, 0, 0.7071067811865476}, { 0.7071067811865476, 0, 0}, {0, 0.7071067811865476, 0}}], Polygon3DBox[{{4, 5, 6}, {4, 6, 2}, {4, 2, 1}, {4, 1, 5}, {5, 1, 3}, {5, 3, 6}, {3, 1, 2}, {6, 3, 2}}]]}, Boxed->False, ViewAngle->NCache[20 Degree, 0.3490658503988659]]], "Output"] }, Open ]], Cell["\<\ Nun schaut das Oktaeder an allen Seiten auf gleiche Weise heraus und muss nur \ noch skaliert werden, so dass Inkugelradius des W\[UDoubleDot]rfels und \ Umkugelradius des Oktaeders \[UDoubleDot]bereinstimmen. Die erforderlichen Informationen k\[ODoubleDot]nnen wir wieder der Datenbank \ entnehmen. Sie sind f\[UDoubleDot]r alle f\[UDoubleDot]nf platonischen K\ \[ODoubleDot]rper in der folgenden Tabelle zusammengestellt.\ \>", "SmallText"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"radii", "=", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{"#", ",", RowBox[{"PolyhedronData", "[", RowBox[{"#", ",", "\"\\""}], "]"}], ",", RowBox[{"PolyhedronData", "[", RowBox[{"#", ",", "\"\\""}], "]"}]}], "}"}], "&"}], "/@", "platonischeKoerper"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Text", "@", RowBox[{"Grid", "[", RowBox[{ RowBox[{"Prepend", "[", RowBox[{"radii", ",", RowBox[{"{", RowBox[{ "\"\\"", ",", "\"\\"", ",", "\"\\""}], "}"}]}], "]"}], ",", RowBox[{"Dividers", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"{", "True", "}"}], "}"}], ",", RowBox[{"{", RowBox[{"True", ",", "True", ",", RowBox[{"{", "False", "}"}], ",", "True"}], "}"}]}], "}"}]}], ",", "\[IndentingNewLine]", RowBox[{"Spacings", "\[Rule]", RowBox[{"{", RowBox[{"1", ",", "2"}], "}"}]}]}], "]"}]}]}], "Input"], Cell[BoxData[ InterpretationBox[Cell[BoxData[ TagBox[GridBox[{ {"\<\"K\[ODoubleDot]rper\"\>", "\<\"Inkugelradius\"\>", \ "\<\"Umkugelradius\"\>"}, {"\<\"Cube\"\>", FractionBox["1", "2"], FractionBox[ SqrtBox["3"], "2"]}, {"\<\"Dodecahedron\"\>", RowBox[{ FractionBox["1", "20"], " ", SqrtBox[ RowBox[{"250", "+", RowBox[{"110", " ", SqrtBox["5"]}]}]]}], RowBox[{ FractionBox["1", "4"], " ", RowBox[{"(", RowBox[{ SqrtBox["3"], "+", SqrtBox["15"]}], ")"}]}]}, {"\<\"Icosahedron\"\>", RowBox[{ FractionBox["1", "12"], " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", SqrtBox["3"]}], "+", SqrtBox["15"]}], ")"}]}], RowBox[{ FractionBox["1", "4"], " ", SqrtBox[ RowBox[{"10", "+", RowBox[{"2", " ", SqrtBox["5"]}]}]]}]}, {"\<\"Octahedron\"\>", FractionBox["1", SqrtBox["6"]], FractionBox["1", SqrtBox["2"]]}, {"\<\"Tetrahedron\"\>", FractionBox["1", RowBox[{"2", " ", SqrtBox["6"]}]], FractionBox[ SqrtBox[ FractionBox["3", "2"]], "2"]} }, ColumnsEqual->False, GridBoxDividers->{ "Columns" -> {{True}}, "Rows" -> {True, True, {False}, True}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{1}}, "Rows" -> {{2}}}, RowsEqual->False], "Grid"]], "Text", "TR"], Text[ Grid[{{"K\[ODoubleDot]rper", "Inkugelradius", "Umkugelradius"}, {"Cube", Rational[1, 2], Rational[1, 2] 3^Rational[1, 2]}, { "Dodecahedron", Rational[1, 20] (250 + 110 5^Rational[1, 2])^Rational[1, 2], Rational[1, 4] (3^Rational[1, 2] + 15^Rational[1, 2])}, { "Icosahedron", Rational[1, 12] (3 3^Rational[1, 2] + 15^Rational[1, 2]), Rational[1, 4] (10 + 2 5^Rational[1, 2])^Rational[1, 2]}, { "Octahedron", 6^Rational[-1, 2], 2^Rational[-1, 2]}, { "Tetrahedron", Rational[1, 2] 6^Rational[-1, 2], Rational[1, 2] Rational[3, 2]^Rational[1, 2]}}, Dividers -> {{{True}}, {True, True, {False}, True}}, Spacings -> {1, 2}]]]], "Output"] }, Open ]], Cell[TextData[{ "Das Oktaeder ist also um den Faktor ", Cell[BoxData[ FormBox[ SqrtBox["2"], TraditionalForm]]], " zu verkleinern. \nDie Szene ist mit wei\[SZ]em Licht ausgeleuchtet, um die \ Eigenfarben der beiden K\[ODoubleDot]rper unverf\[ADoubleDot]lscht zur \ Geltung zu bringen. 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3^Rational[1, 2] + 15^Rational[1, 2]), Rational[ 1, 15] ((Rational[1, 8] + Rational[1, 24] 5^Rational[1, 2]) (10 + 2 5^Rational[1, 2])^(-1) ( 250 + 110 5^Rational[1, 2]))^ Rational[1, 2] (3^Rational[1, 2] + 15^Rational[1, 2])^(-1) ( 3 3^Rational[1, 2] + 15^Rational[1, 2])}, { Rational[ 1, 15] ((Rational[5, 8] + Rational[5, 24] 5^Rational[1, 2]) (10 + 2 5^Rational[1, 2])^(-1) ( 250 + 110 5^Rational[1, 2]))^ Rational[1, 2] (3^Rational[1, 2] + 15^Rational[1, 2])^(-1) ( 3 3^Rational[1, 2] + 15^Rational[1, 2]), Rational[1, 60] (1 + 5^Rational[1, 2]) ((10 + 2 5^Rational[1, 2])^(-1) (250 + 110 5^Rational[1, 2]))^ Rational[1, 2] (3^Rational[1, 2] + 15^Rational[1, 2])^(-1) ( 3 3^Rational[1, 2] + 15^Rational[1, 2]), Rational[ 1, 15] ((Rational[1, 8] + Rational[1, 24] 5^Rational[1, 2]) (10 + 2 5^Rational[1, 2])^(-1) ( 250 + 110 5^Rational[1, 2]))^ Rational[1, 2] (3^Rational[1, 2] + 15^Rational[1, 2])^(-1) ( 3 3^Rational[1, 2] + 15^Rational[1, 2])}, { Rational[-1, 15] ((Rational[3, 4] + 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]], Cell[CellGroupData[{ Cell["Dynamische Grafiken und Animationen", "Section"], Cell[BoxData[ RowBox[{"ClearAll", "[", "\"\\"", "]"}]], "Input"], Cell[TextData[{ StyleBox["Animate", FontWeight->"Bold"], " als automatisches Abspielen, ", StyleBox["Manipulate", FontWeight->"Bold"], " als interaktive Variation." }], "SmallText"], Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"Factor", "[", RowBox[{ SuperscriptBox["x", "n"], "-", "1"}], "]"}], ",", RowBox[{"{", RowBox[{"n", ",", "2", ",", "10", ",", "1"}], "}"}]}], "]"}]], "Input", CellID->2086048250], Cell[BoxData[ RowBox[{"Animate", "[", RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", RowBox[{"x", "+", "a"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "10"}], "}"}]}], "]"}], ",", RowBox[{"{", RowBox[{"a", ",", "0", ",", "5"}], "}"}]}], "]"}]], "Input", CellID->1824107939], Cell[TextData[{ "Das ", StyleBox["Manipulate", FontWeight->"Bold"], "-Objekt kann \[UDoubleDot]ber Schieberegler oder ein Kontrollpanel \ gesteuert werden. 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