(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 806361, 16551] NotebookOptionsPosition[ 798794, 16307] NotebookOutlinePosition[ 799235, 16326] CellTagsIndexPosition[ 799192, 16323] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Turtle Graphics 1: Classical Computational Geometry", "Title"], Cell["\<\ COMP 360 Course Notes Fall 2009\ \>", "Subtitle", CellChangeTimes->{{3.398075349796989*^9, 3.3980753641267643`*^9}, { 3.42842031271552*^9, 3.428420320477256*^9}, 3.46039581584188*^9}], Cell[TextData[{ "In this notebook, we will develop ", StyleBox["LOGO", FontWeight->"Bold"], " inside of ", StyleBox["Mathematica", FontSlant->"Italic"], " and then use ", StyleBox["LOGO", FontWeight->"Bold"], " to build a variety of interesting shapes. In the lab, you will build your \ own version of ", StyleBox["LOGO", FontWeight->"Bold"], " using C++ and OpenGL. Be aware that these two implementations of ", StyleBox["LOGO", FontWeight->"Bold"], " are quite different. 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FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " of all the line segments that the turtle traverses on her journey." }], "Text", CellChangeTimes->{{3.398075468306027*^9, 3.398075481368414*^9}}], Cell[CellGroupData[{ Cell["Implementation", "Exercise"], Cell[TextData[{ "We begin with a program", StyleBox[" ", Background->GrayLevel[1]], StyleBox["init", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " that initializes the turtle ", StyleBox["position", FontSlant->"Italic"], " to the origin, the turtle ", StyleBox["direction", FontSlant->"Italic"], " to the unit vector pointing in the ", StyleBox["x", FontSlant->"Italic"], "-direction, and the ", StyleBox["turtlepath", FontSlant->"Italic"], " to the empty list." }], "Text", CellChangeTimes->{{3.39807550973925*^9, 3.398075580548501*^9}, { 3.398075858162675*^9, 3.398075859303315*^9}}], Cell["\<\ To implement this procedure, click you mouse anywhere in the orange box and \ hit SHIFT-RETURN (or hit ENTER if you have a 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", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "current"], TraditionalForm]]], StyleBox["sin", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(\[Alpha])\n\t", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "new"], TraditionalForm]]], StyleBox[" = v", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["", "current"], " "}], TraditionalForm]]], StyleBox["cos", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(\[Alpha]) + ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["u", "current"], TraditionalForm]]], StyleBox["sin", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["(\[Alpha])\nTypically we shall represent these equations in matrix \ form:\n\t(", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["u", "new"], TraditionalForm]]], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "new"], TraditionalForm]]], StyleBox[") = (", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["u", "current"], " "}], TraditionalForm]]], " ", StyleBox["v", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["", "current"], TraditionalForm]]], StyleBox[") ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}], StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}, { RowBox[{"-", StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}], StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], ".\nVectors have both direction and length. ", StyleBox["The", FontVariations->{"CompatibilityType"->0}], StyleBox[" turn", FontSlant->"Italic"], " command changes the turtle's heading by rotating the turtle's direction \ vector, but we can also change the turtle's step size by changing the length \ of the turtle's direction vector. The ", StyleBox["resize", FontSlant->"Italic"], " command is a function of a single variable, the ", StyleBox["scalefactor", FontSlant->"Italic"], ". The ", StyleBox["resize", FontSlant->"Italic"], " command rescales the turtle step size by setting:\n\t", StyleBox["new direction = scalefactor \[Cross] current direction", FontSlant->"Italic"], StyleBox["\nAgain we can write this transformation in matrix notation:\n\t(", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["u", "new"], TraditionalForm]]], StyleBox[" ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["v", "new"], TraditionalForm]]], StyleBox[") = (", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["u", "current"], " "}], TraditionalForm]]], StyleBox["v", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SubscriptBox["", "current"], TraditionalForm]]], StyleBox[") ", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["0", FontVariations->{"CompatibilityType"->0}]}, {"0", StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], ".\nSometimes, we may want to simultaneously change the turtle direction and \ the turtle step size -- that is, we may want to combine the ", StyleBox["turn", FontSlant->"Italic"], " and ", StyleBox["resize ", FontSlant->"Italic"], StyleBox["commands.", FontVariations->{"CompatibilityType"->0}], " Since ", StyleBox["turn", FontSlant->"Italic"], " and ", StyleBox["resize", FontSlant->"Italic"], " are both represented by 2\[Times]2 matrices, we can combine these \ operations by matrix multiplication into a single matrix.\n\t", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}], StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}, { RowBox[{"-", StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}], StyleBox[ RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["0", FontVariations->{"CompatibilityType"->0}]}, {"0", StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}]} }], ")"}], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{"(", GridBox[{ { RowBox[{" ", StyleBox[ RowBox[{ StyleBox["scalefactor", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}]}], FontVariations->{"CompatibilityType"->0}]}], RowBox[{" ", StyleBox[ RowBox[{ StyleBox["scalefactor", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}]}], FontVariations->{"CompatibilityType"->0}]}]}, { RowBox[{ RowBox[{"-", StyleBox["scalefactor", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}]}], StyleBox[" ", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[ RowBox[{ StyleBox["sin", FontSlant->"Italic"], "(", "\[Alpha]", ")"}], FontVariations->{"CompatibilityType"->0}]}], RowBox[{" ", StyleBox[ RowBox[{ StyleBox["scalefactor", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{ StyleBox["cos", FontSlant->"Italic"], "(", "\[Alpha]", ")"}]}], FontVariations->{"CompatibilityType"->0}]}]} }], ")"}], TraditionalForm]]], ".\nThe advantage of this approach is that it replaces two matrix \ multiplications by a single matrix multiplication. Notice if we reverse the \ order in which we multiply the two matrices, we get the same result. This \ observation corresponds to the fact that if we ", StyleBox["turn ", FontSlant->"Italic"], "first and then ", StyleBox["resize", FontSlant->"Italic"], ", we get the same result as when we ", StyleBox["resize", FontSlant->"Italic"], " first and then ", StyleBox["turn", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell["Implementation", "Exercise"], Cell[TextData[{ StyleBox["Below are four turtle functions ", Background->GrayLevel[1]], StyleBox["forward", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[", ", Background->GrayLevel[1]], StyleBox["move", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[", ", Background->GrayLevel[1]], StyleBox["turn", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[", and ", Background->GrayLevel[1]], StyleBox["resize", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[". THESE ARE THE ONLY PERMISSIBLE TURTLE COMMANDS. A command to \ move to a specific (x,y) location is not allowed, except in the INIT \ command.\nThe ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command takes as input a single parameter, ", Background->GrayLevel[1]], StyleBox["distance", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". The result of the ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is to move the turtle to its new location and to update \ the list in ", Background->GrayLevel[1]], StyleBox["turtlepath", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[" by inserting the new line segment into ", Background->GrayLevel[1]], StyleBox["turtlepath", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[".\nThe ", Background->GrayLevel[1]], StyleBox["move", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is the same as the ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command, except that the ", Background->GrayLevel[1]], StyleBox["move", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command does not update the list in ", Background->GrayLevel[1]], StyleBox["turtlepath", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], StyleBox[".\nThe ", Background->GrayLevel[1]], StyleBox["turn", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command takes as input a single parameter, ", Background->GrayLevel[1]], StyleBox["angle, ", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox["measured in degrees (see ", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox["Degree", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox[" in the Help browser). ", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox["The result of the ", Background->GrayLevel[1]], StyleBox["turn", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is to change the turtle's direction by rotating the \ turtle direction vector by the given ", Background->GrayLevel[1]], StyleBox["angle.\n", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox["The ", Background->GrayLevel[1]], StyleBox["resize", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command takes as input a single parameter, ", Background->GrayLevel[1]], StyleBox["scalefactor", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". ", FontVariations->{"CompatibilityType"->0}, Background->GrayLevel[1]], StyleBox["The result of the ", Background->GrayLevel[1]], StyleBox["resize", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" command is to change the turtle's step size by scaling the turtle \ direction vector by the given ", Background->GrayLevel[1]], StyleBox["scalefactor.", FontSlant->"Italic", Background->GrayLevel[1]] }], "Text", CellChangeTimes->{{3.398075643780702*^9, 3.398075682066102*^9}}], Cell["\<\ To implement these procedurse, click you mouse anywhere in the orange box and \ hit SHIFT - RETURN (or hit ENTER if you have a Mac).\ \>", "Text", CellChangeTimes->{{3.3980757241109657`*^9, 3.398075726834382*^9}, 3.3980757656883087`*^9}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"forward", "[", "dist_", "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"oldposition", "=", "position"}], "}"}], ",", RowBox[{ RowBox[{"position", "=", RowBox[{"oldposition", " ", "+", " ", RowBox[{"dist", RowBox[{"(", "direction", ")"}]}]}]}], ";", RowBox[{"AppendTo", "[", RowBox[{"turtlepath", ",", RowBox[{"Line", "[", RowBox[{"{", RowBox[{"oldposition", ",", "position"}], "}"}], "]"}]}], "]"}]}]}], "]"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"move", "[", "dist_", "]"}], ":=", RowBox[{"position", "=", RowBox[{"position", " ", "+", " ", RowBox[{"dist", RowBox[{"(", "direction", ")"}]}]}]}]}], "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"turn", "[", "angle_", "]"}], ":=", RowBox[{"direction", "=", RowBox[{"N", "[", " ", RowBox[{"direction", " ", ".", RowBox[{"(", GridBox[{ { RowBox[{"Cos", "[", RowBox[{"angle", " ", "Degree"}], "]"}], RowBox[{"Sin", "[", RowBox[{"angle", " ", "Degree"}], "]"}]}, { RowBox[{"-", RowBox[{"Sin", "[", RowBox[{"angle", " ", "Degree"}], "]"}]}], RowBox[{"Cos", "[", RowBox[{"angle", " ", "Degree"}], "]"}]} }], ")"}]}], "]"}]}]}], "\[IndentingNewLine]", "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"resize", "[", "scalefactor_", "]"}], ":=", RowBox[{"direction", "=", RowBox[{"N", "[", " ", RowBox[{"direction", " ", ".", RowBox[{"(", GridBox[{ {"scalefactor", "0"}, {"0", "scalefactor"} }], ")"}]}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.39807574108724*^9, 3.3980757421838703`*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Implementation", "Exercise"], Cell[TextData[{ StyleBox["Later we shall want to display the path of the turtle, so here we \ write two programs ", Background->GrayLevel[1]], StyleBox["displayturtlepath", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " and ", StyleBox["displayturtlecolor", FontColor->GrayLevel[1], Background->RGBColor[0, 0, 1]], " ", StyleBox["to display the path of the turtle by displaying the list of line \ segment in the list ", Background->GrayLevel[1]], StyleBox["turtlepath", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[". The axes are turned off, since we are not interested in seeing \ the coordinate system.", Background->GrayLevel[1]] }], "Text"], Cell["\<\ Again to implement these procedurse, click you mouse anywhere in the orange \ box and hit SHIFT - RETURN (or hit ENTER if you have a Mac).\ \>", "Text", CellChangeTimes->{{3.3980757241109657`*^9, 3.398075726834382*^9}, { 3.3980757656883087`*^9, 3.398075781587788*^9}, {3.398075813892446*^9, 3.3980758159956512`*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"displayturtlepath", "[", "shape_", "]"}], ":=", RowBox[{"Show", "[", RowBox[{ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"Thickness", "[", ".0025", "]"}], ",", "turtlepath"}], "}"}], "]"}], ",", RowBox[{"Axes", "\[Rule]", "False"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"ImageSize", "\[Rule]", "200"}]}], "]"}]}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"displayturtlecolor", "[", "color_", "]"}], ":=", RowBox[{"Show", "[", RowBox[{ RowBox[{"Graphics", "[", RowBox[{"{", RowBox[{ RowBox[{"Hue", "[", "color", "]"}], ",", RowBox[{"Thickness", "[", ".0035", "]"}], ",", "turtlepath"}], "}"}], "]"}], ",", RowBox[{"Axes", "\[Rule]", "False"}], ",", RowBox[{"AspectRatio", "\[Rule]", "Automatic"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"ImageSize", "\[Rule]", "230"}]}], "]"}]}]}], "Input", CellChangeTimes->{{3.394542824417403*^9, 3.394542827637383*^9}, { 3.394542859291737*^9, 3.394542884966013*^9}, {3.394543887656597*^9, 3.394543888151785*^9}, {3.394543921865273*^9, 3.394543924126443*^9}, 3.394547519003417*^9, 3.394547698137738*^9, 3.3945478097337646`*^9, { 3.3980107387756777`*^9, 3.398010767215773*^9}, {3.398010801406927*^9, 3.398010835598792*^9}, 3.39801273521838*^9, 3.3980127684381332`*^9, 3.398013268500551*^9, {3.398013324159527*^9, 3.3980133269508333`*^9}, 3.398013393633882*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Polygons", "Section"], Cell[TextData[{ StyleBox["Using the ", Background->GrayLevel[1]], StyleBox["forward", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" and ", Background->GrayLevel[1]], StyleBox["turn", FontSlant->"Italic", Background->GrayLevel[1]], StyleBox[" commands, you can now make simple shapes by ordering the turtle \ around. 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