(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.1'

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Notebook[{

Cell[CellGroupData[{
Cell[TextData[StyleBox["Geometrica'05",
  FontSlant->"Italic"]], "Title",
  CellTags->"TourOfGeometrica"],

Cell[CellGroupData[{

Cell[TextData[{
  "Calling ",
  StyleBox["Geometrica",
    FontSlant->"Italic"]
}], "Section",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "Each time you need ",
  StyleBox["Geometrica",
    FontSlant->"Italic"],
  ", execute the following command."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(<< Geometrica`Geometrica05`\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "To see the results of the commands contained in that notebook and in any \
other notebook of the ",
  StyleBox["Geometrica",
    FontSlant->"Italic"],
  " documentation, execute the input cells sequentially. If you forget to do \
so, you may have to deal with undefined variables."
}], "Text",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Geometrica versus Cabri or Sketchpad", "Section",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[GridBox[{
        {"Features", 
          StyleBox[\(Cabri\ or\ Sketchpad\),
            FontSlant->"Italic",
            Background->None], 
          StyleBox["Geometrica",
            FontSlant->"Italic"]},
        {
          StyleBox["interactivity",
            FontWeight->"Plain",
            FontSlant->"Italic"], 
          StyleBox["mouse",
            Background->None], \(keyboard\ or\ palette\)},
        {
          StyleBox["geometry",
            FontWeight->"Plain",
            FontSlant->"Italic"], 
          StyleBox["euclidean",
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        {
          StyleBox["functions",
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            FontSlant->"Italic"], 
          StyleBox["\"\<**\>\"",
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        {
          StyleBox[\(multiple\ objects\),
            FontWeight->"Plain",
            FontSlant->"Italic"], 
          StyleBox["no",
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        {
          StyleBox[\(abstract\ objects\),
            FontWeight->"Plain",
            FontSlant->"Italic"], 
          StyleBox["no",
            Background->None], "yes"},
        {
          StyleBox[\(geometrical\ functions\),
            FontWeight->"Plain",
            FontSlant->"Italic"], 
          StyleBox["no",
            Background->None], "yes"}
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      GridFrame->True,
      RowLines->True,
      ColumnLines->True]], "Input",
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}, Open  ]],

Cell[CellGroupData[{

Cell["Geometrica and Mathematica", "Section",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[GridBox[{
        {
          StyleBox[\(list\ processing\),
            FontWeight->"Plain",
            FontSlant->"Italic"]},
        {
          StyleBox[\(symbolic\ facilities\),
            FontWeight->"Plain",
            FontSlant->"Italic"]},
        {
          StyleBox[\(functional\ programming\),
            FontWeight->"Plain",
            FontSlant->"Italic"]},
        {
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            StyleBox[" ",
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              FontSlant->"Italic"], 
            RowBox[{
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              StyleBox[" ",
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              StyleBox["and",
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              StyleBox[" ",
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              StyleBox[" ",
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              StyleBox["commands",
                FontWeight->"Plain",
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        {
          StyleBox[\(many\ new\ geometrical\ functions\),
            FontWeight->"Plain",
            FontSlant->"Italic"]}
        },
      GridFrame->True,
      RowLines->True,
      ColumnLines->True]], "Input",
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}, Open  ]],

Cell[CellGroupData[{

Cell["Euclidean and Analytical geometries", "Section",
  CellTags->"TourOfGeometrica"],

Cell["\<\
Primary concept in EUCLIDEAN geometry: CPoint \[Rule] superb reasoning \
(unfortunately disappearing!)
Primary concept in ANALYTICAL geometry: COORDINATES \[Rule] unification of \
geometry and algebra (Descartes and the theory of curves).\
\>", "Text",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Solving an elementary problem", "Section",
  CellTags->"TourOfGeometrica"],

Cell["\<\
The best navigation consists of choosing a simple problem and to see how it \
is solved. We shall study the generation of a parabola as the envelope of one \
edge of a square ruler whose vertex describes a line and the other edge \
passes through a fixed point.\
\>", "Text",
  CellTags->"TourOfGeometrica"],

Cell[CellGroupData[{

Cell["Point, line and first drawing", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(m = CPoint[1, 0]\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell["The straight line is the y-axis ", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(d = CLine[1, 0, 0]\), "\[IndentingNewLine]", 
    \(\(g1 = Draw[Red, m, d];\)\)}], "Input",
  CellLabel->"In[3]:=",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Bound point and euclidean line", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The edge passing through ",
  StyleBox["m",
    FontSlant->"Italic"],
  " is defined by a second point bound to ",
  StyleBox["d",
    FontSlant->"Italic"],
  ":"
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(m\_1 = Pointer[d, t]\), "\n", 
    \(d\_1 = ELine[m, m\_1]\)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The next drawings are not displayed, they will serve later to show the \
various steps of the construction. The option ",
  StyleBox["DisplayOn \[Rule] False ",
    FontFamily->"Courier New"],
  "replaces the ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " function ",
  StyleBox["DisplayFunction \[Rule] Identity",
    FontFamily->"Courier New"],
  "."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
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    \(g2 = Draw[Blue, m\_1 /. r, DisplayOn \[Rule] False]; \), "\n", 
    \(g3 = Draw[Blue, d\_1 /. r, DisplayOn \[Rule] False]; \)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Geometrical function", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The other edge ",
  Cell[BoxData[
      \(TraditionalForm\`d\_2\)]],
  " is perpendicular in ",
  Cell[BoxData[
      \(TraditionalForm\`\(m\_1\) to\ d\_1. \)]]
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(d\_2 = ELine[m\_1, d\_1]\), "\n", 
    \(g4 = Draw[Blue, d\_2 /. r, DisplayOn \[Rule] False]; \)}], "Input",
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}, Open  ]],

Cell[CellGroupData[{

Cell["Derivative of a geometrical function", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "When the parameter ",
  StyleBox["t",
    FontSlant->"Italic"],
  " varies, the line ",
  Cell[BoxData[
      \(TraditionalForm\`d\_2\ \(\(moves\)\(.\)\)\)]],
  "Between two infinitesimally close positions, the two lines intersect  at a \
point of the envelope."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(d\_21 = \(D[#, t] &\) /@ d\_2\), "\n", 
    \(g5 = Draw[Blue, d\_21 /. r, DisplayOn \[Rule] False]; \), "\n", 
    \(m\_p = d\_2 \[Intersection] d\_21\), "\n", 
    \(g6 = Draw[Red, m\_p /. r, DisplayOn \[Rule] False]; \)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Animation and story board", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "One can now recapitulate the construction by using the command ",
  StyleBox["Movie",
    FontSlant->"Italic"],
  " which produces the six figures by integrating the ",
  StyleBox["n+1",
    FontSlant->"Italic"],
  "-th",
  StyleBox[" ",
    FontSlant->"Italic"],
  "figure to the ",
  StyleBox["n",
    FontSlant->"Italic"],
  "-th one. Double clicking on the right hand bracket which covers all the \
cells regroups the cells into a single one. Then one can animate the film \
with the menu command ",
  StyleBox["Cell`AnimateSelectedGraphics",
    FontFamily->"Courier New"],
  "."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(Movie[g1, g2, g3, g4, g5, g6]; \)], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The movie is adapted to a demo but, to keep the trace of the construction \
it is more appropriate to use ",
  StyleBox["StoryBoard",
    FontSlant->"Italic"],
  " which puts the figures in a table. The last graph recapitulates all the \
construction."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(StoryBoard[{g1, g2, g3, g4, g5, g6}, 3]; \)], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Viewing the envelope process", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "We are now curious to see the motion of the line ",
  Cell[BoxData[
      \(TraditionalForm\`d\_2\)]],
  ". We then make a table of the graph of the line for a whole range of ",
  StyleBox["t",
    FontSlant->"Italic"],
  "."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(g7 = 
      Table[Draw[Red, LineOrigin[m], d\_2, 
          DisplayOn \[Rule] False], {t, \(-2\), 2,  .2}]; \), "\n", 
    \(g8 = Movie[g7]; \)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Study of a conic", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "To find the equation of the envelope, we eliminate ",
  StyleBox["t",
    FontSlant->"Italic"],
  " between the coordinates of ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(m\_p\)\(.\)\)\)]]
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(l\_1 = {x, y};\)\), "\n", 
    \(\(l\_2 = List @@ m\_p;\)\), "\n", 
    \(\(eqs = Thread[Equal[l\_1, l\_2]];\)\), "\n", 
    \(eq = Eliminate[eqs, t]\)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  StyleBox["Geometrica",
    FontSlant->"Italic"],
  " ",
  StyleBox["can recognize the nature of the conic from its cartesian \
equation: r ",
    FontSlant->"Plain"],
  Cell[BoxData[
      \(TraditionalForm\`x\^2\)]],
  " + 2 s x y + t ",
  Cell[BoxData[
      \(TraditionalForm\`y\^2\)]],
  "+ 2 u x + 2 v y + w = 0"
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(p = CConic[0, \ 0, \ 1, \ \(-2\), 0, 0]\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell["and determine its focus and directrix", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \({f, \[CapitalDelta]} = {Focus[p], Directrix[p]}\)], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Summary", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The parabola is drawn together with its directrix and focus and the \
previous graph is added in ",
  StyleBox["Show",
    FontFamily->"Courier New"],
  "."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(g9 = 
        Draw[Thickness[ .01], Green, p, f, \[CapitalDelta], Black, 
          Legend[{"\<f\>", "\<\[CapitalDelta]\>"}, {f, \
\[CapitalDelta]}]];\)\), "\[IndentingNewLine]", 
    \(\(Show[Last[g8], g9];\)\)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Illustration in 3D", "Section",
  CellTags->"TourOfGeometrica"],

Cell["\<\
The previous problem can be visualized in 3D by replacing the parabola by a \
cylinder with a parabolic directrix. \
\>", "Text",
  CellTags->"TourOfGeometrica"],

Cell[CellGroupData[{

Cell["3D conic", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "A 2D object can be converted into a 3D object using the command ",
  StyleBox["To3D",
    FontFamily->"Courier New"],
  StyleBox[".",
    FontSlant->"Italic"]
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(p3D = To3D[p]\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The parabola is defined by its 2D representation and located in the \
horizontal plane. A point belonging to the parabola is still obtained using \
",
  StyleBox["Pointer",
    FontFamily->"Courier New"],
  ","
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(m3D = Pointer[p3D, n]\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "and the tangents using ",
  StyleBox["ELine",
    FontFamily->"Courier New"],
  "."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(tg = ELine[m3D, p3D]\)], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Cylinder", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "A cylinder is defined by a curve, here the parabola, and a generator \
parallel to a fixed direction ",
  StyleBox["Oa ",
    FontSlant->"Italic"],
  "where ",
  StyleBox["a ",
    FontSlant->"Italic"],
  "has been chosen as the point of ",
  StyleBox["Oz",
    FontSlant->"Italic"],
  " of height 6. It can be obtained either in parametric or cartesian form \
using the option ",
  StyleBox["CQuadric",
    FontFamily->"Courier New"],
  ". However, the cartesian form has an internal parametrization which limits \
the generator to a height of 1."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \({cyp, cyc} = {Cylinder[CPoint[0, 0, 6], p3D], 
        Cylinder[CPoint[0, 0, 6], p3D, CQuadric \[Rule] True]}\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The planes tangent to the parabolic cylinder are represented by \
parallelograms of vertices: the contact point ",
  StyleBox["m3D",
    FontSlant->"Italic"],
  " with the parabola, the point of parameter 0 of the tangents ",
  StyleBox["tg",
    FontSlant->"Italic"],
  " to the parabola and the point of height 6 on the vertical line passing \
through the contact point"
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(a = Pointer[tg, 0]\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell["\<\
and the point of height 6 on the vertical line passing through the contact \
point.\
\>", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(b = Translate[m3D, CPoint[0, 0, 6]]\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell["The parallelogram is then given by", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[
    \(pp = Parallelogram[m3D, a, b]\)], "Input",
  CellTags->"TourOfGeometrica"],

Cell["The full figure can now be drawn:", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(ta = Table[pp, {n, \(-3\), 3,  .2}]; \), "\[IndentingNewLine]", 
    \(Draw3D[Paint[cyp, Red], Blue, ta]; \)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["CAD functions", "Section",
  CellTags->"TourOfGeometrica"],

Cell[CellGroupData[{

Cell["Walls as examples of parallel polygons", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "A line ",
  StyleBox["L' ",
    FontSlant->"Italic"],
  "is parallel to a line ",
  StyleBox["L",
    FontSlant->"Italic"],
  " when it is generated by the end of a segment attached and normal to ",
  StyleBox["L",
    FontSlant->"Italic"],
  ". This defintion assumes that ",
  StyleBox["L",
    FontSlant->"Italic"],
  " represents a continuous and derivable function, which is not true for a \
polygon which has no normal at a vertex. A convention is introduced in ",
  StyleBox["Geometica",
    FontSlant->"Italic"],
  " to define a parallel polygon. It is illustrated for the walls of a \
house."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(x = {0, 1, 3, 4, 4, 5, 7, 8, 8, 8, 8, 0, 0, 0};\)\), "\n", 
    \(\(y = {0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 4, 3, 2};\)\), "\n", 
    \(\(m = CPoint[x, y];\)\), "\n", 
    \(\(s = Close[Segment @@ m];\)\), "\n", 
    \(\(s\_p = Parallel[s,  .3];\)\), "\n", 
    \(\(Draw[s, s\_p];\)\)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The wall has been defined with openings which can be seen if the option ",
  StyleBox["Ribbon",
    FontSlant->"Italic"],
  " is used. The inner polygons of the \"ribbon\" can be painted to \
distinguish the full and empty parts of the wall."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(s\_p = Parallel[s,  .3, Ribbon \[Rule] True];\)\), "\n", 
    \(\(col = {Black, White, Black, Black, Black, White, Black, Black, White, 
          Black, Black, Black, White, Black};\)\), "\n", 
    \(\(Draw[Paint[s\_p, col]];\)\)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Putting dimensions on a technical drawing", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell["\<\
The drawing can finally be conpleted with the dimensions of the wall.\
\>", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(dim = Dimension[s];\)\), "\n", 
    \(\(Draw[Paint[s\_p, col], dim];\)\)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Helicoidal staircase", "Subsection",
  CellTags->"TourOfGeometrica"],

Cell["\<\
To have a first view of a helicoidal staircase, one uses the default \
definition.\
\>", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(sc = HelicoidalStaircase[]; \), "\[IndentingNewLine]", 
    \(Draw3D[sc]; \)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "Then various variables and options are at hand to parametrize the \
staircase. The detail of a step is shown by drawing ",
  Cell[BoxData[
      \(TraditionalForm\`1\/36\)]],
  "of the full staircase whose stair width is equal to 1, pitch is 1 and \
angular step is ",
  Cell[BoxData[
      \(TraditionalForm\`\[Pi]\/18\)]],
  ". Each stair has a thickness and a recess which have optional values. \
Optional are also the sense of rotation, the inner radius and the initial \
angle. The stair is represented here from two opposite view points."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(stair = 
      HelicoidalStaircase[1, 1, 1/36, Pi/18]; \), "\[IndentingNewLine]", 
    \(\(Draw3D[
          Paint[List @@ stair, {GrayLevel[ .4], GrayLevel[ .6], 
              GrayLevel[ .4], GrayLevel[ .6], 
              GrayLevel[ .4]}], #] &\) /@ {ViewPoint \[Rule] {\(-5\), \(-5\), 
            5}, ViewPoint \[Rule] {5, 5, 5}}; \)}], "Input",
  CellTags->"TourOfGeometrica"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Optics functions and offset", "Section",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The function ",
  StyleBox["Paraxial",
    FontFamily->"Courier New"],
  " in ",
  StyleBox["Geometrica",
    FontSlant->"Italic"],
  " is inspired by optics but it is also common in CAD applications where it \
is rather called ",
  StyleBox["Offset",
    FontFamily->"Courier New"],
  ". Paraxial lines are generated in the same way as parallel lines but the \
distance between the lines varies according to a prescribed function."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\({m1, m2} = CPoint[{0, 1}, 0];\)\), "\n", 
    \(c = ECircle[m1, m2]\), "\[IndentingNewLine]", 
    \(l = Paraxial[c,  .1\ Sin[2. \ 2  \[Pi]\ #] &]\), "\n", 
    \(\(Draw[Blue, c, Red, l];\)\)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell["\<\
The shape of the paraxial line may take fancier shapes for larger amplitudes \
of the oscillation.\
\>", "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(l = Paraxial[c, Sin[2. \ 2  \[Pi]\ #] &];\)\), "\n", 
    \(\(Draw[Blue, c, Red, l, PlotRange -> All];\)\)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(l = Paraxial[c, 2\ Sin[2. \ 2  \[Pi]\ #] &];\)\), "\n", 
    \(\(Draw[Blue, c, Red, l, PlotRange -> All];\)\)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell[TextData[{
  "The functions ",
  StyleBox["Parallel",
    FontSlant->"Italic"],
  " and ",
  StyleBox["Paraxial",
    FontSlant->"Italic"],
  " has been extended to any curve or surface. This is the paraxial curve of \
an helix."
}], "Text",
  CellTags->"TourOfGeometrica"],

Cell[BoxData[{
    \(\(h = 
        PPoint[2  Cos[#] &, 2  Sin[#] &, # &, 
          PRange \[Rule] {0, 4  Pi, 50}];\)\), "\[IndentingNewLine]", 
    \(l = Paraxial[h, 2\ Sin[8. \ 2  \[Pi]\ #] &]\), "\n", 
    \(\(Draw3D[Blue, h, Red, l];\)\)}], "Input",
  CellTags->"TourOfGeometrica"],

Cell["\<\
Paraxial surfaces can also be obtained by modulating one or both parameters. \
This is a cylinder whose elliptic base is deformed by a sine function.\
\>", "Text",
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