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Notebook[{

Cell[CellGroupData[{
Cell["More on Quadrics", "Title",
  CellTags->"MoreOnQuadrics"],

Cell[TextData[{
  "This notebook illustrates the concept of volume limited by various \
quadrics. The ellipsoid has a closed shape and its volume is finite. The \
other quadrics extend to infinity and the surface of the quadric alone is no \
longer sufficient to define a finite volume. An intuitive approach consists, \
in general, of viewing a quadric as a surface of revolution of axis \
\[CapitalDelta]. This would be exact if the section of the quadric by a plane \
perpendicular to \[CapitalDelta] were a circle. In fact, for a general \
quadric, the section is an ellipse. There is however an exception to that \
approach because the hyperbolic paraboloid has no elliptical section but the \
axis  \[CapitalDelta] still exists. The volume is then limited by the quadric \
and two planes perpendicular to \[CapitalDelta]. The direct access to the \
volume form of a quadric is provided by the ",
  StyleBox["canonical form",
    FontSlant->"Italic"],
  ", the simplest representation of a quadric obtained when the reference \
frame coincides with the axes of symmetry of the quadric. ",
  " Since a volume cannot be represented, it is suggested by a sequence of \
surfaces which fill it",
  ", hence the term of ",
  StyleBox["russian dolls ",
    FontSlant->"Italic"],
  "which are ",
  "objects boxed into one another",
  ". Important also in this notebook is the manipulation of the parameters \
and of their range of variations within the contexts of surfaces or volumes."
}], "Text",
  CellTags->"MoreOnQuadrics"],

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

Cell[CellGroupData[{

Cell["Ellipsoid", "Section",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The default ellipsoid used in the example is not of revolution. All the \
ellipsoids are scaled from their common center. The third variable #3 can be \
seen as a radius since it multiplies all the coordinates.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(cf = CanonicalForm@CEllipsoid[]\)], "Input",
  CellTags->"MoreOnQuadrics"],

Cell[TextData[{
  "As surfaces have to be drawn, the variable #3 has to be fixed and its \
range of variations suppressed from the canonical form. The rule ",
  StyleBox["ru",
    FontSlant->"Italic"],
  " performs this task by deleting the list ",
  StyleBox["c",
    FontSlant->"Italic"],
  ". Three surfaces are selected by imposing the values 1, 0.5, 0.25 to #3."
}], "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(a =. ; b =. ; \), "\[IndentingNewLine]", 
    \(ru = \((PRange \[Rule] {a_, b_, c_, d_})\) \[Rule] \(PRange \[Rule] {a, 
            b, d}\); \), "\[IndentingNewLine]", 
    \(e = cf /. ru\), "\[IndentingNewLine]", 
    \({e1, e2, e3} = {e /. #3 \[Rule] 1, e /. #3 \[Rule]  .5, 
        e /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(Draw3D[Red, e1, Green, e2, Blue, e3]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
To see the interior of a volume, another technique consists of reducing the \
range of parameters. In this example,the first variable is limited to (0, \
\[Pi])  and the objects are painted.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(ru = \((PRange \[Rule] {a_, b_, c_, 
              d_})\) \[Rule] \(PRange \[Rule] {{0, Pi}, b, 
            d}\); \), "\[IndentingNewLine]", 
    \(e = cf /. ru; \), "\[IndentingNewLine]", 
    \({e1, e2, e3} = {e /. #3 \[Rule] 1, e /. #3 \[Rule]  .5, 
        e /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(Draw3D[Paint[{e1, e2, e3}, {Red, Green, Blue}]]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["Default painting can also be used.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Draw3D@Paint[{e1, e2, e3}]; \)], "Input",
  CellTags->"MoreOnQuadrics"],

Cell[TextData[{
  "The volume of the full ellipsoid can be calculated. One notices that it is \
equal to ",
  Cell[BoxData[
      FormBox[
        RowBox[{\(4\/3\), 
          StyleBox[
            RowBox[{"\[Pi]", 
              StyleBox["abc",
                FontSlant->"Italic"]}]], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["when",
            FontSlant->"Plain"], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["the",
            FontSlant->"Plain"], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["equation",
            FontSlant->"Plain"], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["of",
            FontSlant->"Plain"], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["the",
            FontSlant->"Plain"], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["ellipsoid",
            FontSlant->"Plain"], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["is",
            FontSlant->"Plain"], 
          StyleBox[" ",
            FontSlant->"Plain"], 
          StyleBox["written",
            FontSlant->"Plain"]}], TraditionalForm]]],
  "\n",
  Cell[BoxData[
      \(x\^2\/a\^2 + y\^2\/b\^2 + z\^2\/c\^2 = \ 1\)], "Text"]
}], "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@CEllipsoid[]\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Hyperboloids", "Section",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The procedure applied to the ellpsoid can be resumed for the hyperboloids.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[CellGroupData[{

Cell["Hyperboloid of one sheet", "Subsection",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The first variable defines the position of a generator, the second variable a \
point on the generator and the third one is a radius of the ellipse \
intersection of the hyperboloid with a plane perpendicular to the vertical \
axis of symmetry.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(h0 = HyperboloidOfOneSheet[]; \), "\[IndentingNewLine]", 
    \(cf = CanonicalForm@h0; \), "\[IndentingNewLine]", 
    \(h = 
      cf /. \((PRange \[Rule] {a_, b_, c_, 
                d_})\) \[Rule] \(PRange \[Rule] {a, b, 
              d}\)\), "\[IndentingNewLine]", 
    \({h1, h2, h3} = {h /. #3 \[Rule] 1, h /. #3 \[Rule]  .5, 
        h /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(Draw3D[Red, h1, Green, h2, Blue, h3]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The interior of the volume is shown using the canonical view point.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
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              d_})\) \[Rule] \(PRange \[Rule] {{0, Pi}, b, 
            d}\); \), "\[IndentingNewLine]", 
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        h /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(vp = CanonicalViewPoint[h0]; \), "\[IndentingNewLine]", 
    \(Draw3D[Paint[{h1, h2, h3}, {Red, Green, Blue}], 
      ViewPoint \[Rule] vp]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell[TextData[{
  "The same figure can be drawn using the default color function of ",
  StyleBox["Paint",
    FontFamily->"Courier New"],
  ". "
}], "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Draw3D[Paint[{h1, h2, h3}], ViewPoint \[Rule] vp]; \)], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["The entire volume can be calculated analytically.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@h0\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Hyperboloid of two sheets", "Subsection",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The parametric form of a hyperboloid of two sheets specifies the range of the \
first parameter in such a way that the two sheets can be represented \
together.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(h0 = HyperboloidOfTwoSheets[]; \), "\[IndentingNewLine]", 
    \(QuadricPoint /. \((QuadricElements @@ h0)\)\), "\[IndentingNewLine]", 
    \(Draw3D@Paint[h0]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell[TextData[{
  "For the treatment of volumes, the upper sheet only is considered. ",
  "A hyperboloid of two sheets is not a ruled surface and the generators of \
the hyperboloid of one sheet are replaced by hyperbolae. The first variable \
defines the position of a hyperbola, the second variable a point on the \
hyperbola and the third one is a radius of the ellipse intersection of the \
hyperboloid with a plane perpendicular to the ",
  StyleBox["x",
    FontSlant->"Italic"],
  " axis of symmetry. "
}], "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(cf = CanonicalForm@h0\), "\[IndentingNewLine]", 
    \(ru = \((PRange \[Rule] {a_, b_, c_, d_})\) \[Rule] \(PRange \[Rule] {a, 
            b, d}\); \), "\[IndentingNewLine]", 
    \(h = cf /. ru\), "\[IndentingNewLine]", 
    \({h1, h2, h3} = {h /. #3 \[Rule] 1, h /. #3 \[Rule]  .5, 
        h /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(Draw3D@Paint[{h1, h2, h3}]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["The volume can be calculated analytically.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@h0\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Paraboloids", "Section",
  CellTags->"MoreOnQuadrics"],

Cell["The paraboloids may be either elliptic or hyperbolic.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[CellGroupData[{

Cell["Elliptic paraboloid", "Subsection",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The first variable defines the position of a parabola, the second variable a \
point on the parabola and the third variable a radius of the ellipse of cross \
section.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(p0 = EllipticParaboloid[]; \), "\[IndentingNewLine]", 
    \(cf = CanonicalForm@p0; \), "\[IndentingNewLine]", 
    \(p = 
      cf /. \((PRange \[Rule] {a_, b_, c_, 
                d_})\) \[Rule] \(PRange \[Rule] {a, b, 
              d}\)\), "\[IndentingNewLine]", 
    \({p1, p2, p3} = {p /. #3 \[Rule] 1, p /. #3 \[Rule]  .5, 
        p /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(Draw3D@Paint[{p1, p2, p3}]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["The entire volume can be calculated analytically.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@p0\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Hyperbolic paraboloid", "Subsection",
  CellTags->"MoreOnQuadrics"],

Cell[TextData[{
  "The hyperbolic paraboloid is a curious surface generated by a system of \
two lines. It looks like a ",
  StyleBox["pass",
    FontSlant->"Italic"],
  " in a mountain. The first variables determines the position of a \
generator, the second variable the position of a point on that generator and \
the third variable scales the ",
  StyleBox["x",
    FontSlant->"Italic"],
  " and ",
  StyleBox["y",
    FontSlant->"Italic"],
  " coordinates. The canonical form limits the parameter range to a quarter \
of the surface so that the calculation of the volume gives a finite result. \
The next code shows the full surface as the assembly of four patches \
corresponding to four parameter ranges."
}], "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(p0 = HyperbolicParaboloid[]; \), "\n", 
    \(cf = CanonicalForm@p0\), "\n", 
    \(ru1 = \((PRange \[Rule] {a_, b_, c_, 
              d_})\) \[Rule] \(PRange \[Rule] {a, b, d}\); \), "\n", 
    \(ru2 = \((PRange \[Rule] {a_, b_, c_, 
              d_})\) \[Rule] \(PRange \[Rule] {{\(-Pi\), 0}, {0, Pi}, 
            d}\); \), "\n", 
    \(ru3 = \((PRange \[Rule] {a_, b_, c_, 
              d_})\) \[Rule] \(PRange \[Rule] {{0, Pi}, {\(-Pi\), 0}, 
            d}\); \), "\n", 
    \(ru4 = \((PRange \[Rule] {a_, b_, c_, 
              d_})\) \[Rule] \(PRange \[Rule] {{\(-Pi\), 0}, {\(-Pi\), 0}, 
            d}\); \), "\n", 
    \(pa = \(cf /. # &\) /@ {ru1, ru2, ru3, ru4}; \), "\n", 
    \(ph = pa /. #3 \[Rule] 1\), "\[IndentingNewLine]", 
    \(vp = CanonicalViewPoint[p0]; \), "\n", 
    \(Draw3D[Paint[ph, {Blue, Green, Magenta, Yellow}], 
      ViewPoint \[Rule] vp]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["The russian dolls are then constructed.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(ru = \((PRange \[Rule] {a_, b_, c_, d_})\) \[Rule] \(PRange \[Rule] {a, 
            b, d}\); \), "\n", 
    \(pa = cf /. ru; \), "\n", 
    \(ph = {pa /. #3 \[Rule] 1, pa /. #3 \[Rule]  .5, 
        pa /. #3 \[Rule]  .25}; \), "\n", 
    \(Draw3D[Paint[ph, {Red, Green, Blue}], 
      ViewPoint \[Rule] vp]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["Last, the volume is calculated.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume[p0]\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["Quadric cone", "Section",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
A quadric cone can be considered as the limit of a hyperboloid of two sheets \
when the two vertices of the hyperboloid touch each other. The hyperbolae \
become then straight lines. To get a non uniform painting function for the \
defaul color function, one has to improve the resolution by increasing the \
number of points on the generators by using say 25 points instead of 2.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(co0 = QuadricCone[]; \), "\[IndentingNewLine]", 
    \(cf = CanonicalForm@co0\), "\[IndentingNewLine]", 
    \(co = 
      cf /. \((PRange \[Rule] {a_, b_, c_, 
                d_})\) \[Rule] \(PRange \[Rule] {a, 
              b, {25, 25}}\); \), "\[IndentingNewLine]", 
    \(col = {co /. #3 \[Rule] 1, co /. #3 \[Rule]  .5, 
        co /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(vp = CanonicalViewPoint[co0]; \), "\[IndentingNewLine]", 
    \(Draw3D[Paint[col], ViewPoint \[Rule] vp]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The volume of the cone is the same as the one of a pyramid: the third of the \
product of the base by the distance from the vertex to the base.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@co0\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Quadric cylinders", "Section",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
The cylinder is a cone whose vertex is at infinity. There are three different \
types corresponding to the shape of the base: elliptic, hyperbolic and \
parabolic. The first variable determines the position of the generator, the \
second one the position of a point on the generator. The third variable \
depends on the type of the base. The volume of a cylinder is the same as the \
volume of a prismoid. It is the product of the base by the distance between \
the two parallel extreme planes.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[CellGroupData[{

Cell["Elliptic cylinder", "Subsection",
  CellTags->"MoreOnQuadrics"],

Cell["The parametrization is the same as for the cone.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(cy0 = EllipticCylinder[]; \), "\[IndentingNewLine]", 
    \(cf = CanonicalForm@cy0; \), "\[IndentingNewLine]", 
    \(cy = 
      cf /. \((PRange \[Rule] {a_, b_, c_, 
                d_})\) \[Rule] \(PRange \[Rule] {a, 
              b, {25, 25}}\)\), "\[IndentingNewLine]", 
    \({cy1, cy2, cy3} = {cy /. #3 \[Rule] 1, cy /. #3 \[Rule]  .5, 
        cy /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(Draw3D@Paint[{cy1, cy2, cy3}]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["Volume of the cylinder.", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@cy0\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Parabolic cylinder", "Subsection",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
There is one difference with the parametrization of the elliptic cylinder: \
the third variable scales the abscissa. The paraboloids look like the pages \
of a book and it is not necessary to modify the range of the first parameter. \
The range of the second parameter is extended to distinguish the three sheets \
better.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(cy0 = ParabolicCylinder[]; \), "\[IndentingNewLine]", 
    \(cf = CanonicalForm@cy0\), "\[IndentingNewLine]", 
    \(cy = 
      cf /. \((PRange \[Rule] {a_, b_, c_, 
                d_})\) \[Rule] \(PRange \[Rule] {a, {0, 5}, {25, 
                25}}\); \), "\[IndentingNewLine]", 
    \(cyl = {cy /. #3 \[Rule] 1, cy /. #3 \[Rule]  .5, 
        cy /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(vp = ViewPoint[cy0]; \), "\[IndentingNewLine]", 
    \(Draw3D[Paint[cyl]]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
Volume of the parabolic cylinder with the range (0,1) for the second \
parameter.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@cy0\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Hyperbolic cylinder", "Subsection",
  CellTags->"MoreOnQuadrics"],

Cell[TextData[{
  "As for the hyperboloid of two sheets, the canonical form of the hyperbolic \
cylinder retains one sheet only. The third variable scales the ",
  StyleBox["z",
    FontSlant->"Italic"],
  "-coordinate."
}], "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[{
    \(cy0 = HyperbolicCylinder[]; \), "\[IndentingNewLine]", 
    \(cf = CanonicalForm@cy0\), "\[IndentingNewLine]", 
    \(cy = 
      cf /. \((PRange \[Rule] {a_, b_, c_, 
                d_})\) \[Rule] \(PRange \[Rule] {a, {0, 3}, {25, 
                25}}\); \), "\[IndentingNewLine]", 
    \({cy1, cy2, cy3} = {cy /. #3 \[Rule] 1, cy /. #3 \[Rule]  .5, 
        cy /. #3 \[Rule]  .25}; \), "\[IndentingNewLine]", 
    \(g1 = Draw3D@Paint[{cy1, cy2, cy3}]; \)}], "Input",
  CellTags->"MoreOnQuadrics"],

Cell["\<\
Volume of the hyperbolic cylinder with the range (0,1) for the second \
parameter.\
\>", "Text",
  CellTags->"MoreOnQuadrics"],

Cell[BoxData[
    \(Volume@cy0\)], "Input",
  CellTags->"MoreOnQuadrics"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
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(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)