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Cell[CellGroupData[{
Cell["Introduction", "Section",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True,
FontFamily->"Arial",
FontSize->18,
FontWeight->"Bold"],
Cell["\<\
Klein bottle is after projective plane the \"simplest\" closed, \
non-orientable surface.
It is well known fact that one can obtain it by gluing two Moebius strips. \
Although well known, that fact is not obvious.
Here, we are going to make a continuous deformation of a Moebius strip into \
a half-Klein bottle.
Having such a deformation it is easy to obtain an animation as this one \
below.
\
\>", "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True,
FontFamily->"Arial",
FontSize->16,
FontWeight->"Bold",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False}],
Cell["\<\
Evaluate following cell and wait to load the animation (check if file \
m2klein.gif is in the same directory as this notebook). Then double-click on \
picture to see the animation.
\
\>", "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
TextAlignment->Left,
AspectRatioFixed->True,
FontFamily->"Arial",
FontSize->16,
FontWeight->"Bold",
FontSlant->"Plain",
FontTracking->"Plain",
FontVariations->{"Underline"->False,
"Outline"->False,
"Shadow"->False}],
Cell[BoxData[{
\(\(NotebookPath\ = \ \(("\" /.
NotebookInformation[
SelectedNotebook[]])\)[\([1]\)];\)\), "\n",
\(\(PictureName\ = \
StringJoin[
Flatten[Map[If[#\ == "\<\>", \ #, \ List[#, "\<\\\>"]] &, \
NotebookPath]], \ "\"];\)\), "\n",
\(NotebookWrite[SelectedNotebook[],
Cell[CellGroupData[
Map[Cell[
GraphicsData["\",
ExportString[#1, "\"]]] &,
Drop[Import[PictureName], 1]], Closed]]]\)}], "Input"]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["The deformation",
FontSize->18]], "Section"],
Cell["Drawing the Klein bottle", "Text",
FontSize->18],
Cell[BoxData[{
RowBox[{\(Step\ = \ 1; \ Rot\ = \ 0;\), " ",
StyleBox[\( (*\ no\ deformation\ *) \),
FontColor->RGBColor[1, 0, 1]]}], "\[IndentingNewLine]",
RowBox[{\(ParametricPlot3D[
Klein[u, \ v], {u, \ 0, \ 2\ Pi\ , 2\ Pi/50}, \ {v, \ 0, \ 2\ Pi, \
2\ Pi\ /20}, Boxed \[Rule] False,
ViewPoint \[Rule] {1, \(-3\), 2.5}, \ Axes \[Rule] False];\),
"\[IndentingNewLine]"}]}], "Input"],
Cell["Drawing the Moebius strip and the half-Klein bottle", "Text",
FontSize->18],
Cell[BoxData[{\(TheMoebius\ = \
ParametricPlot3D[
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By theory, those two objects are homeomorphic. Lets find a continuous \
deformation from one to another!\
\>", "Subtitle",
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Step 1: How to get red of self-intersection? Of course, by rotating the \
half-Klein bottle\
\>", "Text",
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Cell["Step2: Now, we shrink the half-bottle into the strip? ", "Text",
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When Step goes to infinity we obtain a very flat half-Klein bottle\
\>", "Text",
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FontSize->18],
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Cell["\<\
Step 3: Now we deform this singular Moebius strip into smooth one by changing \
the parameter s, i.e. deforming the curve fMeb[u,s] over which Moebius strip \
is made. \
\>", "Text",
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Cell[CellGroupData[{
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We realize Klein bottle as a tube around certain curve f[u].
In order to do that we have to calculate normal and binormal vector of the \
curve f[u]\
\>", "Text",
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\
\>", "Text"],
Cell["\<\
Because of esthetical reasons the radius of a tube (bottle) have to be \
variable.
Here are some data neccesary for variable radius (Rad[u]) of a Klein \
bottle.\
\>", "Text",
FontSize->18],
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Trigonometric fitting of the data gives us a satisfactory function for the \
radius\
\>", "Text",
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\
\>", "Text"],
Cell["\<\
Things which follows will be more clear later. They are necessary for \
deformation of a half-Klein bottle into a Moebius strip. \
\>", "Text",
FontSize->18],
Cell["\<\
The following function gives rotation angle of the half-Klein bottle in \
order to get red of self-intersections. The start value is for Rot=0 (no \
rotation), and end value is Rot=5 (rotation for 90 degrees).\
\>", "Text",
FontSize->18],
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RowBox[{\(Rot = 0;\), " ",
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The following function gives shrinking of the half-circles of the half-Klein \
bottle through ellipses into a segment of a Moebius strip. The Step variable \
controls shrinking (Step = 1 for no shrinking). When Step goes to infinity \
the smaller diameter of the ellipse becomes 0, while larger goes to 4-the \
length of the segment of a Moebius strip. \
\>", "Text",
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\
\>", "Text"],
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Finally, here is the realization of the Klein bottle as a tube around the \
curve f[u]. \
\>", "Text",
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RowBox[{" ",
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\
\>", "Text"],
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The realization of the Moebius strip \"around the curve\" fMeb[u,s], (s= \
const, fMeb[u,0] = f[u]).
More precisely, for any curve fMeb[u,s] in a Oxy plane one can realize a \
Moebius strip \"over it\" in a following way. In a point fMeb[u=Pi/2,s] take \
a segment of a line perpendicular to Oxy plane and continuously rotate it \
over the curve as parameter u goes. \
\>", "Text",
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FontSize->18],
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FontSize->18],
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RowBox[{\(AngleMeb[u_] := \
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2 Pi\ , \ \(-Pi\)\ \ Sin[\((u\ - \ Pi/2)\)/2]/2, \
Pi\ \ Sin[\((u\ - \ Pi/2)\)/2]/2];\), " ",
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Cell["\<\
Here is the realization of the family (parametrized by s) of the Moebius \
strips \
\>", "Text",
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FontSize->18]], "Section"],
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How the Klein bottle is made?
As a tube around a curve f[u]. Notice that we needed a cusp (which happens \
for u = 3Pi/2) in order to let bottle go through itself.\
\>", "Text",
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FontSize->18],
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FontSize->18],
Cell[BoxData[{
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Cell["This is the intersection of a Klein bottle with Oxy plane. ", "Text",
FontSize->18],
Cell[BoxData[
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Here is the family (with parameter s) of curves fMeb[u,s] (fMeb[u, 0] = \
f[0], fMeb[u,5] = circle). Over each curve the Moebius strip is made. In such \
way the singular Moebius strip is deformed into smooth one. \
\>", "Text",
FontSize->18],
Cell[BoxData[
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}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox["About",
FontSize->18]], "Section"],
Cell[BoxData[{
\(Version : \ 1.0\), "\[IndentingNewLine]",
\(Date : \ September\ 2000\), "\[IndentingNewLine]",
\(Author : \ Srdjan\ Vukmirovic\), "\[IndentingNewLine]",
\(Address : \ Studentski\ Trg\ 16, \ Matematicki\ Fakultet, \
11000\ BELGRADE, \ YUGOSLAVIA\), "\[IndentingNewLine]",
\(Organization : \ Faculty\ of\ Mathematics, \ Belgrade, \
Yugoslavia\), "\[IndentingNewLine]",
\(e - \(mail : \ vsrdjan@matf . bg . ac . yu\)\), "\[IndentingNewLine]",
\(Scientific\ \(interests : \ differential\ geometry\), \
field\ theory, \ programming\)}], "Input"],
Cell[BoxData[
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