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Notebook[{
Cell["Experiments in the Theory of Surfaces", "Title",
PageWidth->PaperWidth,
FontColor->RGBColor[1, 0, 0]],
Cell[TextData[{
StyleBox["Yoshihiko Tazawa\nTokyo Denki University",
FontFamily->"I Times Italic",
FontSize->14,
FontColor->RGBColor[0, 0, 1]],
StyleBox["\n",
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StyleBox["e-mail: tazawa@cck.dendai.ac.jp",
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FontSize->14,
FontColor->RGBColor[0, 0, 1]]
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Cell["Abstract", "Section",
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Cell[TextData[{
"The purpose of this article is to show the advantage of using ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" in the theory of surfaces. ",
StyleBox["\n\n",
FontFamily->"\.8d\[Times]\.96\.be\.92\[Copyright]\.91\[CapitalIGrave]"],
"The examples in the classical differential geometry, namely the theory of \
curves and surfaces, have been confined to a small group of calculable \
objects, because of the difficulty in evaluating geometrical quantities and \
solving differential equations explicitly. But ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" has made it possible to deal with wide range of objects and to perform \
experimental treatment of them, based on the power of numerical calculation, \
the ",
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FontFamily->"Courier",
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" and ",
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FontFamily->"Courier",
FontWeight->"Bold"],
" commands in particular, and graphical visualization by the ",
StyleBox["ParametricPlot",
FontFamily->"Courier",
FontWeight->"Bold"],
" command. I would like to introduce here several animations and figures \
that I use in my differential geometry class for undergraduate students. \
First of all, these graphics help students understand the basic notions of \
differential geometry. Secondly, we can experiment on geometry through these \
graphics.",
StyleBox["\n\n",
FontFamily->"\.8d\[Times]\.96\.be\.92\[Copyright]\.91\[CapitalIGrave]"],
"This article is one of the serial talks given by the author at Developers \
Conference 95, IMS 97 in Rovaniemi, and WMC 98 in Chicago. They are all \
targeted for the experimental usage of ",
StyleBox["Mathematica",
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" on differential geometry. This time the topics are focused on the \
surfaces in the 3-dimensional Euclidean space. \n\nThe commands are contained \
in six notebooks. If the commands in other notebooks cause any trouble, \
please restart ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". See also the explanation of the notebooks."
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Cell["1. Decomposition of the curvature vector.", "Section",
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"Let ",
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" be a curve on a surface ",
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", ",
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" be its unit binormal vector, and ",
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" be the curvature. Then, the vector ",
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\`\[GothicK](t) = \(\[Kappa](t)\) \(\(\[GothicE]\_2\)(t)\)\)]],
", called the curvature vector, is decomposed into the direct sum ",
Cell[BoxData[
\(TraditionalForm
\`\[GothicK](t) = \(\[GothicK]\_n\)(t) + \(\[GothicK]\_g\)(t)\)]],
" of the normal part (normal curvature vector) and the tangential part \
(geodesic curvature vector). If two curves on ",
Cell[BoxData[
\(TraditionalForm\`f(u, v)\)]],
" that are tangential at some point on the surface, then their normal \
curvature vectors at this point coincide. Denote by \[GothicN](u,v) the unit \
normal vector of the surface, and call the inner product ",
Cell[BoxData[
\(TraditionalForm
\`\(\(\[Kappa]\_n\)(t) =
\(\(\[GothicK]\_n\)(t)\)\[CenterDot]\(\[GothicN](c(t))\)\ \)\)]],
"is called the normal curvature of the curve. Enter the commands of \
Animation 1 in the notebook Tazawa1.nb.\n\nKey ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions : numerical calculation.\nKey user commands : ",
StyleBox["mywireframe, normalcurvatureanim.",
FontWeight->"Bold"]
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Cell["2. Principal curvatures.", "Section",
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Cell[TextData[{
"Let ",
Cell[BoxData[
\(TraditionalForm\`p\)]],
" be a point on a surface ",
Cell[BoxData[
\(TraditionalForm\`f(u, v)\)]],
", and \[GothicN] be the unit normal vector of the surfce at ",
Cell[BoxData[
\(TraditionalForm\`p\)]],
" . Consider the curve of the section of the surface by a plane including \
\[GothicN] , and orient it so that the unit principal normal vector of the \
curve coincides with \[GothicN]. The minimum ",
Cell[BoxData[
\(TraditionalForm\`k\_1\)]],
" and the maximum ",
Cell[BoxData[
\(TraditionalForm\`k\_2\)]],
" of the normal curvature of the section curve are called the principal \
curvatures. The mean curvature and the Gaussian curvature are defined by ",
Cell[BoxData[
\(TraditionalForm\`H = \(k\_1 + k\_2\)\/2\)]],
" and ",
Cell[BoxData[
\(TraditionalForm\`K = \(k\_1\) k\_2\)]],
". Enter the commands of Animation 2 in the notebook Tazawa2.nb.\n\nKey ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions : numerical culculation.\nKey user commands :",
StyleBox[" planesectionanim22.",
FontWeight->"Bold"]
}], "Text",
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Cell["3. Variation of a piece of a catenoid.", "Section",
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Cell[TextData[{
"A surface with vanishing mean curvature is called a minimal surface. A \
minimal surface is characterized as the surface with the minimal area bounded \
by an assigned boundary. Animation 3 in Tazawa3.nb shows the change of the \
area under a variation of a piece of catenoid, a well known minimal surface.\n\
\nKey ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions : ",
StyleBox["NIntegrate",
FontWeight->"Bold"],
"."
}], "Text",
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Cell["4. Minimality of a geodesic.", "Section",
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Cell[TextData[{
"The shortest curve jointing two given points on a surface is a geodesic. \
For an assigned point and direction, there is a unique geodesic, obtained by \
solving a system of ordinary differential equations. Animation 4 in \
Tazawa4.nb shows the change of the length under a variation of a geodesic on \
a randomly generated surface.\n\nKey ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions : ",
StyleBox["NIntegrate, NDSolve",
FontWeight->"Bold"],
".\nKey user commands :",
StyleBox[" mywireframe77.",
FontWeight->"Bold"]
}], "Text",
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Cell["5. Approximation of a Geodesic", "Section",
FontColor->RGBColor[0, 0, 1]],
Cell[TextData[{
"A geodesic is also characterized as a curve on a surface with vanishing \
geodesic curvature vector, which means, a geodesic looks infinitesimally like \
a line, if observed from the direction of the normal vector. Therefore, a \
geodesic can be approximated by jointing the pieces of curves that are the \
inverse images of segments on the tangent planes under the projections to the \
tangent planes. Animation 5 in Tazawa5.nb shows an approximation of the \
geodesic in Animation 4.\n\nKey ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions : ",
StyleBox["NIntegrate, NSolve, NDSolve",
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".\nKey user commands :",
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Cell["6. Surfaces with Assigned fundamental quantities", "Section",
FontColor->RGBColor[0, 0, 1]],
Cell[TextData[{
"The fundamental theorem of surfaces states that if a set of functions \
satisfy the integrability conditions of Gauss and Mainardi-Codazzi, then \
there exit surfaces whose first and second fundamental quantities coincide \
with these assigned functions, and they are unique up to isometries. Commands \
in Tazawa6.nb check the integrability conditions and plot the solution \
surface numerically.\n\nKey ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" functions : ",
StyleBox["NIntegrate, NDSolve",
FontWeight->"Bold"],
".\nKey user commands :",
StyleBox[" integrabilitycheck, numintegrabilitycheck, numplotfundsol",
FontWeight->"Bold"]
}], "Text",
PageWidth->PaperWidth],
Cell["References", "Section",
FontColor->RGBColor[0, 0, 1]],
Cell[TextData[{
StyleBox["Introduction to the theory of curves and surfaces with ",
FontWeight->"Bold"],
StyleBox["Mathematica",
FontWeight->"Bold",
FontSlant->"Italic"],
", by Yoshihiko Tazawa, in preparation to be published in Japanese language \
from Addison-Wesley Publishers Japan"
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