(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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", 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 ", StyleBox["NDSolve", FontFamily->"Courier", FontWeight->"Bold"], " and ", StyleBox["NIntegrate", 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", FontSlant->"Italic"], " 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." }], "Text", PageWidth->PaperWidth], Cell["1. Decomposition of the curvature vector.", "Section", FontColor->RGBColor[0, 0, 1]], Cell[TextData[{ "Let ", Cell[BoxData[ \(TraditionalForm\`c(t)\)]], " be a curve on a surface ", Cell[BoxData[ \(TraditionalForm\`f(u, v)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\[GothicE]\_2\)(t)\)]], " be its unit binormal vector, and ", Cell[BoxData[ \(TraditionalForm\`\[Kappa](t)\)]], " be the curvature. Then, the vector ", Cell[BoxData[ \(TraditionalForm \`\[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"] }], "Text", PageWidth->PaperWidth], Cell["2. Principal curvatures.", "Section", FontColor->RGBColor[0, 0, 1]], 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", PageWidth->PaperWidth], Cell["3. Variation of a piece of a catenoid.", "Section", FontColor->RGBColor[0, 0, 1]], 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", PageWidth->PaperWidth], Cell["4. Minimality of a geodesic.", "Section", FontColor->RGBColor[0, 0, 1]], 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", PageWidth->PaperWidth], 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", FontWeight->"Bold"], ".\nKey user commands :", StyleBox[" mywireframe77.", FontWeight->"Bold"] }], "Text", PageWidth->PaperWidth], 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" }], "Text", PageWidth->PaperWidth] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 832}, {0, 604}}, CellGrouping->Manual, WindowSize->{520, 506}, WindowMargins->{{76, Automatic}, {Automatic, 13}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, StyleDefinitions -> "ArticleModern.nb", MacintoshSystemPageSetup->"\<\ 00<0004/0B`000003;H8`_mooh/=<`Tk0fL5N`?P0080004/0B`000000]P2:001 0000I00000400`<30?l00BL?00400@MNIn000000000006P801T1T00000000000 00000000004000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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