(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing 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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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 2151618, 88796]*) (*NotebookOutlinePosition[ 2158213, 88964]*) (* CellTagsIndexPosition[ 2156689, 88927]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["DGeometrica", "Title", PageWidth->PaperWidth], Cell[TextData[{ "A ", StyleBox["Mathematica", FontSlant->"Italic"], " package for differential geometry" }], "Subtitle", PageWidth->PaperWidth] }, Open ]], Cell[CellGroupData[{ Cell["Yoshihiko TAZAWA", "Author", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["\<\ School of Information Environment Tokyo Denki University 2-1200 Muzai-gakuendai Inzai, Chiba Pref. 270-1382 JAPAN tazawa@cck.dendai.ac.jp\ \>", "TextAboutAuthor", PageWidth->PaperWidth], Cell[CellGroupData[{ Cell["1. Abstract", "Section", PageWidth->PaperWidth], Cell[TextData[{ StyleBox[" This notebook is an introduction of the first half of ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["DGeometrica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[", a package for doing differential geometry of curves and \ surfaces by ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Mathematica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[".\n The geometry of curves and surfaces is a nice topic to deal \ with computer algebras. Alfred Gray's book [5] is a comprehensive \ introduction of how to utilize ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Mathematica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" in differential geometry. He gave a tutorial talk [1] in the \ first IMS. His ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Mathematica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" package is still downloadable from [4]. John Oprea published a \ book [9] based on Maple. Richard Palais and his group created a freeware \ 3D-XplorMath [5] which works on Macintosh OS and deals with many topics from \ differential geometry.\n In the third IMS the author gave a talk [2] and \ introduced several examples from the book [10] written in Japanese. The main \ part of ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["DGeometrica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" is a set of ideas and ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Mathematica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" commands contained in the [10]. The feature of ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["DGeometrica", PageWidth->PaperWidth, FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" compared to other packages is that the usage of numerical \ functions such as ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["NDSolve", PageWidth->PaperWidth, FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" or ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["NIntegrate", PageWidth->PaperWidth, FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" to do experimental approach and to extend the range of examples \ in differential geometry of curves and surfaces.\n This article consists of \ three materials: the present notebook, ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["DGeometrica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[" package files contained in a folder named DGeometrica, and a \ pair of ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Mathematica", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[" notebooks ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Plane- CurvesTest.nb", PageWidth->PaperWidth, FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[" and ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["SpaceCurvesTest.nb", PageWidth->PaperWidth, FontFamily->"Courier New", FontWeight->"Bold"], StyleBox[". 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", StyleBox["dg-", PageWidth->PaperWidth, FontFamily->"Courier", FontWeight->"Bold"], StyleBox["commands without ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Anim", PageWidth->PaperWidth, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" or ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["Fig", PageWidth->PaperWidth, FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" like ", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox["dgCurvature", PageWidth->PaperWidth, FontFamily->"Courier", FontWeight->"Bold"], " are just for calculation. ", StyleBox["\n The present notebook is an outline of the package and \ introduces several commands in it", PageWidth->PaperWidth, FontFamily->"Times New Roman"], StyleBox[". All Input cells in this notebook are set non-evaluatable in \ order to make the appearance stable. 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If the curve is of the form ", StyleBox["c", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")=(", StyleBox["x", FontSlant->"Italic"], ", ", StyleBox["f", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], ")), namely the graph of a function ", StyleBox["y ", FontSlant->"Italic"], "= ", StyleBox["f", FontSlant->"Italic"], "(", StyleBox["x", FontSlant->"Italic"], "), it is plotted by ", StyleBox["Plot", PageWidth->PaperWidth, FontFamily->"Courier", FontWeight->"Bold"], " of ", StyleBox["Mathematica", FontSlant->"Italic"], ", but it fails to show the discontinuity of functions. 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Segments with length greater than the 0.1 \ times width or the 0.1 times height of the box containing the curve are not \ shown. 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FontFamily->"Courier New", FontWeight->"Bold"], ". We can calculate the length of a fairly complicated curve by ", StyleBox["dgNumArcLength", PageWidth->PaperWidth, FontFamily->"Courier New", FontWeight->"Bold"], " numerically." }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["c[t_]=\n {1.18445+0.215722*t-10.2437*t^2-13.3893*t^3+\n \ 25.9661*t^4+30.553*t^5-36.084*t^6,\n \ -0.6248+6.4332*t+0.151186*t^2-52.3013*t^3+\n \ 11.1881*t^4+139.806*t^5-105.033*t^6};\n", AspectRatioFixed->True, FontFamily->"Courier New"], "dg", StyleBox["NumArcLength", PageWidth->PaperWidth, FontFamily->"Courier New", FontWeight->"Bold"], "[0,1][c]" }], "Input", CellLabel->"In[31]:=", AspectRatioFixed->True], Cell[BoxData[ \(TraditionalForm\`7.180770467685628`\)], "Output", CellLabel->"Out[32]="] }, Open ]], Cell["\<\ The geometric meaning of the length of a curve is the limit of the \ length of the broken line jointing points on the curve, as the number of the \ points tends to infinity. 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