(*********************************************************************** 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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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[ 592403, 30326]*) (*NotebookOutlinePosition[ 593806, 30369]*) (* CellTagsIndexPosition[ 593762, 30365]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["Experiments in Differential Geometry with ", FontSize->16, FontColor->GrayLevel[0]], StyleBox["Mathematica", FontFamily->"Palatino", FontSize->16, FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\n\n", FontColor->GrayLevel[0]], StyleBox["Yoshihiko Tazawa", FontSize->14, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["\n", FontSize->24, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["tazawa@cck.dendai.ac.jp", FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[0]] }], "Title", CellMargins->{{19, Inherited}, {Inherited, Inherited}}, FontColor->RGBColor[1, 0, 0], Background->GrayLevel[1]], Cell[TextData[StyleBox["Introduction", FontSize->16]], "Subsubsection", CellMargins->{{18, Inherited}, {Inherited, Inherited}}, Evaluatable->False, AspectRatioFixed->True, FontFamily->"Helvetica", FontSize->24], Cell[TextData[{ StyleBox[ "The classical theory of curves and surfaces was established long ago. But \ there has always been difficulty in applying the general theory to individual \ curves or surfaces, because calculating integrations,, or solving \ differential equations explicitly is not possible in general. So the examples \ appearing in this field have been confined to a small group of calculable \ objects. But ", FontSize->12], StyleBox["Mathematica", FontFamily->"Palatino", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[ " has made it possible to deal with a more generic group of objects. \nIn \ this talk, I will introduce some experimental visualizations that I have \ produced by ", FontSize->12], StyleBox["Mathematica", FontFamily->"Palatino", FontSize->12, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[ " and used in my differential geometry class. 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Since the curvature \[Kappa](", FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["t", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") is 1 over the radius of the osculationg circle \[Rho](", FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["t", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox["), \[Kappa](", FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox["t", FontSize->12, FontWeight->"Plain", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[") = \[PlusMinus]", FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[0]], Cell[BoxData[ \(TraditionalForm\`1\/\(\[Rho](t)\)\)], FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[0]], StyleBox[ " and the osculating circle approximates the curve, so, we can approximate \ the curve by jointing together pieces of circles with radii determined by the \ curvature function.\n", FontSize->12, FontWeight->"Plain", FontColor->GrayLevel[0]], "\n", StyleBox[ "Let us consider the simplest case: the curvature is \[Kappa](s) = s , \ where s is the arclength. 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.66828 .57405 L .66589 .57809 L .66339 .58206 L .66078 .58596 L .65807 .58978 L .65525 .59354 L .65233 .59721 L .64931 .6008 L .6462 .60431 L .64299 .60773 L .63969 .61106 L .63629 .6143 L .63281 .61745 L .62925 .6205 L .6256 .62345 L .62188 .6263 L .61807 .62905 L .6142 .63169 L .61025 .63423 L .60623 .63665 L .60215 .63897 L .59801 .64117 L .59381 .64326 L .58955 .64524 L .58525 .64709 L .58089 .64883 L s P P p 0 0 1 r p .004 w .58089 .64883 m .57648 .65044 L .57202 .6519 L .56752 .65322 L .56298 .65439 L .5584 .65542 L .55379 .65629 L .54916 .65702 L .5445 .6576 L .54216 .65783 L .53983 .65802 L .53749 .65818 L .53632 .65824 L .53514 .6583 L .53397 .65834 L .5328 .65838 L .53163 .6584 L .53104 .65841 L .53075 .65842 L .53045 .65842 L .53016 .65842 L .52987 .65843 L .52972 .65843 L .52957 .65843 L .52943 .65843 L .52928 .65843 L .52913 .65843 L .52899 .65843 L .52884 .65843 L .52869 .65843 L .52855 .65843 L .5284 .65843 L .52811 .65843 L .52781 .65842 L .52752 .65842 L .52693 .65841 L .52635 .6584 L .52576 .65839 L .52459 .65836 L .52342 .65832 L .52107 .65821 L .51873 .65806 L .51639 .65788 L .51173 .6574 L .50708 .65676 L .50245 .65598 L .49786 .65505 L .49329 .65397 L .48876 .65274 L .48428 .65137 L Mistroke .47984 .64985 L .47545 .64819 L .47112 .64639 L .46685 .64445 L .46264 .64237 L .4585 .64016 L .45444 .63781 L .45045 .63534 L .44655 .63273 L .44273 .63001 L Mfstroke P P p 0 0 1 r p .004 w .44273 .63001 m .43901 .62715 L .4354 .62415 L .43191 .62102 L .42853 .61777 L .42528 .61439 L .42215 .61089 L .41916 .60727 L .4163 .60355 L .41359 .59972 L .41102 .5958 L .4086 .59178 L .40633 .58768 L .40421 .58349 L .40225 .57923 L .40046 .57489 L .39882 .57049 L .39735 .56604 L .39605 .56153 L .39492 .55698 L .39395 .55239 L .39316 .54776 L .39254 .54311 L .3923 .54078 L .3921 .53844 L .39194 .5361 L .39188 .53493 L .39183 .53376 L .39179 .53259 L .39178 .532 L .39176 .53141 L .39175 .53083 L .39175 .53053 L .39174 .53024 L .39174 .52995 L .39174 .52965 L .39174 .52951 L .39174 .52936 L .39174 .52921 L .39174 .52907 L .39174 .52892 L .39174 .52877 L .39174 .52863 L .39174 .52848 L .39174 .52833 L .39174 .52819 L .39174 .52789 L .39174 .5276 L .39175 .52731 L .39176 .52672 L Mistroke .39177 .52614 L .39178 .52555 L .39182 .52438 L .39187 .52321 L .39192 .52203 L .39207 .51969 L .39227 .51735 L .3925 .51502 L .39311 .51037 L .39389 .50574 L .39483 .50115 L Mfstroke P P p 0 0 1 r p .004 w .39483 .50115 m .39597 .49659 L .39729 .49209 L .3988 .48765 L .4005 .48328 L .40238 .47898 L .40444 .47477 L .40668 .47064 L .40909 .46662 L .41167 .4627 L .41441 .45889 L .41731 .45521 L .42036 .45164 L .42356 .44821 L .42691 .44492 L .43039 .44178 L .434 .43878 L .43773 .43594 L .44158 .43326 L .44554 .43074 L .4496 .42839 L .45376 .42622 L .458 .42422 L .46233 .42241 L .46673 .42078 L .47119 .41934 L .47572 .41809 L .48029 .41703 L .4849 .41616 L .48722 .4158 L .48954 .41549 L .49187 .41523 L .49421 .41502 L .49655 .41486 L .49772 .41479 L .49889 .41474 L s P P p 0 0 1 r p .004 w .49889 .41474 m .50006 .41471 L .50065 .41469 L .50094 .41469 L .50124 .41468 L .50153 .41468 L .50168 .41468 L .50182 .41468 L .50197 .41468 L .50212 .41467 L .50226 .41467 L .50241 .41467 L .50256 .41467 L .5027 .41467 L .50285 .41467 L .503 .41467 L .50314 .41467 L .50329 .41468 L .50358 .41468 L .50388 .41468 L .50417 .41469 L .50475 .4147 L .50534 .41471 L .50593 .41473 L .5071 .41477 L .50827 .41483 L .51061 .415 L .51295 .41521 L .5176 .41581 L .52222 .41663 L .5268 .41766 L .53132 .41891 L .53577 .42037 L .54016 .42204 L .54446 .42392 L .54866 .426 L .55277 .42827 L .55676 .43073 L .56063 .43339 L .56437 .43622 L .56798 .43922 L .57143 .44239 L .57474 .44572 L .57788 .4492 L .58085 .45283 L .58365 .4566 L .58627 .46049 L .58871 .4645 L .59095 .46862 L .59299 .47284 L Mistroke .59483 .47716 L .59646 .48156 L .59789 .48603 L .5991 .49056 L .6001 .49514 L .60088 .49977 L Mfstroke P P p 0 0 1 r p .004 w .60088 .49977 m .60119 .50209 L .60143 .50443 L .60153 .5056 L .60161 .50677 L .60168 .50794 L .60173 .50911 L .60175 .50969 L .60177 .51028 L .60179 .51087 L .60179 .51116 L .60179 .51145 L .6018 .51175 L .6018 .51189 L .6018 .51204 L .6018 .51219 L .6018 .51233 L .6018 .51248 L .6018 .51263 L .6018 .51277 L .6018 .51292 L .6018 .51307 L .6018 .51321 L .6018 .51351 L .60179 .5138 L .60178 .51439 L .60177 .51497 L .60175 .51556 L .60173 .51614 L .60168 .51732 L .60161 .51849 L .60142 .52083 L .60118 .52316 L .60051 .5278 L .59959 .5324 L .59844 .53695 L .59705 .54143 L .59543 .54583 L .59358 .55014 L .59151 .55435 L .58922 .55845 L .58672 .56242 L .58402 .56625 L .58112 .56994 L .57803 .57347 L .57476 .57684 L .57132 .58003 L .56772 .58303 L .56397 .58585 L .56007 .58846 L Mistroke .55605 .59087 L .5519 .59306 L .54764 .59504 L .54329 .59679 L .53885 .59831 L .53434 .5996 L .52977 .60064 L .52746 .60108 L .52515 .60145 L .52049 .60202 L Mfstroke P P p 0 0 1 r p .004 w .52049 .60202 m .51932 .60212 L .51815 .60221 L .51698 .60228 L .5164 .60231 L .51581 .60233 L .51522 .60235 L .51464 .60237 L .51435 .60237 L .51405 .60238 L .51376 .60238 L .51361 .60238 L .51347 .60238 L .51332 .60239 L .51317 .60239 L .51303 .60239 L .51288 .60239 L .51273 .60239 L .51259 .60239 L .51244 .60239 L .51229 .60239 L .51215 .60239 L .512 .60239 L .51171 .60238 L .51141 .60238 L .51112 .60237 L .51053 .60236 L .50995 .60234 L .50878 .60229 L .5076 .60223 L .50643 .60215 L .5041 .60193 L .50177 .60165 L .49714 .6009 L .49256 .59987 L .48805 .59859 L .48362 .59705 L .47928 .59526 L .47505 .59323 L .47095 .59095 L .46698 .58845 L .46317 .58572 L .45951 .58278 L .45603 .57964 L .45273 .5763 L .44963 .57278 L .44674 .56909 L .44406 .56524 L .44161 .56124 L .43939 .5571 L Mistroke .43741 .55285 L .43568 .54849 L .4342 .54404 L .43297 .53951 L .43201 .53492 L .43162 .53261 L .43131 .53028 L .43106 .52795 L .43096 .52678 L .43087 .52561 L .43081 .52444 L .43078 .52386 L .43076 .52327 L .43074 .52268 L .43072 .5221 L .43072 .5218 L .43071 .52151 L .43071 .52136 L .43071 .52122 L .43071 .52107 L .43071 .52092 L .43071 .52078 L .43071 .52063 L .43071 .52048 L .43071 .52034 L .43071 .52019 L .43071 .52004 L .43071 .5199 L .43071 .51975 L .43071 .51946 L .43072 .51916 L .43073 .51858 L .43074 .51799 L .43076 .51741 L .43081 .51623 L .43088 .51506 L .43096 .51389 L .43118 .51156 L Mfstroke P P p 0 0 1 r p .004 w .43118 .51156 m .43147 .50923 L .43183 .50691 L .43277 .50232 L .43399 .49779 L .43549 .49334 L .43726 .489 L .4393 .48477 L .4416 .48068 L .44415 .47675 L .44693 .47297 L .44995 .46938 L .45318 .46598 L .45662 .46279 L .46025 .45982 L .46406 .45708 L .46803 .45458 L .47215 .45234 L .4764 .45035 L .48077 .44863 L .48523 .44719 L .48977 .44603 L .49207 .44555 L .49438 .44515 L .4967 .44482 L .49904 .44456 L .5002 .44446 L .50137 .44437 L .50254 .44431 L .50313 .44428 L .50372 .44426 L .5043 .44424 L .5046 .44424 L .50489 .44423 L .50518 .44423 L .50533 .44422 L .50548 .44422 L .50562 .44422 L .50577 .44422 L .50592 .44422 L .50606 .44422 L .50621 .44422 L .50636 .44422 L .5065 .44422 L .50665 .44422 L .5068 .44422 L .50694 .44422 L .50723 .44423 L .50753 .44423 L .50782 .44424 L Mistroke .50841 .44425 L .50899 .44427 L .50958 .4443 L .51075 .44436 L .51192 .44444 L .51309 .44454 L .51542 .44479 L .51775 .44511 L .52236 .44597 L .52691 .44712 L .53137 .44854 L .53575 .45025 L .54 .45222 L Mfstroke P P p 0 0 1 r .01 w .33049 .19079 Mdot P p 0 0 1 r .01 w .47292 .21859 Mdot P p 0 0 1 r .01 w .59968 .28804 Mdot P p 0 0 1 r .01 w .68164 .40606 Mdot P p 0 0 1 r .01 w .68023 .5486 Mdot P p 0 0 1 r .01 w .58089 .64883 Mdot P p 0 0 1 r .01 w .44273 .63001 Mdot P p 0 0 1 r .01 w .39483 .50115 Mdot P p 0 0 1 r .01 w .49889 .41474 Mdot P p 0 0 1 r .01 w .60088 .49977 Mdot P p 0 0 1 r .01 w .52049 .60202 Mdot P p 0 0 1 r .01 w .43118 .51156 Mdot P p 0 0 1 r .01 w .54 .45222 Mdot P p 0 0 1 r .01 w .67003 .51738 Mdot P p 0 0 1 r .01 w .80006 .58255 Mdot P p 0 0 1 r .01 w .93009 .64772 Mdot P p 0 0 1 r p .004 w .54 .45222 m .54411 .45427 L .54821 .45633 L .55232 .45839 L .55643 .46045 L .56053 .46251 L .56464 .46456 L .56875 .46662 L .57285 .46868 L .57696 .47074 L .58106 .4728 L .58517 .47485 L .58928 .47691 L .59338 .47897 L .59749 .48103 L .60159 .48309 L .6057 .48514 L .60981 .4872 L .61391 .48926 L .61802 .49132 L .62213 .49338 L .62623 .49543 L .63034 .49749 L .63444 .49955 L .63855 .50161 L .64266 .50366 L .64676 .50572 L .65087 .50778 L .65497 .50984 L .65908 .5119 L .66319 .51395 L .66729 .51601 L .6714 .51807 L .67551 .52013 L .67961 .52219 L .68372 .52424 L .68782 .5263 L .69193 .52836 L .69604 .53042 L .70014 .53248 L .70425 .53453 L .70835 .53659 L .71246 .53865 L .71657 .54071 L .72067 .54277 L .72478 .54482 L .72889 .54688 L .73299 .54894 L .7371 .551 L .7412 .55305 L Mistroke .74531 .55511 L .74942 .55717 L .75352 .55923 L .75763 .56129 L .76173 .56334 L .76584 .5654 L .76995 .56746 L .77405 .56952 L .77816 .57158 L .78226 .57363 L .78637 .57569 L .79048 .57775 L .79458 .57981 L .79869 .58187 L .8028 .58392 L .8069 .58598 L .81101 .58804 L .81511 .5901 L .81922 .59216 L .82333 .59421 L .82743 .59627 L .83154 .59833 L .83564 .60039 L .83975 .60244 L .84386 .6045 L .84796 .60656 L .85207 .60862 L .85618 .61068 L .86028 .61273 L .86439 .61479 L .86849 .61685 L .8726 .61891 L .87671 .62097 L .88081 .62302 L .88492 .62508 L .88902 .62714 L .89313 .6292 L .89724 .63126 L .90134 .63331 L .90545 .63537 L .90956 .63743 L .91366 .63949 L .91777 .64155 L .92187 .6436 L .92598 .64566 L .93009 .64772 L Mfstroke P P p 1 0 0 r .015 w .54 .45222 Mdot P p 1 0 0 r p .004 w .57856 .51467 m .57856 .51512 L .57856 .51556 L .57855 .516 L .57854 .51644 L .57853 .51689 L .57851 .51733 L .57847 .51821 L .57842 .5191 L .57836 .51998 L .5782 .52174 L .578 .5235 L .57776 .52525 L .57713 .52874 L .57633 .53219 L .57536 .53559 L .5729 .54222 L .56978 .54858 L .56604 .55458 L .55683 .56531 L .55146 .56992 L .54565 .57396 L .53947 .5774 L .53296 .58019 L .52961 .58133 L .52621 .58231 L .52276 .58311 L .51928 .58373 L .51753 .58398 L .51577 .58418 L .51401 .58433 L .51312 .5844 L .51224 .58445 L .51135 .58449 L .51091 .5845 L .51047 .58451 L .51003 .58452 L .50959 .58453 L .50914 .58453 L .5087 .58454 L .50826 .58453 L .50782 .58453 L .50737 .58452 L .50693 .58451 L .50605 .58449 L .50516 .58445 L .50428 .5844 L .5034 .58433 L .50163 .58418 L .49987 .58398 L Mistroke .49812 .58373 L .49464 .58311 L .49119 .58231 L .48444 .58019 L .47793 .5774 L .47175 .57396 L .46594 .56992 L .46057 .56531 L .45569 .56018 L .45136 .55458 L .44762 .54858 L .4445 .54222 L .44319 .53894 L .44204 .53559 L .44107 .53219 L .44027 .52874 L .43964 .52525 L .4394 .5235 L .4392 .52174 L .43904 .51998 L .43898 .5191 L .43893 .51821 L .43889 .51733 L .43887 .51689 L .43886 .51644 L .43885 .516 L .43884 .51556 L .43884 .51512 L .43884 .51467 L .43884 .51423 L .43884 .51379 L .43885 .51335 L .43886 .5129 L .43889 .51202 L .43893 .51114 L .43898 .51025 L .43904 .50937 L .4392 .50761 L .4394 .50585 L .43964 .50409 L .44027 .50061 L .44107 .49716 L .44204 .49376 L .4445 .48712 L .44762 .48077 L .45569 .46917 L .46057 .46404 L .46594 .45943 L .47175 .45539 L .47793 .45195 L Mistroke .48444 .44916 L .48779 .44802 L .49119 .44704 L .49464 .44624 L .49812 .44562 L .49987 .44537 L .50163 .44517 L .5034 .44501 L .50428 .44495 L .50516 .4449 L .50605 .44486 L .50649 .44485 L .50693 .44483 L .50737 .44482 L .50782 .44482 L .50826 .44481 L .5087 .44481 L .50914 .44481 L .50959 .44482 L .51003 .44482 L .51047 .44483 L .51135 .44486 L .51224 .4449 L .51312 .44495 L .51401 .44501 L .51577 .44517 L .51753 .44537 L .51928 .44562 L .52276 .44624 L .52621 .44704 L .53296 .44916 L .53947 .45195 L .54565 .45539 L .55146 .45943 L .55683 .46404 L .56171 .46917 L .56604 .47476 L .56978 .48077 L .5729 .48712 L .57536 .49376 L .57633 .49716 L .57713 .50061 L .57776 .50409 L .578 .50585 L .5782 .50761 L .57836 .50937 L .57842 .51025 L .57847 .51114 L .57851 .51202 L .57853 .51246 L Mistroke .57854 .5129 L .57855 .51335 L .57856 .51379 L .57856 .51423 L .57856 .51467 L Mfstroke P P P p 0 .7 .7 r p .004 w .18519 .18519 m .18822 .18519 L .19125 .18519 L .19428 .18519 L .19731 .18519 L .20034 .18519 L .20337 .18519 L .2064 .1852 L .20943 .1852 L .21549 .18522 L .21852 .18523 L .22155 .18524 L .22761 .18528 L .23367 .18532 L .23973 .18538 L .24579 .18546 L .2579 .18565 L .27002 .18593 L .28214 .18629 L .29425 .18676 L .30635 .18735 L .33054 .18892 L .35468 .19112 L .37875 .19403 L .4027 .19777 L .42648 .20244 L .45005 .20811 L .47332 .21489 L .51862 .23208 L .56154 .25457 L .60097 .2827 L .6356 .31655 L .66395 .35579 L .67528 .37722 L .68444 .39965 L .68815 .41118 L .69125 .4229 L .69372 .43477 L .69471 .44074 L .69554 .44675 L .6962 .45277 L .69646 .45579 L .69669 .45881 L .69687 .46184 L .697 .46486 L .6971 .46789 L .69715 .47092 L .69715 .47395 L .69712 .47698 L .69704 .48001 L Mistroke .69691 .48304 L .69652 .48908 L .69626 .4921 L .69596 .49512 L .69521 .50113 L .69428 .50712 L .69188 .519 L .68875 .5307 L .68488 .54219 L .67496 .56429 L .6622 .58488 L .64675 .60353 L .62883 .61982 L .61903 .62695 L .60874 .63334 L .59799 .63894 L .58685 .6437 L .57537 .6476 L .56953 .6492 L .56363 .65058 L .55768 .65172 L .55168 .65263 L .54868 .65299 L .54566 .65329 L .54264 .65353 L .53962 .65371 L .53659 .65382 L .53356 .65388 L .53053 .65387 L .5275 .6538 L .52447 .65367 L .52145 .65348 L .51843 .65322 L .51542 .6529 L .50941 .65207 L .50345 .65099 L .49169 .64808 L .48022 .64417 L .46913 .63929 L .45852 .63345 L .44846 .62669 L .43905 .61905 L .43038 .61059 L .42252 .60137 L .41555 .59146 L .40955 .58094 L .40457 .56989 L .40068 .55841 L .39916 .55255 L .39793 .54662 L Mistroke .39742 .54363 L .39699 .54063 L .39663 .53762 L .39635 .5346 L .39614 .53158 L .39601 .52855 L .39595 .52553 L .39597 .5225 L .39606 .51947 L .39624 .51644 L .39648 .51342 L .39681 .51041 L .39769 .50441 L .39887 .49847 L .40215 .48681 L .40662 .47555 L .41225 .46482 L .41897 .45474 L .42674 .44545 L 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.47586 L .88454 .47987 L .88478 .48388 L .88497 .4879 L .88513 .49191 L .88519 .49392 L .88524 .49593 L .88528 .49794 L .8853 .49895 L .88531 .49995 L .88532 .50096 L .88533 .50196 L .88534 .50247 L .88534 .50297 L .88534 .50322 L .88534 .50347 L .88534 .50372 L .88534 .50397 L .88534 .50422 L .88534 .50448 L .88534 .50473 L .88534 .50498 L .88534 .50523 L .88534 .50548 L .88534 .50573 L .88534 .50598 L .88534 .50649 L .88534 .50699 L .88533 .50799 L .88532 .509 L .88531 .51 L .88528 .51201 L Mistroke .88524 .51402 L .88518 .51603 L .88505 .52005 L .88487 .52407 L Mfstroke P P p 0 0 1 r p .004 w .88487 .52407 m .88438 .53209 L .88369 .5401 L .88281 .5481 L .88173 .55607 L .88046 .564 L .87899 .57191 L .87733 .57978 L .87548 .5876 L .87344 .59538 L .87122 .60311 L .8688 .61078 L .8662 .61839 L .86341 .62593 L .86044 .6334 L .85729 .6408 L .85396 .64812 L .85045 .65535 L .84677 .6625 L .84292 .66956 L .83889 .67652 L .83469 .68338 L .83033 .69013 L .82581 .69678 L .82112 .70331 L s P P p 0 0 1 r p .004 w .82112 .70331 m .81627 .70972 L .81123 .71599 L .80603 .72212 L .80065 .7281 L .79511 .73392 L .7894 .73959 L .78354 .74509 L .77752 .75042 L .77136 .75559 L .76505 .76058 L .75861 .76539 L .75204 .77002 L .74533 .77446 L .73851 .77871 L .73156 .78276 L .72451 .78662 L .71735 .79028 L .71009 .79374 L .70274 .79699 L .69529 .80003 L .68777 .80287 L .68017 .80549 L .67249 .80789 L .66476 .81008 L s P P p 0 0 1 r p .004 w .66476 .81008 m .65696 .81204 L .6491 .81374 L .64119 .8152 L .63324 .8164 L .62526 .81735 L .62125 .81772 L .61725 .81804 L .61323 .81829 L .61123 .81839 L .60922 .81848 L .60721 .81854 L .6052 .8186 L .60419 .81862 L .60319 .81864 L .60269 .81864 L .60218 .81865 L .60168 .81865 L .60118 .81866 L .60093 .81866 L .60068 .81866 L .60043 .81866 L .60017 .81866 L .59992 .81866 L .59967 .81866 L .59942 .81866 L .59917 .81866 L .59892 .81866 L .59867 .81866 L .59842 .81866 L .59816 .81866 L .59766 .81866 L .59716 .81865 L .59615 .81864 L .59515 .81862 L .59314 .81858 L .59113 .81852 L .58912 .81845 L .58511 .81825 L .58109 .81799 L .57709 .81766 L .56909 .81682 L .56112 .81572 L .5532 .81437 L .54532 .81277 L .53749 .81092 L .52973 .80882 L .52204 .80647 L .51443 .80388 L .50691 .80105 L Mistroke .49947 .79798 L .49214 .79467 L .48492 .79113 L .47782 .78737 L Mfstroke P P p 0 0 1 r p .004 w .47782 .78737 m .47085 .78336 L .46402 .77911 L .45735 .77462 L .45085 .7699 L .44451 .76495 L .43836 .75977 L .43239 .75438 L .42662 .74878 L .42106 .74298 L .4157 .73698 L .41056 .73079 L .40565 .72443 L .40096 .7179 L .39651 .7112 L .39231 .70435 L .38834 .69735 L .38463 .69022 L .38118 .68296 L .37799 .67558 L .37506 .66809 L .3724 .6605 L .37001 .65282 L .3679 .64506 L .36607 .63724 L s P P p 0 0 1 r p .004 w .36607 .63724 m .36453 .62934 L .3633 .6214 L .3628 .61741 L .36238 .61341 L .36205 .6094 L .36191 .6074 L .36179 .60539 L .36168 .60338 L .3616 .60138 L .36154 .59937 L .36152 .59836 L .3615 .59736 L .3615 .59685 L .36149 .59635 L .36148 .59585 L .36148 .5956 L .36148 .59535 L .36148 .59509 L .36148 .59484 L .36148 .59459 L .36148 .59434 L .36148 .59409 L .36148 .59384 L .36148 .59359 L .36148 .59334 L .36148 .59308 L .36148 .59283 L .36149 .59233 L .36149 .59183 L .3615 .59133 L .36151 .59032 L .36154 .58932 L .36159 .58731 L .36167 .5853 L .36189 .58128 L .36218 .57727 L .36255 .57327 L .36353 .56529 L .36483 .55735 L .36643 .54948 L .36835 .54167 L .37057 .53394 L .37309 .5263 L .37591 .51878 L .37903 .51136 L .38243 .50408 L .38612 .49693 L .39009 .48994 L .39433 .48311 L Mistroke .39883 .47645 L .4036 .46997 L .40861 .46369 L .41387 .45761 L Mfstroke P P p 0 0 1 r p .004 w .41387 .45761 m .41938 .45175 L .42513 .44613 L .43112 .44076 L .43734 .43566 L .44376 .43084 L .4504 .42629 L .45722 .42203 L .46422 .41807 L .47138 .41442 L .47869 .41108 L .48614 .40806 L .49372 .40536 L .5014 .40299 L .50918 .40095 L .51704 .39926 L .52496 .3979 L .52894 .39735 L .53294 .39688 L .53694 .3965 L .53894 .39635 L .54095 .39621 L .54296 .3961 L .54497 .39601 L .54597 .39597 L .54697 .39594 L .54798 .39591 L .54898 .39589 L .54949 .39588 L .54999 .39587 L .55049 .39586 L .55099 .39586 L .55125 .39586 L .5515 .39586 L .55175 .39586 L .552 .39585 L .55225 .39585 L .5525 .39585 L .55275 .39585 L .553 .39585 L .55326 .39586 L .55351 .39586 L .55401 .39586 L .55451 .39587 L .55501 .39587 L .55602 .39589 L .55702 .39591 L .55903 .39597 L .56104 .39605 L .56506 .39628 L Mistroke .56906 .39659 L .57306 .39699 L .58103 .39805 L .58895 .39945 L Mfstroke P P p 0 0 1 r p .004 w .58895 .39945 m .5968 .40121 L .60455 .40333 L .6122 .40582 L .61972 .40866 L .6271 .41184 L .63433 .41537 L .64138 .41924 L .64824 .42343 L .6549 .42794 L .66133 .43275 L .66754 .43787 L .6735 .44326 L .6792 .44893 L .68463 .45486 L .68977 .46104 L .69462 .46746 L .69917 .47409 L .70339 .48093 L .7073 .48796 L .71087 .49516 L .71409 .50253 L .71697 .51003 L .7195 .51767 L .72166 .52541 L s P P p 0 0 1 r p .004 w .72166 .52541 m .72344 .53325 L .72418 .5372 L .72483 .54117 L .72537 .54516 L .72581 .54915 L .72599 .55115 L .72615 .55316 L .72628 .55516 L .72639 .55717 L .72643 .55818 L .72647 .55918 L .7265 .56018 L .72652 .56119 L .72654 .56219 L .72655 .5627 L .72655 .56295 L .72655 .5632 L .72655 .56345 L .72655 .5637 L .72656 .56395 L .72656 .5642 L .72656 .56446 L .72656 .56471 L .72656 .56496 L .72656 .56521 L .72655 .56546 L .72655 .56571 L .72655 .56621 L .72654 .56672 L .72653 .56722 L .72651 .56822 L .72649 .56923 L .72642 .57124 L .72632 .57325 L .72605 .57726 L .72568 .58126 L .72463 .58923 L .72318 .59714 L .72133 .60497 L .71909 .61269 L .71646 .62028 L .71345 .62774 L .71006 .63503 L .70631 .64215 L .70221 .64906 L .69776 .65576 L .69298 .66222 L .68788 .66843 L .68247 .67438 L Mistroke .67676 .68005 L .67078 .68542 L .66453 .69048 L .65803 .69521 L Mfstroke P P p 0 0 1 r p .004 w .65803 .69521 m .6513 .6996 L .64433 .70362 L .63716 .70725 L .6298 .71049 L .62228 .71332 L .61461 .71574 L .60682 .71774 L .59894 .71932 L .59497 .71994 L .59098 .72046 L .58698 .72087 L .58498 .72103 L .58297 .72117 L .58096 .72128 L .57996 .72132 L .57895 .72136 L .57795 .72139 L .57695 .72141 L .57644 .72142 L .57594 .72143 L .57544 .72144 L .57519 .72144 L .57493 .72144 L .57468 .72144 L .57443 .72144 L .57418 .72144 L .57393 .72144 L .57368 .72144 L .57343 .72144 L .57318 .72144 L .57292 .72144 L .57242 .72143 L .57192 .72143 L .57142 .72142 L .57091 .72141 L .56991 .72139 L .56891 .72136 L .5669 .72127 L .56489 .72116 L .56288 .72103 L .55888 .72067 L .55489 .72021 L .55091 .71964 L .543 .71817 L .53519 .71627 L .52749 .71395 L .51993 .71122 L .51253 .70808 L .50531 .70455 L Mistroke .49829 .70062 L .4915 .69632 L .48495 .69166 L .47866 .68665 L Mfstroke P P p 0 0 1 r p .004 w .47866 .68665 m .47266 .68129 L .46699 .67559 L .46166 .66958 L .45668 .66326 L .45208 .65667 L .44788 .64982 L .44407 .64273 L .44069 .63544 L .43773 .62796 L .43521 .62033 L .43314 .61256 L .43153 .60468 L .4309 .60071 L .43038 .59673 L .42997 .59273 L .42982 .59072 L .42969 .58872 L .42959 .58671 L .42955 .5857 L .42952 .5847 L .4295 .5842 L .42949 .58369 L .42948 .58319 L .42948 .58269 L .42947 .58244 L .42947 .58219 L .42947 .58193 L .42947 .58168 L .42947 .58143 L .42946 .58118 L .42946 .58093 L .42946 .58068 L .42946 .58043 L .42947 .58018 L .42947 .57992 L .42947 .57967 L .42947 .57917 L .42948 .57867 L .42949 .57817 L .4295 .57766 L .42953 .57666 L .4296 .57465 L .42971 .57264 L .43001 .56863 L .43042 .56463 L .43095 .56065 L .43159 .55668 L .43323 .54881 L .43532 .54104 L Mistroke .43785 .53342 L .44083 .52595 L .44423 .51866 L .44806 .51159 L Mfstroke P P p 0 0 1 r p .004 w .44806 .51159 m .45229 .50476 L .45695 .4982 L .462 .49195 L .46742 .48601 L .47321 .48043 L .47933 .47521 L .48575 .47039 L .49247 .46596 L .49944 .46197 L .50665 .45841 L .51407 .4553 L .52166 .45266 L .5294 .45049 L .53332 .44959 L .53726 .44881 L .54123 .44815 L .54521 .44761 L .54721 .44739 L .54921 .4472 L .55122 .44704 L .55322 .44691 L .55423 .44686 L .55523 .44681 L .55624 .44678 L .55724 .44675 L .55774 .44674 L .55825 .44673 L .55875 .44672 L .559 .44671 L .55925 .44671 L .5595 .44671 L .55975 .44671 L .56 .44671 L .56026 .44671 L .56051 .44671 L .56076 .44671 L .56101 .44671 L .56126 .44671 L .56151 .44671 L .56176 .44671 L .56227 .44672 L .56277 .44672 L .56327 .44673 L .56428 .44676 L .56528 .44679 L .56729 .44688 L .5693 .447 L .5733 .44733 L .5773 .44779 L Mistroke .58523 .44907 L .59308 .45084 L .60079 .45309 L .60836 .45582 L .61574 .45901 L Mfstroke P P p 0 0 1 r p .004 w .61574 .45901 m .6229 .46266 L .62981 .46677 L .63643 .47133 L .64274 .47631 L .64871 .4817 L .65431 .48746 L .65953 .49358 L .66433 .50003 L .6687 .50678 L .67261 .5138 L .67606 .52106 L .67902 .52854 L .68149 .53619 L .68345 .54399 L .68424 .54793 L .68489 .55189 L .68542 .55588 L .68563 .55788 L .68582 .55988 L .68596 .56189 L .68608 .56389 L .68612 .5649 L .68616 .5659 L .68619 .56691 L .6862 .56741 L .68621 .56791 L .68622 .56841 L .68622 .56866 L .68622 .56892 L .68623 .56917 L .68623 .56942 L .68623 .56967 L .68623 .56992 L .68623 .57017 L .68623 .57042 L .68623 .57067 L .68622 .57093 L .68622 .57143 L .68621 .57193 L .6862 .57243 L .68619 .57294 L .68616 .57394 L .68613 .57494 L .68608 .57595 L .68597 .57796 L .68582 .57996 L .68542 .58396 L .6849 .58795 L .68424 .59191 L Mistroke .68254 .59977 L .68033 .6075 L .67761 .61507 L .67441 .62244 L Mfstroke P P p 0 0 1 r p .004 w .67441 .62244 m .67071 .62958 L .66653 .63645 L .66189 .64301 L .6568 .64923 L .65129 .65509 L .64539 .66054 L .63912 .66558 L .63252 .67017 L .62562 .67429 L .61845 .67793 L .61104 .68105 L .60343 .68366 L .59567 .68573 L .59173 .68656 L .58777 .68726 L .58379 .68782 L .5818 .68805 L .57979 .68824 L .57779 .6884 L .57578 .68852 L .57478 .68857 L .57378 .68861 L .57277 .68864 L .57227 .68865 L .57177 .68866 L .57126 .68867 L .57101 .68867 L .57076 .68868 L .57051 .68868 L .57026 .68868 L .57001 .68868 L .56976 .68868 L .5695 .68868 L .56925 .68868 L .569 .68868 L .56875 .68868 L .5685 .68868 L .56825 .68867 L .56775 .68866 L .56724 .68866 L .56674 .68864 L .56574 .68861 L .56473 .68858 L .56373 .68853 L .56172 .68841 L .55972 .68825 L .55572 .68784 L .55174 .68728 L .54778 .68659 L Mistroke .53994 .6848 L .53224 .68247 L .52473 .67961 L .51743 .67624 L .51039 .67236 L Mfstroke P P p 0 0 1 r p .004 w .51039 .67236 m .50363 .668 L .49722 .66316 L .49117 .65786 L .48553 .65213 L .48033 .64601 L .47558 .63952 L .47132 .6327 L .46757 .62559 L .46435 .61822 L .46167 .61064 L .45956 .60289 L .45872 .59895 L .45802 .595 L .45747 .59101 L .45725 .58902 L .45706 .58701 L .45691 .58501 L .4568 .583 L .45676 .582 L .45673 .58099 L .45671 .58049 L .4567 .57999 L .45669 .57949 L .45669 .57923 L .45669 .57898 L .45668 .57873 L .45668 .57848 L .45668 .57823 L .45668 .57798 L .45668 .57773 L .45668 .57748 L .45668 .57722 L .45668 .57697 L .45669 .57672 L .45669 .57647 L .4567 .57597 L .45671 .57547 L .45672 .57496 L .45675 .57396 L .45679 .57295 L .4569 .57095 L .45705 .56894 L .45723 .56694 L .4577 .56295 L .45832 .55898 L .45908 .55503 L .46104 .54723 L .46356 .5396 L .46663 .53217 L Mistroke .47024 .52498 L .47436 .51808 L .47898 .5115 L .48406 .50527 L Mfstroke P P p 1 0 0 r .01 w .19292 .00434 Mdot P p 1 0 0 r .01 w .38454 .02599 Mdot P p 1 0 0 r .01 w .56893 .08152 Mdot P p 1 0 0 r .01 w .73151 .18401 Mdot P p 1 0 0 r .01 w .84724 .33679 Mdot P p 1 0 0 r .01 w .88487 .52407 Mdot P p 1 0 0 r .01 w .82112 .70331 Mdot P p 1 0 0 r .01 w .66476 .81008 Mdot P p 1 0 0 r .01 w .47782 .78737 Mdot P p 1 0 0 r .01 w .36607 .63724 Mdot P p 1 0 0 r .01 w .41387 .45761 Mdot P p 1 0 0 r .01 w .58895 .39945 Mdot P p 1 0 0 r .01 w .72166 .52541 Mdot P p 1 0 0 r .01 w .65803 .69521 Mdot P p 1 0 0 r .01 w .47866 .68665 Mdot P p 1 0 0 r .01 w .44806 .51159 Mdot P p 1 0 0 r .01 w .61574 .45901 Mdot P p 1 0 0 r .01 w .67441 .62244 Mdot P p 1 0 0 r .01 w .51039 .67236 Mdot P p 1 0 0 r .01 w .48406 .50527 Mdot P p p 0 1 0 r p .004 w 0 0 m .00309 0 L .00619 0 L .00928 0 L .01237 0 L .01546 0 L .01856 0 L .02165 0 L .02474 1e-05 L .02783 1e-05 L .03093 1e-05 L .03711 2e-05 L .0433 3e-05 L .04948 5e-05 L .05567 7e-05 L .06185 .0001 L .07422 .00017 L .08659 .00026 L .09896 .00039 L .11133 .00056 L .1237 .00076 L .14844 .00132 L .17317 .00209 L .19789 .00312 L .22259 .00445 L .24728 .0061 L .29656 .01053 L .34565 .01672 L .39444 .02492 L .44279 .03543 L .49051 .0485 L .5831 .08324 L .62738 .10532 L .66982 .13073 L .71002 .15956 L .74749 .19184 L .78175 .22752 L .81225 .26646 L .83844 .30842 L .85975 .35304 L .86841 .37622 L .87565 .39987 L .8814 .42393 L .8837 .43609 L .88561 .44831 L .88712 .46059 L .88773 .46674 L .88823 .47291 L .88863 .47908 L .88879 .48217 L Mistroke .88893 .48526 L .88904 .48835 L .88912 .49144 L .88918 .49453 L .88921 .49762 L .88921 .50072 L .88919 .50381 L .88914 .5069 L .88907 .50999 L .88897 .51308 L .88884 .51617 L .8885 .52235 L .88806 .52852 L .88751 .53468 L .88608 .54697 L .88422 .5592 L .88192 .57135 L .876 .59537 L .86832 .61888 L .85889 .64175 L .83489 .68497 L .80431 .7238 L .76776 .75707 L .74749 .77124 L .72606 .7836 L .70362 .794 L .68034 .80234 L .66843 .80571 L .65639 .80852 L .64422 .81076 L .6381 .81167 L .63197 .81244 L .62581 .81305 L .62273 .81331 L .61964 .81353 L .61656 .81371 L .61347 .81385 L .61038 .81396 L .60729 .81403 L .60419 .81406 L .6011 .81406 L .59801 .81401 L .59492 .81393 L .59183 .81382 L .58874 .81366 L .58257 .81324 L .57641 .81267 L .57026 .81195 L .55804 .81005 L .54593 .80755 L Mistroke .53395 .80445 L .51055 .79646 L .48808 .78612 L .4668 .77352 L .44696 .75876 L .41256 .72332 L .39845 .70301 L .38668 .68126 L .38173 .66993 L .37742 .65833 L .37379 .64651 L .37083 .6345 L .36962 .62844 L .36857 .62234 L .36771 .61621 L .36703 .61007 L .36675 .60699 L .36652 .6039 L .36634 .60081 L .3662 .59773 L .36611 .59463 L .36606 .59154 L .36606 .58845 L .3661 .58536 L .36619 .58227 L .36632 .57918 L .3665 .57609 L .36673 .573 L 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length of the Gauss map. 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