(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 41609, 876] NotebookOptionsPosition[ 40372, 829] NotebookOutlinePosition[ 40892, 851] CellTagsIndexPosition[ 40807, 846] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[{ StyleBox[ RowBox[{"Math", " ", "105"}], "Title"], "\[IndentingNewLine]", StyleBox[ RowBox[{"Parametric", " ", "Plots"}], "Title"]}], "Input", CellChangeTimes->{{3.47643248915625*^9, 3.476432500609375*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ "These", " ", "simulations", " ", "show", " ", "how", " ", "circles", " ", "on", " ", "circles", " ", RowBox[{"(", RowBox[{ RowBox[{ "big", " ", "circles", " ", "have", " ", "little", " ", "circles"}], ",", " ", RowBox[{"on", " ", "their", " ", "paths", " ", "to", " ", "spiral"}], ",", " ", RowBox[{ "and", " ", "little", " ", "circles", " ", "have", " ", "littler", " ", "circles"}], ",", " ", RowBox[{"and", " ", "so", " ", "on", " ", "ad", " ", "infinitum"}]}], ")"}], " ", "can", " ", "lead", " ", "to", " ", "an", " ", "incredible", " ", "wealth", " ", RowBox[{"ofshapes", ".", " ", "Essentially"}], " ", "we", " ", "have", " ", "a", " ", "Taylor", " ", "series", " ", "of", " ", "paths", " ", "through", " ", RowBox[{"circles", "!"}]}], "Subtitle"]], "Input", CellChangeTimes->{{3.476433782890625*^9, 3.4764338275625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"?", "ParametricPlot"}]], "Input", CellChangeTimes->{{3.476432585328125*^9, 3.47643258890625*^9}}], Cell[BoxData[ RowBox[{ StyleBox["\<\"\\!\\(\\*RowBox[{\\\"ParametricPlot\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \ \\\"TI\\\"], StyleBox[\\\"x\\\", \\\"TI\\\"]], \\\",\\\", \ SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"y\\\", \ \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\", \ RowBox[{StyleBox[\\\"u\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) generates \ a parametric plot of a curve with \\!\\(\\*StyleBox[\\\"x\\\", \\\"TI\\\"]\\) \ and \\!\\(\\*StyleBox[\\\"y\\\", \\\"TI\\\"]\\) coordinates \ \\!\\(\\*SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"x\\\", \\\ \"TI\\\"]]\\) and \\!\\(\\*SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"y\\\", \\\"TI\\\"]]\\) as a function of \ \\!\\(\\*StyleBox[\\\"u\\\", \\\"TI\\\"]\\). \ \\n\\!\\(\\*RowBox[{\\\"ParametricPlot\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"x\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"y\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", \ RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"g\\\", \\\"TI\\\"], \ StyleBox[\\\"x\\\", \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"g\\\", \ \\\"TI\\\"], StyleBox[\\\"y\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", \ StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\"}], \\\",\\\", RowBox[{\ \\\"{\\\", RowBox[{StyleBox[\\\"u\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) plots \ several parametric curves. \\n\\!\\(\\*RowBox[{\\\"ParametricPlot\\\", \ \\\"[\\\", RowBox[{RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"x\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"y\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", \ RowBox[{\\\"{\\\", RowBox[{StyleBox[\\\"u\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\ \", RowBox[{StyleBox[\\\"v\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"v\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"v\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) plots a \ parametric region. \\n\\!\\(\\*RowBox[{\\\"ParametricPlot\\\", \\\"[\\\", \ RowBox[{RowBox[{\\\"{\\\", RowBox[{RowBox[{\\\"{\\\", \ RowBox[{SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], StyleBox[\\\"x\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"f\\\", \\\"TI\\\"], \ StyleBox[\\\"y\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", \ RowBox[{\\\"{\\\", RowBox[{SubscriptBox[StyleBox[\\\"g\\\", \\\"TI\\\"], \ StyleBox[\\\"x\\\", \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"g\\\", \ \\\"TI\\\"], StyleBox[\\\"y\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", \ StyleBox[\\\"\[Ellipsis]\\\", \\\"TR\\\"]}], \\\"}\\\"}], \\\",\\\", RowBox[{\ \\\"{\\\", RowBox[{StyleBox[\\\"u\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"u\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}], \\\",\\\", RowBox[{\\\"{\\\ \", RowBox[{StyleBox[\\\"v\\\", \\\"TI\\\"], \\\",\\\", \ SubscriptBox[StyleBox[\\\"v\\\", \\\"TI\\\"], StyleBox[\\\"min\\\", \ \\\"TI\\\"]], \\\",\\\", SubscriptBox[StyleBox[\\\"v\\\", \\\"TI\\\"], \ StyleBox[\\\"max\\\", \\\"TI\\\"]]}], \\\"}\\\"}]}], \\\"]\\\"}]\\) plots \ several parametric regions. \"\>", "MSG"], "\[NonBreakingSpace]", ButtonBox[ StyleBox["\[RightSkeleton]", "SR"], Active->True, BaseStyle->"Link", ButtonData->"paclet:ref/ParametricPlot"]}]], "Print", "PrintUsage", CellChangeTimes->{3.476432589921875*^9}, CellTags->"Info3476414589-7267634"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", " ", RowBox[{"Sin", "[", "t", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", " ", "Pi"}]}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.476432596390625*^9, 3.476432616828125*^9}}], Cell[BoxData[ GraphicsBox[{{}, {}, {Hue[0.67, 0.6, 0.6], LineBox[CompressedData[" 1:eJw1m3c41f///42KyChKpaKFFBWl3aOiYWRENLyzUkJmkbJSUih7ZCayZWbz sHf23hzjHM45rxSlUL/X5/pdX/8ct+vFuc7r+Xw8Hs/77bpeZ6eh5TVjNhYW lkfsLCz/e/3/P9/g/34zPHBAePkfAS/XNCgE+OuBaFym+TeSn49ceujjbwms EDA/SLLXZ7+qqTR7WCnoyion+dX6M3yu/i5wMs8lIYrkDw9su14XvQDO0vft tiQ7raF1NaV5wMRJ6SPnSd4vtEuA+4MXWLsoNq0h2bXwQsgFfx9wqPgdVvWX AKGbx+VOVPjBhIjah6ckK7c8aFYpCoCi0NuDEiRrRCq9v50TBDsEZNVaVgio bfwmbJwWAhduzS4+JPluqu2ASfx7MN0R0beK5KFfs7r3osNhWUftnPAyAekm 4oUqftHAnuBZE7ZEQBvHBvWt1R9ATuzQyw0kn8hQkPYsj4Gas+fiZ34TULja K1e3MBbE+r5zKZNcYVw5WZobB6mfohNiFwlwjnLg2pb9CXaner8584sAnk25 Z2tSEsCMZ0+L808CtHpNOzYkJsJDwuhq/gIB20P/3L8ZlwRPdT7MC8wT0FUq FN4RkQJaHxt/PPxGQD2+t3n8Nh2Usm+usiYIuNWfm1CtkAFE8WU1cyYBaYa+ H5oqMsDGRD5ImU7AmvMTcs2lmWBfJ0pLnSbgtalHeERuNvzhXNtpPUXAY5cR t5dHcqC25vmS9CQB5QP3XO5n5cDJSX1W/3ECBNufnNnx+QsURU/M/Rkk4Ffb 3kfKn/LA1O1aw/MBAnC9eQrn7nx4q/FnjKWfgPdUBlvZh3woVrRqne4mQOWZ Qui2iAJYVk/JudFKgKjRHLunfxEoeJ78WVNOQICz8C5hZ4TvHMG2CUjA0kcU WGIrgxUTmUHnEgI8J9jZJi6VQXBiovWGAnJ/rP07EprLIPHSE2xKJ2CK//T+ haFy+LDk4/wunAALA74vQUuVUP+g5N9QKAF/3/L3HYIquNMrb78nmIBSu1cy NW5V4HPbUSXYl4AH91yOj6ytBp296+N53Ml6UvtqWydUAyWxM23K5uT+PqPU 3JOpgybdDmHuBwSImKnLRTyug6u0igflxgSwPCmOb8ivg3Iex40CegQ003LP bIB6oF68WqukQcDbmGRJJeUGkDP0erTuCAHvwsRPqRg2gdCqHMfR70ygvQ0s W3uxFQ6U301pYjIhmNad/c2oFepvau7OnGECh3GRb4dbK/yYW/XDcJwJn2Ii qN5lrXBu35i0aRsTXEx1Tnw91QbyTqs4OT8zQZ4YjXc63A7xBl6ibwyY0P26 W8pkSyfofN0V6qrLBHbaH6eSY51gXmjrZKXDhKPFppU82p1wpV89+MRVJjRs nuOL9u+E7YL1e6yPM+E/m7ODL7i7QHS7cZ8dDxPU9iilqC13QYzspX1TmQww v/4vp6ivBxKGZZa4UxnAyLTQaPrZA19vWGVIxjNA3fzhwR6BXmj96NZwLYwB shIXAgeu9sLZ41MTV90YcLw592xUeS9Ydl359U2dAefkh9esTuyD//aI8S9Q 6XAzqv+QhNUA6JeslTg5TodYbZkoW+8BcOqN1HwyQAfpv3ut8xMH4Em+Jn3k Kx1Y/qs7ID02AF978zR0s+iQ6fzpTL3GIMwG9+txP6VDbaDlaN7hITi0ye2Z 6io6nKD5a1+hD8Ot2u12gsuzoMQlzb2GawRy/0h4t/+YBUOjz38LxUegcHFU 4BhlFjYGD7lyGY3AW7H3HTHls9B89DcH9I1AoOezfSxOs3Agrcx8yWAUxLVH VOYYM1AwSKXJKo7BZ4qiEU7MgFSgWpqzzhjctaZT3AdmoDUpxLvKeAwe9MpL L9fNwAbx7Hfn3cbIegs/9C52Bn6l5h1iKRoDV8rYGSWdGZDZGPDQe/847Iln czhaQAOC58Wuak4K8PWdPVKZToNq6vInt40U2PpT5INSPA24+685nNpFAa0p 0YBLATTQWhBgjz5FAfMfseI/zGmQFpqlLGVBAWokQ3inCA3Yr9osGbVT4HWk F/eYMxU6IkuN9oRMgF71umdJj6jwr8J89mLsBFyJCqOamVKhu+2cgWH6BNj0 MWqHr1PBRO8xt3ftBEzwwbDjASqskV7P8fHXBHwyKI2L6J2GO9VnOLq1JyH1 fmWp44FpiDFqaVFcPwUc64+xRotOw5Ug5u6C7VMgJeoeUig4DbdH3m/aJTkF pl/Mvg4tT4HsFYX60QtTsJ+VUdrZNAV/a7axrH80BbM3r1V6mJHXJcPDT3RN QZz9d07huEno5W+UDPCfhkoDMT+LkEl4tjL08mDUNIhQ4oaLPCdBT9ePtzpp GvRHJHvlrSchdqNT7EDZNMjq6KWuPTsJnx3U7XKZ09AQbChk1T0Bjd3JQyKK VHgRaSlfwTYBKo2FXCmLVNgjPPYgcIECv8XrFo+w02BQxYbdgEqBO6Envufx 0ODntbgo2lcKuJi8X0rcRYMcf6vB/FAK9N9hbjqmQoP8vT+zr0mR607Tn2WJ okGZZ67Au2vjsPDXXPXY6Rlos9/d+0VhHK6ab987dHEGpJ/zve6VG4dt95+k PVObgU1X2wt4hceBl+5Y99FwBsr3q7WdpYxBc8StodDXZN3pbTmWaTMG/Rt/ HN3VMQMf0qT++/N2FCjDSikhhrPQFcZNdXEdhV81A0v5prOgX1DxiNV2FB7I HkrusJkFemfTh0WdUbCQv7p5wW0WbNQXo/J2joLHShc79eMsXL6xXuKnyQi8 ENB94D02C/E3w/fd+TwEIrfuX624SQeuY0xXO/8haHwjm8RtQIcd29wzPOyG QD5jn7iqCR0sXN9kB54dgrXd79xK7egw+iJ4za3mQXB+aPdKJYAOeeofO0Lo A2DTverCtkby/20DfvHt6Qfp3x/6nY8yIOyBkTVlVT+Y7BK3sD3NALND+16m T/ZB9rBWhp48AxSit4YeTuiDtKC64yIa5HVqos3wvj7Yb+y6+6Q5OYdOt5aP S/VCF8fT7KEPDPic35HSeagbUhtvdzWzM2H/+fyoJv5uMC0QDv+ylglpJS+U Sr91QU3O9q4APibAaZW5dxldEJd7OlpOmAljHRbPpg52wT3b8v82yjAhwX7N Kzspcg67Fm+Vu8MEvbPe9Ft72sm5M9n4O50JkV9YFutWt0N/X4Ha0xwmsL6w 95KZbgPli15yP/KZ0P+40mQxqQ30X+VLNJczwevq4za1g23wY2eCq0QHE1Lp texHTrRCoN8dfot5JizU9Ijc2doK1Te3yj9aZMLFfPdnnL9aoCthsdZmmQmK aW9OvXnWArJhElU3VhHQytn4RuplM+yf32BWJ0CAu6ViE29gE/waPxN99DAB h0ymLGeuN0GIpd3zMvIci7CfWodCTfBHcDhe/jgB3q6FVprhjUBpLsuTAQKi 4op9JD82gNObmb5sFTLn+H7PiP5cB7Yh7ffO3yPAzOu4xAarOuiuShdSJ8/V xAMqvs6H66A6P/uvNnnuKu1hWl3MqQVW7cPmV20I6EybFXtbUANNyg4Zg04E mKcRLBsqq+BQiMK5lwEEZNirx913r4Kdg6Gia8lznt+w3yj/chU8Stn89BWZ A3qM5WyUmyqh88iZYNNIAr69pt4/0VUBr14wDXvjyes2Rq8fhFSAqcj9ZdEk AvIvlZ4MulkBvJ7OXIYpBAhksPuOkDkjx9fY7SuZO1iOhbw/P1UGXm7P2hXz CTD42DGpllgGKTsusmkXkrlKY+bELdMyEBoM+HirmIBiffMHNYMIklu7Za6W EaCe2r7Hl14Cv6WpL3tqCdApvlrcyVUIGhpsb127yPULvL/f9V0BWK697SPQ Q8D0E8cXezcUQPUnvaWoXgKCHUSN72zJh+22C7zRZA5bEdA74C2eCzKs9lsV xsj1S9HayZXyBRw6L+qHkTlOh/mU7ib9BRrusvLQKATM1K9zND6aA1u9WcMs yRx4b8IC5y5kgQ7viMPOWQL4tnB9UqzJhEq5MuZJMkce0bmlGqmYCZs7L9Sp MAjYZvHN8Lh6Brgd25d4k8yh56cuC7AkpAN9hu+zFplT50NaujzSPsMaZqTQ lTkCOhq5E58XpoJvRr3g5h8E6HMkS8+XpYDbyGLCPMnj505v1q9NBvkz3yPr ydw71beTbW9nIojdfJqpS+bkK+/MHJz7EyCCK6l5K5mj9zQHnW8fjYe4trGr bSRbUTyyHjDiYJVztbU4mcPvsR/ntB2OBaFtR1U7Sc4o52i0a/kIfw+qyzz5 Q+bi6qVLj8ti4B9StwuQuX5RemudReYHCNLu4P9E8jhP15FahWhwZ5vmkia9 4LrirECiWCRMaK9wfya56HzjmMZMKEgMxO0KID2Cn6WKZVtTMByqXTi2SPLG Xz6FI2mBsLqzTlOL9JLlJ8oJkT7+QJNcZZdA8rUngnra1r6Q6uAbNUeySLHg vGC8NyR8v/FVhvSegeJw0ccPXkOTijKbGck24ayPmg+8hE+VOmfCSPbtEzqj MOoCt4SsnctIvpfPHjN+zx5yvb2rh0h+ciOdOt//ECQYMRu+k2we9P7O8dab UPMqxXiF5BbW8ViW/edQ/UB86T+S/9T9uhO+chcPHvXb/ptkUfWqG8cXbdDO ycKNRvLFnRXpkdbPcNT4AqOV5LCCuSSnhefY0SFwJ53kdRdVbMRlXuHNw/TO lyT3nU1mabTwxJ0PezQ0SF77vWbP/ZR36PCH0ilIsubIQZ0VWz9Ubtt1p4W8 /0a5MLkL/gG41SKB4Ury08oXoc8zgvBPm8eLfSQf02qZKGwOQS+3QZF6cn09 zQ1eMOjvce5NWYUeyeGWooObuSOwxU7DnEHuz2DDvPbpfVF4Sjd9mw3JN30n 0lJ2f8CCtuUOBrm/GScuJuXIxKD0jzv++iQ/FQ9syz//EXv0V3QayPqQfdrQ n6ceiwBse/eTHLSxT++TxSdMfjzf1UZ6XfnuFPkAp3jsSrlQKETyJ5UbvU+9 SYPhLYjXIuvRVldp4mhyEm6pXgrNJr1OfELsmOZkGn5OtaWt/k7AKVO9a2vm PyPDxHvjKrI/ruvd0rvBmoEnXteTGknOj2pXqQW+TFRpHe6pI/stqsBIcv5A NloYytsvUwl4MVDcJZaTjTxPT61kkV6nd0jgj+apHKRzLvjok/3sHca1KVLx C1Y0Sw59IPudwlrzeMo4Dx1vxuzMHSIgVmf8xCglD90ERo6eI73ulrW8RKdB PvJv5narID1OPLCOM/m/AmRpiatMJ+ePx57xZ1zXi3DP8uufB9oI4DLacbS9 vQhbKrs6n7QQoLozY1eAejGu7TrILP5K9mfrHxEWlRJ8YowdUg3/8/qIyy8u IFpYRttlVRCQs5Z6eX0ZoolZxIkych5KHXerbNxRhsI74VRNKQExFh67L0WU 4bMfxsuF5DwNyTxduDakHAe/Hnl5OYucv8EzQ6Velci6mybCiCbA0UA94VxD Japf1OR2Iec7R6J2eAlnFc6Y/zuzlvTCc34nahLdq/DA4qtna8jz4exfVccL LtUoVxZhEudFgE/l64121rVom38n3OIxWW+9JjG2Go043FKieF6egKQXVRsf BjaiiInSkUDyfLux82f0nZ5G3BiwT3z4FNkPmquT9v3XhOWVeyR1yPPxMZ/j o/9MviKLk1vl0B6y/mNGpfY7teAd/mk4S563XG+t2SOxBb8SHEfZWMj6t1Me ZmNtxVURBQolS0xovJJd8598K2o55edv/sGEd+brlQZqW/HtXoFNxmNM6DgX 30hvacNi13U1WsVMmNnqd6f3Wxsmsrcve+YxIV/wUWPp+nZs5Z03zM9iwr4m PR5HzXYMsu+rWExkwsCvln153e3Iv+fNIl8QE858F/cUH+zAIodS5/dmTFhm O2OcPt6F48Hmgbs3MOHjYTntq2zduCS4IH1nHROuz48cmtzZjTS5RQffNUyg Kmm6shl0o4b2oPjgHwYE7X0W8H24G7W2RPHzUxgwK/2O+aGvB9Ntwt/Mkt6o u63gx+/6Psxl5kg5KTLgx+oJ1SpqHz6vnhfJvMCA2JFQFw+OfuQJtvszeIoB 42vDYFGhHzVut1puk2aAa+MZFTfsx28Cyn4nNzBgq3nrocPZA6hyaafhQi8d hELab2HgELaXZ1601afDk1jGZ7+MIeRQ9q6TJvNoaFL6sG7TEHpJJjpQNOhw kd08d4h9GEU0kh8el6eD5ZO/Xq9thlE9Wb/Zdy8d3FV5/jmqjuDLJNlVK9RZ uH3X3PKm6ygulbjubiXzMrv1/Ez921GMkNCZzyHz9Ink9WZy4aPIuFkgHXBr FlS+7TRnzx3Fg5UL06eUZmF+2kzo4ewoTgg/iT27bxZuKJooOF4fw0H1LStc U2Te7z9UaCs2ju+eB7p+vTUDz2ruSNkdGUfJFQbPLo0Z8HpU5GB7geSDzRKW l2fgb8uBd3fujKPSeZENP2RnIDXN9tjPoHH8G2zA8nHdDAzS3m2TWEXBldSD JVHFpDeuW8X5bJCC1uMs2eJCNBgyt9nQRqPgJmaXacA6Guy7qvJ05y8KrvJa u26RlQaa22U0M9dPoBqv76ZUOhX64lLLvC9NYMSl2A9YToUq6dFLB9MnkP1Y pmwI6YXKuh4n1zpOYtbL/QI78qbh4Gmlv00ekxhAqClZpU7Dk1WnbL0CJrFr Qj+lOGYamiY3GC8mT2LUo54Fee9psE8YEvLvm8TtUeo/1xlMg851heMVR6fw 6UH6kufaaZhV7YtknZnC6ou+QkraU8B7S3j1poUp5Nh6rWpKaQrGQ50q97BM o3KJxHcnmAKDT8UnD2+axjx+7cFwiSlg27CvdvWFafw9UHsi+M8kMMyeLYeF TmNz6d5gj8hJOPW0S9HoPBWFr8qxW49MwPdb6ZFhKlRseOcmb9A+ARJSBLVB h4qK9ymOytUT8Evj841NFlQMEuCe4UmZgKHi5Qs3wqh4gP/hZmW7CRCX2PfE fY6Km3/+2pHMPQGihgqOlmE0jNq3bnGjDAVSi15GYRwNv3Mnt6vsocA3t7da nOk0XDXXt8t5EwXeX9xg9aKShm+YDNnWP6QHFm7aIEqnIUVW9o1M5Tg8Lg6x +XZqBksO1nj8IL3x0cu4iL6uGbz9StTohcUYKBk0cXGPzOBf+aCDWvpj8OVC xO8j1Bn095ncJXptDM4tL6na/JnBiIPzb5OOjkHF9kQ//x2zKGctOOm2PAo6 i2wX7xvPYoTCxk2ab0Zh05u8i+qMWQyiX37KDSMQ8CVxxXthFm/e0rjxaucI 2HZ5U8tXZvGcT9mrRfYREDXgUBXkpWOdf/NCRd0wHCMOb74tTUfVwN2r5jSG ofsMM9viIR1jn17bW2A4BPZXFNuiJuk4yR7Jf8R5AA6/crM0oNNxSezkvy8G A3DDo5Rj+w86npfPuHz44gA8MMoVcWFhoPblQVbBdQOwsE2pjLmVgbPKLkOe 7/thU271naeqDBQ94qKokN0Hm9uizx/PYOCvkU+sQuM9EGXmu0kol4EyetY5 bVU9cLnodQ6ziIEfC1NjXiX0wEn3yYuvaxn47yb18oh5DzhrrQ65O8zA4nM/ PVV+kV7YpVZaxsXEvzEsB7Zzd0OrtGHfyn9MLHRc2F0v3QlXuzfhf0ZMpHgf eKjJ2wkf3I4u5pgwcc3ztuUeRgcMaoTdUbRlYpOok2BragfceMZsW/uaiTjz 3z4nyQ6I/XSVi5rOxD2OnrwdpDdKse58XJzDxPHZwq1rVrVD3I09214XMNF+ 2ZLtCKUNflXLb1ldyUTb9vE3rjFtoGd2MDmoi4l6W0Ndhra3wfA7BVb+30yc /aJ0Ln1jK9Qz1X11Vph40FWuV4LWArzpHZeCWQl0b1kRmIlvgWM89/b+Xkug Jj3uhcauFpBcsPTQ2krgqwXz67mbm6Fncn5Q/wSBXP3Nt6nsTTBevnS+zopA k8E1x2w7a2EL4/f2ElsCn73ZoZcTVAtXlGzEku0IzJ3sSSS0a8F09UyBuSOB DzOTR7T6amDcm6cnwIPAxg4n34GhakAJFY0HEQRedWjKmJ6shKP5HTGjUQSG rlO50h5fCVbPtwiqxxD4xX1xKe9+JfhKP74tFE+gpWjiz0e0CjD+9O3u5XQC f0lozeXSy0Ev+eXkgTICXftuOLdNIBjFP+cRqyBQIfnRt833EJT57FSFqgh8 Gmvpf2O6FPwO39gyXksgT5TURBmtBD6LL1RvbiFwDd9/MieYRRB2LCCxvZVA x+RfwRcti6B17vmHl+3k9aCf4srfCoHrYWxzaxeBibbuVy98L4CiDbGbuQYJ vNHwT3hiIQ/egnHYyyECDxyrv1Filwdl99Skfw0T6NFKLfX9lQs7qg94VY0R WKhV5i72+wvYSdvwsUyT+7Pj2KP1y9kw19i0+RSVwJ+cvDbomA3ji3GHLWgE Zvx6E2yykgUlGlG+xbMEqjsdak/9mwlBQ11LZM5EpucXLxbWDDDvfxbx7RuB N23G2DdzpkP+kaD3I3MEfmp7+lKc9zP4yEFi/A8CZ38cOnx4ayrMrjUqcJ4n sHzXSOd+0RR44i8qprZA4CW+jiURsWQQORq7vu8ngW0PyyoWDifChq1cvv6/ CFz3Rae/51gCsCmpPlBYJJDzitXdnDPxMHy6Zr/fbwKD/ljif4pxsMUsWfDA HwJZff5x7FeLhaXCJh0kefVpg6wfWh/hRs45evMSgeaai8JP9D9AhanGGeVl Asf5Sz+ZNEVBpe4OKpI8FTn/pa8nAkJivegHVsj7Oa40Kk8Jg/YDLkr+JGfZ 2EUkMEPBrHCU9RvJqvdlDFf/CYaS8fcbLv0lsCHqeNJ/q4PAnz/YMZDk8TkF WgZ/AOx6UQEDJIeZC4b9E/aDbaq8ulv/EXjyXsO9QNV3EH/TtI30HpSto9/c ZewJxob10c9JPvuTbyXp2Stw/i1Sm0Qy0/QLHPB3g9CXdy83kCzUibu0NJxg 9IyPKIVkn3dilVecHwPHkbBr8yR3Zau7WGmbgbLS8yHS45CXF4b89HXA6viF ctLjMCjL6SF3OmC15+AK6XG4Nd69hyvCCPd81Aig/+/9g43rhGKskZkd97yH 5OrdXxh7HzkgX0N/XSHJA6Waf4NjXXGp4o9JMMnbebZyC719ickaPIamJFvv T5byt3uNYSvbcuRIdv/53I5D3xsbd8npLZHrEanDv1S4xxfz5+4Z5ZP8I+zF HYst/njbEssektyya/Dhdt5A3P1Yw16Y5NPXdTNr2YLxR8Yhj3Jy/ekllzPN f4Xgtk8230ivw25JygI36XlVk04t7uR+fj+nYHGsKxKHdJ9d5CFZuiMsq6I+ Gh+NyW98S9YDFBpI7x75gN89sioek/WS9GvkeVjHR5yzC/8wTNZXRtT71y1f Y9GHp6sXSI4fHmj/VxuH4jMdN2bJ+oxO/3RBozgeC6JmPhwlOYrr1Tfr3ARc 3Tsv/4Ss56nPH757ZyRiWntY+BRZ/zyEwqrMuGQ02xoYfJ/sl43nBtuTvUmv 26tco0f2EzQ/FWsKT8fz/qmLqmS/3ZglJ9/1DHx4LO7eBgaBC4NPax3vZmIt q7mVFNm/GsH67dfdsnEu+UHp0BRB5n+RwtK1pNc1lK53nySwmZ37706/HNzB /rM+f5wg33eZtSf6C5b1dXjlkPPEuCSv5WdxHnLx5PFtHyDnSe9ZC2mFfJT2 Ohfh2Edgq/eV43qN+cjesL1NtJvAd76ZuSl9BXhOwSZvpZnAGeaMLH2hCK2u qM+sRQJ3/qj89WhTGXoxeI5Sigl06b95su1mGRLVP3yzC8n1udIwLhFZhhOf MpxP5JLzt9t0S8Pucmz5Vivfn0rgKbNL3O2HKnCbgBW3VSjZfyyjNtOKVfhJ e3V+YRCB3IoxsnPeVWioVue67E/g7YmY/QstVbja/tsD47fkfiDxePZ6NVrI r85sek5g7zG2uz4GNbhtXq1pzoSs/9wuqTj7OtzB4pQ+ZUygfJeEQNeXOqQK Ph3tMCRQTPg77d+POnRNpEmH6JL9PDsvLW9VjyOyUgHlagQOb7M2dzZpwAIB 2v74I+R8vsVrUqjThIO8EUqihwlstw+RfxHUhHEfeav8pQh8w/dy+8WOJvx+ OKNBX4xAo+Rln2zVr5i/pQRihAjUP81Sd+JiM7K2jwXlLDLxC6e5uOXmVmzm EfuaM89Esyv/VoSOtaK++O+Had+YeKl400Dx9VYsXXX+mweVidux1vOPfyv2 0mwExnuYeGK1KFOBpw2rH2sU7f/CRMb0g2N8rO14blfIvm2mTMx/2BVzcbID Lz08fbX8Lvn3he4+U6s7sb8kJuM/PSZGbbpV8FysE392UU47apGsQwmPv9+J zEDtEPWzTOzyWBWcROtE+4j8Yh1+Jn689+qQN70LBfmy/WfTGCi0sp3uxujB aj+n0WcJDNyulK12k6cXfaZfxbHHMNA9waZGQqoX9y42Bv8OYODT/ua6LPNe VH07fsPFgcxPY5w7bWZ70fK0Puv8BQYmwnux4Ok+3MN+o9OnhY4rLklSR/oH 8K/FYO3ROjrec/dY7bU4gBW8SaUdZXS8laW6tX/TIE4Jq5X+yqSjC2OhV19z EB8l5cFoAB3l7dORr2kQj75w3XlUm4577imMzxYNoa/uztcrnbO49N8eC6Og EbR6Y8eR2DSL9g5vKjKzRvBs7HiLUtUsLrvxKf9qGcF1b5UuOmTPYpcZK112 9SgK6g+xP/WbxYslOv8sYBS/PNhPWacyiw9fX8nnzh7FjdNiMm+KZtB7PO1Z ZfAYXs1MejuWNYPOodnTlNgxVPHcZC+bPIN1M07rWTLHcKpp/XJl6Azuepjv t79xDJfPRK4NfTyD2nz36ar/xvCFtw8/r/QMrv8goVx1bxwFJyK5zpN+wEiZ pWnKUPCTc5h6iy8N7Xbzsk6dpWBD6mtTHQ8a8moV2dgqU7D+t97u649o6NYR ctP1LgUlV92krVGlYbdAzahBMAUf2Sk0X2Sh4aLX5iPnf1Mw46eJx3l9Kiq3 yezRLZrAbcqCm2SuU7GtXVZBu24Ck+bFioWVqPhKM8FJqWsC1bOFb/fLUvGM 5mHpncQEPr08fno1BxW3Cp0T8dg1ifQ7cFswdRpNONQ3yLyeRHf/uQrpuSl0 sG7fMas6hUcuXHe6MjmFp3eyWOy8PYWdLkF5t/umkPJV/q7G/SlcsjteYFE+ hRqxx45HuUzhZzZF9ct+U7jlrP722fQpvN0d8rX64BQmx4uPqPJNY7D2JyN+ k0kMPrbqi1j9NDaYTP89rjuJ4pUt3vUd06iSe0VDV30SXUIdXO4NT2Okq7h4 wPFJlGmjJr/7MY0yYp20cs5JvHdxXVvIdtLjJLYaiSVOoG/W3EYvKypyzdTv E6VQcDdbRCplPQ0r99lZf+qmYFi9w1SVMA1V5mNb9zZQUDGXfyZ6Lw172ihj QpkUVMhIVpA/QcP4ua74cmcKftNIZgd9Gu6uc3wsvYWCvsM17zjTaMij7nPq hdI4Uk+t3Oo4P4Pnt8jZZ58Zx8n8apsipRk8KxWmO3KI9HcHBaVIzRns8Gg+ JCk0jn7VPw6qGs+gTY6PljdlDP1Hr4fffT2DzwQ4Tro/G0MHvnlRw68zaKnd elQvcRTfGoil+V2bxTPPnT9xhY9ii5xuedqtWWwTe7Qr6+0oCv+J1qo0nEWX tVT1BZtR7Ar7+K3fZhZZHFf3qZwZxQXYU1/kP4utqjaVrz6O4PbA7OsX2mYx Q21/ZT19COMn+JevK5J9PNjd0N08hEKBjemnNej4/vTlpwMZQ2i2V8RO+CYd F1y5vzc+GsJ7HJWHy03ouF4575DS8iDacPaJp3nQUc2gVrVy7SAO+JlKra6i 44XmC3GUHf3ofNp2dYIcA1dNGAn8/NeHbqP9373OMMg5L3uKdawPrRnZGSYK DDxt793DGtuHPno8wWuvkddZL8ulifVhLduedqo5A7PevN+fc6AXc47EPBQm 5xb1b/GNMtluvKBv+l6BnYk+uQ5twpu6UWWgfaCIk4lx1WpU68UuJEKdm/bz MnGySJaHt6QL79RrZS9sZuJ6++C/ay514ZLbNZvD0kw0VW7vY9XpxGsnxYMv 6zAxaFLuherjdrzcdqR3MpaJinWr+7h12tHcwvuRVSITtf7F51Ydb8eHVadC f6QycdWlSbF9y21IuXfGmkLO+SGLg8klbm345sBj5Qe1pJeJ6ecK+7Tiv9Hf 2etoTPzTn92Pil+Ra9dHtRTyHBJt55hfx/EVtSQCf/pJkp6QKlWiUdWEGgc9 MyylyXP1olFKxdkm1GcXuskvR+ALRzE+BdlGfPJA686UAoG2MuwFXVvrUWSs UXVCn8y5Hi318T11+NW3MP7UXQI/XFcosQqsw874faye98ncPif+nuCpQ1xX UcBtQaDFitK5cJZa/HN6SST2KYEpT79JFU1U4fWbbH9EyXNdWfO27e6YKoyG FR5qIIEl9aqt7v9VYQLb603xIQQedLi040R3Jbo47WfjiCTw3PuN+3RrKzCE XV1aN4FAFiFtxZAXFdhg/l9bfxKBL08f2/YVKvB3/x9DdTJn3LrgLr4vvxyz 9wmpiWYSWC92eT4usQyZX8uOaZI55SyXQ33e3TJMtRQ//YbMMdbNVfRq0TI8 Ylu1L7eUQHtutp7Ew4i2p8RK50kP3N1o3vVkfQkmzxYWrmoksK947I9/Sz7K C152tyRz1DmBkNHT6vloJL9eabGfwJUHa3JGW/Nwk/mjrQ5kDuPXUo7d1J6L Ly57DBqOkPtzWHEHdOZgZWHfQPcEgVL1UfQiLZKNbvwQIXOed6F8gmxXNjYw 4gUMSK/bnjmiL9idhZdGHz1sJD3OPUPPIKonA+uVvJ3OMQn0KxLj/amTgT3C RzoUyVz537HLKgEZ6XiFzfCgMpk7c+zeSjdkpmHksvqfo9/J3Jh3T/9WVip2 PA8xESE9bj9H9X1KVgpK/7MYZCFz7FHrsz9Hs5Ow+fNcRwrpcWFR88vXcxLR 54KMzmMy9w6OGPdW5SSgp5/0iByZixd2la/z+/IJr/GHL8eQHvfQ89ok80sc Wgdk+CmTuVosk+F3OTcWnz9pkmSQPJkUdW8mNwaVV+xNNpM53ULuHu/RvA8Y RCle95Hk7M9HBX5ZROOqApuc3WSur7ly6JCjZSTKbHfQiyT5hv7jH78tw/Hq 50wePtILvL/joJXVexRZy4JPSH6Z1WQ2ZhWCwvWKtv0kO07fFFWxDsKbr+0k j5CeMawq/jTTOgATRiwn3Ek2+ruwuN7GDz3u7YxpIfnqS/4TgxPvUC77if56 0mtcFy7Ue0964pdbxruUSca/3B/lpl5h9bfeKUeSK/8tVPRPueEcR21aPMnR Xlt05pud0PfwXvs6klt9CnP3tz7GmpWZC+Mkj6d0XS76aIavltav/5/HKR3a YX9ISQeD8oPG/udxZ7d4TsgZngfvctuc/3mcmhVbqIz2XeiISXnzP4/j2GR6 mLbPBlIoZw1mSZY5+fltjsRT2M1/+FQXyav8ezOKBZ6DSLqHUB7J5//u79kq 4A7XLiv99CWZsuZioe2GNwB33HoMSbaIDL1Vs/4tyD45VXSA5JYY1cHXr33h yuYXMUxyPaITjE0lX/uDbpmJZyLJv/ybdWo9AoFvbuXxLZKdimv36HkEA+7T vruK5IAqX425V6HgTn11PZ5cfw7R0Ea2VxEgMdN2rp3cv6ITB7WfPosCdxae 6Dskez2ZvPz4+Af4sWVWjULu/xU7zZA+qY/g4cuZ3EV629rEcA13iVgQi3+t KU+y0b1cmwO746DgoJR6EllPB0xw0WxzPHhpOsjfIetPzPfOJLtAAmwwsjqZ RtZnlqTMlWCeROBrk3SfJ+tXsXDKJYktGU4vRGwyJ+vbdMfU7hf0NCByKqX/ kd4m3+AqTZ36DIKSm+r5SfbaodN8cSwdPI9W5G8l+0maSGXOz2RA2+6N74XI /sN6Zc37I1lg8N3n7heyP4tirSee6GZDWpNXxhvS4y5MzLW492XD2RMMTW2y nyNnf5zx68iB8dy1H3vJ/l9aGJEyq82F1FTpu8/J+SCtmK2opZAHbLEttpuH CVT/NdFyrDwPTjXpDCWQ82Tgs1MsozAfnpk/x3Ry/pQ2eYqu/1wIqZG3Gow6 CDzJqXqlQbII1nMpZpe0kfV/u37GMaEIttVt/M3XSqBvyWGJ5g/F8OADNSKy icCee2KwPaAU2g4d1PivmsBjQcs/o/gQcitLDzytJOd9+ZWTW7yRdJr3Jj7l BJ6xHnv87EIZcLFzsMeWEOi8ZcPAvlPlcP3G9laLL+T9/V5/v3h/JYxED596 H0t65bpV4X8sKsH7274KzhgCLRM5rslmVcLPjYM3LaMIrBLRXgk4XgUb194v 2/me9N7U2aP8F6uBMSrvIvyO/PzPLodTdGvBgw0fG9mR8/fb9Pby6FrofsRn pWBL4Daz+PmQ8VooMM1+vc2K5MW/1w6b1IHi39ebsx6QHsy1sZLfth4sqMuN NqT3KR4R+GXt0QjHtVq0lc8RmGvfNzVQ3wg7blTFR50mPc85deXMuiZYDHy2 h3acwEOhztdnfJtArNYozYj0wGPRDjFj77/CpSZuAWIn6Z2rPqidTG6BV+n3 9+qzEtjBz97NOd4GPTK1n7ILmEiNObrgytIOa4Y4gs6R57lsWXX8/I52OHjr xc/qDCYaSq6Nbb7dDntGAwULE5joJRknrdLVDh1vrQYvBzFxeW4z1/3aDvDZ 70+uIxO179KapxO6oJ895nT1NiZOdUY1VVZ3Qcp5zwJDISamRp6zDKN0QcWH Fy2L65l4/jnT/ZhIN5w7t9zIy8FEk72V+w8Fd4PsxfID498Y2E+cmRd80QMm 56zKxStJr7O68t7hRh+oXR4+ctyQgTvtt7RzP+mD5d1blO/eZmDhHfv04JA+ CJe9NeWhxUDhoAeywd19ENU3nP3lEgMdDr4zOKXVD8KZXJQUSQY+SJV5sEl1 AFbPp99aNUdHRpCldOupIUgp9nj0yJ6Od/QuOxrfGgLXa0mHgyzpuOaXhdr3 J0Ow04jIS7tPx2OyRobfvwzBF1OefTU6dMwNNlDwOjgMixTB4cBjdORIWFTZ vHMEOAalolMWZvH8oUCVTOdRePDlgU6L2SyqnqsRE387CiZFG68LGc1icO8h 49DwUWh2isjSIfMvUyT8r3n+KOQV/AkuvzKLHt2OssNzo3DiaI7iyb2zqH34 pfiQ0RhcVet9UT44g7Y/BXwOK4zDppqEHScukR5XoQ3rro3DW9bQRb/TM2jQ ntc1pjcO5/TuvqTIzGBo6cNPTs/GQYJ6u9hKZAYPFIgb22eNw77Ast+yizTc tMuD10qEAmHiHfz6iTRU+BAafnuBAu+O56gYstPwWlCVSBLrBPCnsyl7/6ai 0w72xjmeCWConzidQVBR3bTN85HYBBBVTJapASqWmSbogs4E2HMVaq/NoaLP j8qc6bwJ2P428NphIyoy003bLR5PQtzUmq9CJdP425LiJfx8Evi6L0++zJrG zuVXfJXek3BfIw3oCdNIhVWmbHGTUKpyoDrJfxq34KbtF9onQTryyO3R+9OY dKlmbaXUFAhW1Pw+s34aB41cs/JGp+AU9fyzr3pTmF7y1O0TfQpGLkmZBWpO YaNQdoP34hS8ZJ/7q3V5Cr9ZquVe5p8GDd5FkUrpKcwrLQy3gGmgZYkla65M Ip9iQqZTxDSkXv29cOX9JE40CLF4alDhEtHiEtk0gdka700/6FKBd09FSzRO YMxP94n0+1Qo3xwtF5E1gQoXOP8UOFEhRZTQeRE6gfxvFzSfJlGhNXCHBvfd CXy2c8+6pH9UcPGZurv6DwXPyhw9kxRPg7vpujHFOyl4R072pXQGDc5WScl8 F6Tgm/EHpmkFNGhqFV2zm5OC1l6C3KFNNAjIEHtpxxzHwqd4W/Q7DdSlOWUm C8dxS0OyzNKpGRjt+rR74to4Cggrf61vmIGMt6q8Li5jGLSO/Y1zxwycTBGV 320zhgkPosMPDM6AXBP/7sq7Y7hRjlr8iDEDcx2BcwuKY/hfZEN2Dt8snP9l JiYhOIY5rF9D567NgmZxy2B5wiieCEioVe+cheP59meqQkZQ0mKid9XgLDgs HTrk6ziCVh+vKGRRZqGm2CVay2AEb+Yf2fX7+yzUlw7x1EqOYClnTq3cejoo fPg3cq1kGJ+Hq4pbqdCBdql8w7qxIdxcK7fSXEIHhjatQ2b3IHJF5kwmV9Eh L+jkBmGOQfy3uvubSyMdOPaYfVieGcAbNS4mG/vo8N3nQm1K1gCa/Xcog/GD Dro/eJYKzw9gtab96WsSDHj65tr4xzv9eDCE69moFwPczz8+URfQiyn1te/1 /BnAIrRGw/ZRL4od2WXaHcoAEYPdnkLXe5Gr8eHmlE8MOJF17qXypl58GbzN 4VspAyQ6d/gYhvagccbDMhpB8oBOqVZ4N2ZES/p4KjNhNiqtzyOiE0vZnnsu qzNhn/hDHRPHTpQ0mWG9p80E1YRtgvK6nYisE3dEDJgg+9fUYlq4EzXWOCTs t2OCic2fWHpYB4Zcg/1JUUxyPmaxzoW04/ev5qelaUzIut6rsvNdKwq27Te/ wGCC9UW+p96WrdhLBHCrzTGBq+fExx/qrRh62ixI4zcTPkbfZM0VaMU7dKJg iZOAwXf874VkWtDYRM6GIkbA13XlaqsufcX6npJ9zpIENIVX8RtwfsXjnvPn 10sTUNTJkZ/b0IRdt7/7ih4lICDw44KKWhOmJEr6US4QIL1tuEPsRiN27PDQ /6NLQL6Xh06kST2eHTlYulWfgEv+S34+kvUocMbMUtaIAEOZqrhn9Do0lQus v/qAgJ5IPt/zVnVoJTB3WewxAbL6nOKv7GsxM3L9V2MvAtxDl+P/vazGaDNR Hu53BLjO13kEX6rGw+/dgxJ9CZA46HpEgrMa035VOzYFEcDfUSl92qsKFatT A/KiCZB6/PiahF8l8n3N/W9rFgEd+tsKdkSUY+ZeSv6LHAJWc3+W26Fdjskr 90wpuQQoWsWbbeYvx39KAV/diwg4O7pPY+VFGa458dNVu4qAwuyWsk09pXi3 +fwzmxoCkvtEh6uVSjGx6mHNyzoC/uvlLDMrLUFWdYqKdxMBKlEFjhGfivFR rMZa8U4CTnQueQTbFmLZzVCHhS4CMs9cfcIyXYAbj66WK+whIHU+64XhrQKs vnvAT2yAgI2d+0L4zucjb6hmhecYAXICKgYbeHPxes5Nd24KAZq2X/rPP/+C /ouvfd0myPv1Uaoxnc9B58IIJ81pApglKdFJ/dmoe6sj3pVOQLt5FM9ofCaW Hb4Snc0gYKgxpWFkSya+L4sfHGQSYC3pqNHvnYGvPxTs2TBHwEx0bKvr5Gdk JJVL7/hOwLrUmIgbjaT3DdxyFP1BwA2XdVrimano38g5xrFAgNFH6aFkx2Tk +ff6F51kzkDilr5hElpctFav+0nAQd1D23ivJKJehHWdwSIBw7p65ZoC8Tgy tm5a5DcBHIKmGvTFOFSL23T5f89VsppJOTkPx6JSTkSDyBIBV7dKsfglxSBl MZuRS3LxqyD2DT4f8GOipYbCMgEPebf/FKZFoVxq3EItyefu3nXgyonAgQnJ /vMrBHwRPn7mm3MYPjgw9DeTZAF6G+OrYijqXX9vtPkvuZ+sFsfiBINxVvIc 52OS1ZqUamxHAvCn1Rd6LckTdYmXziT74YGJEX7S4yAlKLCebeYdrj6Z9Eid ZJmAqffPXTyxTvSv0CuSDXXbtRcFX2EgNP/JJPnaickbJsnP0Vhu3fbO/z2n +FN37pPLUxSKSnOm/+85SGnn7bbJ1viAM0VkieR/qz81P042wP/7vhyHrqCs WHxm8f8DJChnmQ== "]]}}, Axes->True, AxesOrigin->{0, 0}, PlotRange->{{-0.9999998831351729, 1.}, {-0.9999998592812047, 0.9999998782744886}}, PlotRangeClipping->True, PlotRangePadding->{ Scaled[0.02], Scaled[0.02]}]], "Output", CellChangeTimes->{3.476432617375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", " ", RowBox[{"Sin", "[", "t", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", " ", "Pi", " ", "c"}]}], "}"}], ",", RowBox[{"PlotRange", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1.1"}], ",", "1.1"}], "}"}]}], ",", " ", RowBox[{"Epilog", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", "0.04", "]"}], ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", RowBox[{"2", " ", "Pi", " ", "c"}], "]"}], ",", " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "Pi", " ", "c"}], "]"}]}], "}"}], "]"}]}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"c", ",", "0.01", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.476432642125*^9, 3.476432792984375*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`c$$ = 0.06878125, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`c$$], 0.01, 1}}, Typeset`size$$ = { 360., {178., 182.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`c$1999$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`c$$ = 0.01}, "ControllerVariables" :> { Hold[$CellContext`c$$, $CellContext`c$1999$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ParametricPlot[{ Cos[$CellContext`t], Sin[$CellContext`t]}, {$CellContext`t, 0, 2 Pi $CellContext`c$$}, PlotRange -> {-1.1, 1.1}, Epilog -> { PointSize[0.04], Point[{ Cos[2 Pi $CellContext`c$$], Sin[2 Pi $CellContext`c$$]}]}], "Specifications" :> {{$CellContext`c$$, 0.01, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{405., {235., 240.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.476432659796875*^9, 3.476432683671875*^9}, { 3.47643275034375*^9, 3.47643280359375*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ParametricPlot3D", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", " ", RowBox[{"Sin", "[", "t", "]"}], ",", RowBox[{"t", "/", "20"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", " ", "Pi", " ", "c"}]}], "}"}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"c", ",", "1", ",", "10"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.476432813109375*^9, 3.476432833359375*^9}, { 3.47643289984375*^9, 3.47643291171875*^9}, {3.476432952375*^9, 3.47643296475*^9}, {3.4764330020625*^9, 3.476433059984375*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`c$$ = 8.734375, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`c$$], 1, 10}}, Typeset`size$$ = {336., {214., 218.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`c$5975$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`c$$ = 1}, "ControllerVariables" :> { Hold[$CellContext`c$$, $CellContext`c$5975$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ParametricPlot3D[{ Cos[$CellContext`t], Sin[$CellContext`t], $CellContext`t/20}, {$CellContext`t, 0, 2 Pi $CellContext`c$$}], "Specifications" :> {{$CellContext`c$$, 1, 10}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{381., {271., 276.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.4764328293125*^9, 3.476432833984375*^9}, { 3.476432867671875*^9, 3.4764328746875*^9}, {3.47643291240625*^9, 3.476432965046875*^9}, {3.476433005265625*^9, 3.476433069015625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", " ", RowBox[{"Sin", "[", "t", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "t", "]"}], "+", RowBox[{"r", " ", RowBox[{"Cos", "[", RowBox[{"t", " ", "s"}], "]"}]}]}], ",", " ", RowBox[{ RowBox[{"Sin", "[", "t", "]"}], " ", "+", " ", RowBox[{"r", " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "s"}], "]"}]}]}]}], "}"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", "0", ",", " ", RowBox[{"2", " ", "Pi", " ", "c"}]}], "}"}], ",", " ", RowBox[{"PlotRange", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], ",", " ", RowBox[{"Epilog", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", "0.04", "]"}], ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"2", " ", "Pi", " ", "c"}], "]"}], "+", " ", RowBox[{"r", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "Pi", " ", "c", " ", "s"}], "]"}]}]}], ",", " ", RowBox[{ RowBox[{"Sin", "[", RowBox[{"2", " ", "Pi", " ", "c"}], "]"}], "+", " ", RowBox[{"r", " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "Pi", " ", "c", " ", "s"}], "]"}]}]}]}], "}"}], "]"}]}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"c", ",", ".01", ",", "10"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"s", ",", ".1", ",", "10"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"r", ",", ".1", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.4764330735625*^9, 3.476433306984375*^9}, { 3.476433352953125*^9, 3.476433422*^9}, 3.476433498125*^9, { 3.47643357921875*^9, 3.476433605875*^9}}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`c$$ = 6.7159063163999875`, $CellContext`r$$ = 0.1, $CellContext`s$$ = 4, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`c$$], 0.01, 10}, { Hold[$CellContext`s$$], 0.1, 10}, { Hold[$CellContext`r$$], 0.1, 1}}, Typeset`size$$ = {360., {180., 184.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`c$29622$$ = 0, $CellContext`s$29623$$ = 0, $CellContext`r$29624$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`c$$ = 0.01, $CellContext`r$$ = 0.1, $CellContext`s$$ = 0.1}, "ControllerVariables" :> { Hold[$CellContext`c$$, $CellContext`c$29622$$, 0], Hold[$CellContext`s$$, $CellContext`s$29623$$, 0], Hold[$CellContext`r$$, $CellContext`r$29624$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ParametricPlot[{{ Cos[$CellContext`t], Sin[$CellContext`t]}, { Cos[$CellContext`t] + $CellContext`r$$ Cos[$CellContext`t $CellContext`s$$], Sin[$CellContext`t] + $CellContext`r$$ Sin[$CellContext`t $CellContext`s$$]}}, {$CellContext`t, 0, 2 Pi $CellContext`c$$}, PlotRange -> {-2, 2}, Epilog -> { PointSize[0.04], Point[{Cos[2 Pi $CellContext`c$$] + $CellContext`r$$ Cos[2 Pi $CellContext`c$$ $CellContext`s$$], Sin[2 Pi $CellContext`c$$] + $CellContext`r$$ Sin[2 Pi $CellContext`c$$ $CellContext`s$$]}]}], "Specifications" :> {{$CellContext`c$$, 0.01, 10}, {$CellContext`s$$, 0.1, 10}, {$CellContext`r$$, 0.1, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{405., {276., 281.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.47643360678125*^9, 3.47643366840625*^9}}] }, Open ]], Cell[BoxData[""], "Input", CellChangeTimes->{{3.476433174703125*^9, 3.476433175421875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Manipulate", "[", RowBox[{ RowBox[{"ParametricPlot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], ",", " ", RowBox[{"Sin", "[", "t", "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Cos", "[", "t", "]"}], "+", RowBox[{"r", " ", RowBox[{"Cos", "[", RowBox[{"t", " ", "s"}], "]"}]}]}], ",", " ", RowBox[{ RowBox[{"Sin", "[", "t", "]"}], " ", "+", " ", RowBox[{"r", " ", RowBox[{"Sin", "[", RowBox[{"t", " ", "s"}], "]"}]}]}]}], "}"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", "0", ",", " ", RowBox[{"2", " ", "Pi", " ", "c"}]}], "}"}], ",", " ", RowBox[{"PlotRange", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "2"}], ",", "2"}], "}"}]}], ",", " ", RowBox[{"Epilog", " ", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"PointSize", "[", "0.04", "]"}], ",", RowBox[{"Point", "[", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"Cos", "[", RowBox[{"2", " ", "Pi", " ", "c"}], "]"}], "+", " ", RowBox[{"r", " ", RowBox[{"Cos", "[", RowBox[{"2", " ", "Pi", " ", "c", " ", "s"}], "]"}]}]}], ",", " ", RowBox[{ RowBox[{"Sin", "[", RowBox[{"2", " ", "Pi", " ", "c"}], "]"}], "+", " ", RowBox[{"r", " ", RowBox[{"Sin", "[", RowBox[{"2", " ", "Pi", " ", "c", " ", "s"}], "]"}]}]}]}], "}"}], "]"}]}], "}"}]}]}], "]"}], ",", " ", RowBox[{"{", RowBox[{"c", ",", "1", ",", "10"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"s", ",", "8", ",", "10"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"r", ",", ".2", ",", "1"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.47643367975*^9, 3.47643370359375*^9}, 3.476433751703125*^9}], Cell[BoxData[ TagBox[ StyleBox[ DynamicModuleBox[{$CellContext`c$$ = 2.0837603519561734`, $CellContext`r$$ = 0.2, $CellContext`s$$ = 8, Typeset`show$$ = True, Typeset`bookmarkList$$ = {}, Typeset`bookmarkMode$$ = "Menu", Typeset`animator$$, Typeset`animvar$$ = 1, Typeset`name$$ = "\"untitled\"", Typeset`specs$$ = {{ Hold[$CellContext`c$$], 1, 10}, { Hold[$CellContext`s$$], 8, 10}, { Hold[$CellContext`r$$], 0.2, 1}}, Typeset`size$$ = {360., {180., 184.}}, Typeset`update$$ = 0, Typeset`initDone$$, Typeset`skipInitDone$$ = True, $CellContext`c$47129$$ = 0, $CellContext`s$47130$$ = 0, $CellContext`r$47131$$ = 0}, DynamicBox[Manipulate`ManipulateBoxes[ 1, StandardForm, "Variables" :> {$CellContext`c$$ = 1, $CellContext`r$$ = 0.2, $CellContext`s$$ = 8}, "ControllerVariables" :> { Hold[$CellContext`c$$, $CellContext`c$47129$$, 0], Hold[$CellContext`s$$, $CellContext`s$47130$$, 0], Hold[$CellContext`r$$, $CellContext`r$47131$$, 0]}, "OtherVariables" :> { Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$, Typeset`animator$$, Typeset`animvar$$, Typeset`name$$, Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$, Typeset`skipInitDone$$}, "Body" :> ParametricPlot[{{ Cos[$CellContext`t], Sin[$CellContext`t]}, { Cos[$CellContext`t] + $CellContext`r$$ Cos[$CellContext`t $CellContext`s$$], Sin[$CellContext`t] + $CellContext`r$$ Sin[$CellContext`t $CellContext`s$$]}}, {$CellContext`t, 0, 2 Pi $CellContext`c$$}, PlotRange -> {-2, 2}, Epilog -> { PointSize[0.04], Point[{Cos[2 Pi $CellContext`c$$] + $CellContext`r$$ Cos[2 Pi $CellContext`c$$ $CellContext`s$$], Sin[2 Pi $CellContext`c$$] + $CellContext`r$$ Sin[2 Pi $CellContext`c$$ $CellContext`s$$]}]}], "Specifications" :> {{$CellContext`c$$, 1, 10}, {$CellContext`s$$, 8, 10}, {$CellContext`r$$, 0.2, 1}}, "Options" :> {}, "DefaultOptions" :> {}], ImageSizeCache->{405., {264., 269.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.476433696890625*^9, 3.4764337145625*^9}, { 3.476433752296875*^9, 3.476433767109375*^9}}] }, Open ]] }, WindowSize->{616, 750}, WindowMargins->{{55, Automatic}, {41, Automatic}}, ShowSelection->True, FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{ "Info3476414589-7267634"->{ Cell[1884, 57, 4366, 67, 152, "Print", CellTags->"Info3476414589-7267634"]} } *) (*CellTagsIndex CellTagsIndex->{ {"Info3476414589-7267634", 40702, 840} } *) (*NotebookFileOutline Notebook[{ Cell[545, 20, 228, 5, 113, "Input"], Cell[776, 27, 958, 22, 304, "Input"], Cell[CellGroupData[{ Cell[1759, 53, 122, 2, 31, "Input"], Cell[1884, 57, 4366, 67, 152, "Print", CellTags->"Info3476414589-7267634"] }, Open ]], Cell[CellGroupData[{ Cell[6287, 129, 362, 10, 31, "Input"], Cell[6652, 141, 18497, 310, 372, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[25186, 456, 1096, 30, 92, "Input"], Cell[26285, 488, 1845, 39, 492, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[28167, 532, 688, 17, 52, "Input"], Cell[28858, 551, 1782, 36, 564, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[30677, 592, 2196, 59, 152, "Input"], Cell[32876, 653, 2602, 52, 574, "Output"] }, Open ]], Cell[35493, 708, 92, 1, 31, "Input"], Cell[CellGroupData[{ Cell[35610, 713, 2107, 58, 152, "Input"], Cell[37720, 773, 2636, 53, 550, "Output"] }, Open ]] } ] *) (* End of internal cache information *)