(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 11.1' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 43333, 1867] NotebookOptionsPosition[ 39202, 1758] NotebookOutlinePosition[ 39555, 1774] CellTagsIndexPosition[ 39512, 1771] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Deltoid Pedal Curve", "Title",ExpressionUUID->"3a093c23-21b1-42fe-be59-2e04ec1ed63f"], Cell[CellGroupData[{ Cell["Author", "Subsection",ExpressionUUID->"8dd131b5-72a0-4860-9944-38ec753a56e6"], Cell["\<\ Eric W. Weisstein March 16, 2008\ \>", "Text",ExpressionUUID->"bf5b769f-cb56-4e6d-a2a1-fe2e9ecbc0c6"], Cell[TextData[{ "This notebook downloaded from ", ButtonBox["http://mathworld.wolfram.com/notebooks/PlaneCurves/\ DeltoidPedalCurve.nb", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://mathworld.wolfram.com/notebooks/PlaneCurves/DeltoidPedalCurve.\ nb"], None}], "." }], "Text",ExpressionUUID->"c054bfdf-349a-44f1-b105-2f99155186e4"], Cell[TextData[{ "For more information, see Eric's ", StyleBox["MathWorld", FontSlant->"Italic"], " entry ", ButtonBox["http://mathworld.wolfram.com/DeltoidPedalCurve.html", BaseStyle->"Hyperlink", ButtonData:>{ URL["http://mathworld.wolfram.com/DeltoidPedalCurve.html"], None}], "." }], "Text",ExpressionUUID->"6af2bf5a-0bfc-4ccf-b1cc-99f48f5ff884"], Cell["\<\ \[Copyright]2008 Wolfram Research, Inc. except for portions noted otherwise\ \>", "Text",ExpressionUUID->"115309c7-3e85-4526-be42-a1cf9e01c38d"] }, Open ]], Cell[CellGroupData[{ Cell["Plot", "Section",ExpressionUUID->"39775dc4-969e-4e9c-891f-a8271effbe61"], Cell[BoxData[ RowBox[{ RowBox[{"deltoid", "=", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"2", "/", "3"}], " ", RowBox[{"Cos", "[", "t", "]"}]}], "+", RowBox[{ RowBox[{"1", "/", "3"}], " ", RowBox[{"Cos", "[", RowBox[{"2", "t"}], "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"2", "/", "3"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], "-", RowBox[{ RowBox[{"1", "/", "3"}], " ", RowBox[{"Sin", "[", RowBox[{"2", "t"}], "]"}]}]}]}], "}"}]}], ";"}]], "Input",ExpressionUU\ ID->"4bf43b1a-4359-46d2-aa19-4b1306eeb764"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"pedal", "=", RowBox[{"FullSimplify", "[", RowBox[{"PedalCurve", "[", RowBox[{"deltoid", ",", RowBox[{"{", RowBox[{"x", ",", "y"}], "}"}], ",", "t"}], "]"}], "]"}]}]], "Input",Exp\ ressionUUID->"06fefbe0-d424-4ef0-ab2b-a26cd3b8cffb"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", "x"}], "+", RowBox[{"Cos", "[", "t", "]"}], "+", RowBox[{"3", " ", "x", " ", RowBox[{"Cos", "[", "t", "]"}]}], "-", RowBox[{"Cos", "[", RowBox[{"2", " ", "t"}], "]"}], "-", RowBox[{"3", " ", "y", " ", RowBox[{"Sin", "[", "t", "]"}]}]}], ")"}]}], ",", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ RowBox[{"3", " ", "y"}], "-", RowBox[{"3", " ", "y", " ", RowBox[{"Cos", "[", "t", "]"}]}], "+", RowBox[{"Sin", "[", "t", "]"}], "-", RowBox[{"3", " ", "x", " ", RowBox[{"Sin", "[", "t", "]"}]}], "+", RowBox[{"Sin", "[", RowBox[{"2", " ", "t"}], "]"}]}], ")"}]}]}], "}"}]], "Output",Expressio\ nUUID->"feabd876-028d-454d-8c17-55727153d517"] }, Open ]], Cell[CellGroupData[{ Cell["Cusp", "Subsection",ExpressionUUID->"b200d088-c8e3-4e15-8e58-384b386f2b7c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"deltoid", "/.", RowBox[{"t", "\[Rule]", "0"}]}]], "Input",ExpressionUUID->"be14451f-380b-\ 415e-934b-636d8934a683"], Cell[BoxData[ RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]], "Output",ExpressionUUID->"4de7ccff-1a24-4b43-b37d-e2a673d66acd"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"pedal", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "1"}], ",", RowBox[{"y", "\[Rule]", "0"}]}], "}"}]}], "]"}]], "Input",ExpressionUUID-\ >"c087fdd1-dff0-4abe-9b14-0abdd296abd2"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{"3", "+", RowBox[{"4", " ", RowBox[{"Cos", "[", "t", "]"}]}], "-", RowBox[{"Cos", "[", RowBox[{"2", " ", "t"}], "]"}]}], ")"}]}], ",", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], " ", RowBox[{"Sin", "[", "t", "]"}]}], "+", RowBox[{"Sin", "[", RowBox[{"2", " ", "t"}], "]"}]}], ")"}]}]}], "}"}]], "Output",Expressio\ nUUID->"8f89d18e-6e84-49f6-84bd-b3aae729d99b"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PedalCurvePlot", "[", RowBox[{"deltoid", ",", RowBox[{"{", RowBox[{"3", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "Pi", ",", ".2"}], "}"}]}], "]"}]], "Input",ExpressionUU\ ID->"b4b26b37-90c9-4b2a-b289-1f919f4bb845"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.15469 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.341269 0.21164 0.577349 0.21164 [ [ 0 0 0 0 ] [ 1 1.15469 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 .57735 m 1 .57735 L s .34127 0 m .34127 1.15469 L s 0 0 m 1 0 L 1 1.15469 L 0 1.15469 L closepath clip newpath 1 0 0 r .5 Mabswid .97619 .57735 m .97619 .57735 L .97619 .57735 L s .97619 .57735 m .97611 .57651 L .95105 .57903 L s .97619 .57735 m .97487 .57084 L .87859 .59036 L s .97619 .57735 m .96973 .55648 L .76731 .61909 L s .97619 .57735 m .95672 .5313 L .62999 .66944 L s .97619 .57735 m .93147 .49548 L .4819 .74108 L s .97619 .57735 m .89014 .45157 L .33859 .82891 L s .97619 .57735 m .83038 .40424 L .2138 .92357 L s .97619 .57735 m .75201 .35962 L .11763 1.0128 L s .97619 .57735 m .65745 .32442 L .05531 1.08322 L s .97619 .57735 m .55175 .30482 L .02678 1.12241 L s .97619 .57735 m .44215 .30554 L .02712 1.12097 L s .97619 .57735 m .33735 .32898 L .04766 1.07409 L s .97619 .57735 m .24645 .37476 L .07772 .98253 L s .97619 .57735 m .17783 .43965 L .10659 .85274 L s .97619 .57735 m .13808 .51791 L .12544 .69622 L s 0 g [ 1 0 ] 0 setdash .97619 .57735 m .97619 .57735 L .97619 .57735 L .97619 .57735 L .97619 .57735 L .97619 .57734 L .97619 .57733 L .97619 .5773 L .97619 .57726 L .97618 .57715 L .97616 .57695 L .97613 .57667 L .97597 .57563 L .97561 .57382 L .97501 .57136 L .97212 .56244 L .96696 .55031 L .95719 .53209 L .94121 .50809 L .92033 .48234 L .85621 .42294 L .81318 .393 L .75829 .36262 L .69838 .33746 L .66588 .32684 L .62871 .31712 L .61058 .31328 L .59318 .31015 L .5772 .30773 L .56008 .30565 L .5512 .30477 L .54136 .30396 L .53229 .30336 L .52396 .30294 L .51919 .30275 L .51416 .3026 L .51155 .30254 L .50911 .3025 L .50644 .30246 L .50359 .30243 L .50111 .30242 L .49843 .30242 L .49589 .30244 L .49356 .30246 L .48913 .30253 L .48445 .30264 L .48006 .30278 L .47529 .30298 L .4666 .30345 L .45816 .30404 L Mistroke .45035 .30471 L .43275 .30666 L .41417 .3094 L .3806 .31623 L .35106 .32441 L .31828 .3362 L .28966 .34918 L .25843 .36683 L .23173 .38556 L .18889 .426 L .1705 .44983 L .1544 .47683 L .14268 .50376 L .13767 .51934 L .13411 .53393 L .13184 .54678 L .13091 .55413 L .13026 .56103 L .13001 .56457 L .12991 .5665 L .12982 .5683 L .12971 .57148 L .12964 .57496 L .12963 .57692 L .12963 .57876 L .12968 .58222 L .12974 .58415 L .1298 .58593 L .13 .58996 L .13024 .59347 L .13052 .59679 L .13119 .60302 L .13211 .6097 L .13336 .61696 L .13622 .62993 L .1396 .64182 L .14989 .66856 L .16463 .6959 L .18069 .71874 L .20108 .74191 L .24537 .77919 L .2719 .79598 L .30321 .812 L .33807 .82597 L .37215 .83635 L .38952 .84053 L .40864 .84435 L .42571 .84709 L .44397 .84935 L .46053 .85084 L Mistroke .46997 .85145 L .47452 .85168 L .47884 .85187 L .48273 .85201 L .48697 .85212 L .49083 .8522 L .49441 .85225 L .49867 .85228 L .50253 .85227 L .50698 .85223 L .51117 .85216 L .51539 .85206 L .51991 .85192 L .52806 .85157 L .53773 .851 L .54652 .85034 L .56555 .84844 L .58343 .84608 L .61451 .84063 L .64748 .83298 L .71167 .81231 L .77309 .78464 L .8227 .75559 L .86226 .72702 L .92352 .66871 L .94415 .64254 L .95899 .61954 L .96778 .60264 L .97241 .59146 L .97496 .58354 L .97564 .58077 L .97597 .57908 L .97611 .57815 L .97615 .57783 L .97617 .57759 L .97619 .57745 L .97619 .57741 L .97619 .57738 L .97619 .57736 L .97619 .57736 L .97619 .57735 L .97619 .57735 L .97619 .57735 L .97619 .57735 L .97619 .57735 L Mfstroke [ .05 .02 ] 0 setdash .97619 .57735 m .97615 .57735 L .97605 .57735 L .97586 .57735 L .97559 .57736 L .97494 .57737 L .97397 .57739 L .97257 .57744 L .97072 .57752 L .96645 .57775 L .96079 .57815 L .95434 .57871 L .93561 .5808 L .91094 .58441 L .88377 .58933 L .80827 .60716 L .72944 .63142 L .63374 .66786 L .43554 .76737 L .26005 .88617 L .11944 1.01094 L .04337 1.09871 L .02814 1.12032 L .02445 1.12613 L .02405 1.1268 L .02385 1.12714 L .02381 1.1272 L .02381 1.1272 L .02385 1.12713 L .02393 1.12699 L .0259 1.12334 L .0304 1.11433 L .04561 1.07931 L .06443 1.02681 L .08294 .96307 L .10155 .88033 L .11004 .83185 L .11759 .77778 L .12292 .72743 L .12527 .69856 L .12697 .67201 L .12819 .64699 L .12872 .63281 L .1291 .6197 L .12935 .60821 L .12945 .60167 L .12953 .59554 L .12959 .58909 L .12962 .5832 L .12963 .57639 L Mistroke .12961 .56993 L .12957 .56295 L .12953 .55903 L .12949 .55542 L .12939 .54878 L .12926 .54178 L .12896 .52965 L .12853 .51638 L .12722 .4871 L .12555 .46005 L .1215 .4122 L .11577 .36253 L .10822 .31163 L .09865 .25969 L .07886 .17629 L .05865 .11062 L .03975 .06112 L .02761 .03468 L .02478 .02925 L .02402 .02787 L .02384 .02754 L .02381 .02749 L .02382 .02751 L .02387 .0276 L .02398 .02778 L .02428 .02829 L .02534 .03002 L .0299 .03704 L .05038 .0652 L .13312 .15754 L .26548 .27273 L .44808 .39464 L .63771 .48849 L .73525 .52525 L .81763 .55005 L .88527 .56567 L .93269 .57352 L .95089 .57566 L .96362 .57676 L .96894 .57709 L .97256 .57726 L .97396 .57731 L .9751 .57733 L .97553 .57734 L .97572 .57734 L .97587 .57735 L .97598 .57735 L .97607 .57735 L .97617 .57735 L .97619 .57735 L Mistroke Mfstroke .05 w .97619 .57735 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", CellDingbat->None, Evaluatable->False, AspectRatioFixed->True, ImageSize->{173, 199.75}, ImageMargins->{{34, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}},ImageCache->GraphicsData["CompressedBitmap", "\<\ eJy9XeuOFccRbs797B0whgCGhQWbBAffIME2QbIUOVKkPASyLNk/rFiOXyvP kcea7HTPVFVXf9VTvWeXkXbPOX2pqfq6qrr6Mj3/evf7Tz/+8u73n394d/qP 3979+tPPP/zn9Pt//3aeNL8WQvjv+d//TkP/vTv/Ovz7JnRdF+bpx/m3/u/X +D+8z6wFyPpqovisz3k5SZNlrdNccnHK+tIuvgLFv7CLr0HW53bxDaD+2QT1 WZ8xzQLR7K8GFvqrwkIqvsg4Pr9e2DW2uMafp2ossxoja59OVcvBrPC1B4rP QNqzSRLdDiROQPFP7OLXQfGP7eI3QPGndvF9UBzZy5M2EsiGzmwSB04Sj9tI PG8rjrTpoU3i0EniQRsJpKQftZFALXK/jQTC555N4iNQ/K5d/AEo/ge7+BEo /se24sjoPmwjgQzxlk3i2EnigzYSyL5vtpG4aZP1kvjAFs5L4pbdSpfgNVHx 2yDt0Cbx3DYLVPxTUHzfLo5UIZpA7OwbK96jitu2ishuV20kGvuiB8Tqsq3i KUibARJR1273Sf+0iT0iLhCJGE9tYtKip9D/7D879gwyOdFY9d/+bt/yjG6p 7jbv/y/6pHVOm0WcddTPLrLcMW9FHLy1OXgK0iIr++C+C6K9sXJVWlS9SO/b SR66FNql8P0QEFvnoqncJQKlC/z3OnHwmlhKHI+NyDWBCJHuHORuKHebN8X4 Ge/0Kt36Fd96TrcmhUuqZ6F7kOp2GJpkN0utB/H3PLF4/i2Nh6Cvv0+siWYk lZwTfzF9CTg4IiAO6RvSCATijG7+uc3hXRJjDThcCkuSrUq2XPCaW01HZlwk s9otiYUXNp93qJBqxsjrhvkstO+kYswM6h4QaEEAPrcZu83GnauvYnFvIBly RWY2TK9wUGVSaM1okkPEBtm9RTjugXYJY7hewDVnY7TcB8o1M5QSJ6Xq/31i 836TeBdoFAJIbzNeSxIB+Tj2RSgXmRYLnXJXGi5u3vPP5Ik9kRzb1pxsxJIw b6IQCqaApzNz2c2rxoqfS8zHE1uu0ehEWjeaeqSTHEjpWtlUUOdMYsbM6/Rt j9hHTh451tgwj20BjrEAe0qAqt1LRc+Ir3IRRPc6bSRAJ0jRUuA26NsyY22s fgBImka6ItCBBpAMyHuxgQqVqwRUqLUVuCmAhtMjqDeC3ojrdtSTkV6iGKSc KONGMF3y3MqNn0je1NDQi/RJaRAPp3WQ6Kb9oLqpGPXuMeOQ2n1LRRDbKxKU XSiKrBeDQkoYWPnO/923DRGNz5AHQ+09B3XZBemebri01tb7ChzXQPnXqHnD 8DFM+UAI0EyoOSzwQlDt4ktLJWRiFsdNZuDJGKFyrHQqTLprw4AG3Mhw22AA LRVFRAAvcihi5hF9Qy2v7iHErocMd9pg8GoDWiIy6y5YH9C0BdYfHlDVx5UJ BDy0Qr7qlg2I6BGlZ/eqhheTA8LkCOTqAAr3dYwJ24jyvcpXTgKTEIFzfGM7 qR4sal5tKDHlag9A2ozQ4TEj6ps2pB+m+ZmdzCL/VvpeZOU3bXyEvERpTQgh /hBCSEqEEMet7EZUkX0qopRC2ZKJ32wSF8q60YaLGbWsSK8viAv3KkLo/joh oXWXjK3DBmSjAck+Q6gpiXAPeXPAcFGNm4RUI00vLMfU0FFTUJDNHTYrhDIU FXh3g29EhpKGbHrpKdZE4xpzoBUVArF7nPMy/o1OUAJs+mn2oguyhuaeBY86 r9vyo/Y2x0sK1hj1IbBQLyJ71uGiljXdST0EU1qRii2lASh08hnA1GhDWFYM hx2GIAymqL4ifVH+7gTQSJF37FdADRR3GHGuJx4lTlMrwdXHGRaqDZNc0Q4J EOQSDggBHr8scpG5jVkDZk5RD6Wo5exBCJWpI0M2hEEhL7LfJalHxWrGi8ex CGcGQkVfChIwBpf5yDhyxAJCDLl7e7bBRCzXc2QZkfGToe2UxkfOeGhjtiIP 9NlGrFnOCjYx60Bik21QUvp3ZbAo5YqfERrkupP8pOMxcx8godZBOAN0LfRd hnQjEGleY9wZMeKzs5Uh2dDUojmGWVoasiWEdMdMmKowBHZM8ddGgpg10PBR okPAXTY6aCSE0NknK5Pz7+JiA2MIlLpwhhj/ZStWSp9yf7MHUYnVImNo+DQB g9LHqgWZaTOCBnGANu2xu0HDXzFKYiiwaSkIR/3BSJn6c8zNihC/GCpbMife Wsomhoa1KRHZBNCUPmFri+nteCJvUX5kHF5Z2VWyLiRHGsijgTaurgdxeSx8 6geGBVi82oEiCNsgeJg00DpSakHlUXeDym0EGEEOXEqjn+Nk2OkM/6XEq8sD Y5ODMQJiTqtG5UGG4nAp8VONAOMvhkqYR4HOGiCxzGyCHKRlE+aUU11Faqgg 7ZiTPpk1eP1ylYwmK8ITbPHbFtvE6mqE35TWSXZibqvxonJMciP9GYa4o5pw J6uMAqnCog0Nc6W/AQ1U3osGspcl6Y29sMTRI9kRu+K9iBrhNaKxzNAY65qL fI6wHE1Oow7HKzlP+qEOZ0W46Ng8jFdawOwT4na9WSF+1nhXIb65jOAV31GX YdpSc0fJY7cTt1ZeS5Lf7cirRAKB7CVPWzvTtiDtAKTtg7Q9q1zaC0HftvRt 040VeAfQnNmotitSQ2+7oqlS5Ci8YYJ3gMYLzWrJeRhhjBotd+eNSK5Eq18G LhXXk9FFTnUXXFi31QYfDh5F2BTEFctvdoDBOxmxqNA1R5EXHJXyrlj+JlfC xMWqEZkoHSDemnVBXRj3ySIQzNGCKRyPJRAIektBIfqHyUiC6vCuQOCCZJOs TQ3Oy375GLEbHGh/DY/1BNkKWQTkgsA7laLWOtrb2uvv0NC3w2vhdwAC2XDg yhFoGhg63IBa+NUrH8nUA34qLBsL5DOwHZ6p8Uouli4o3YxNHJIjx8bePbJ+ 6hJQtsVlC+gN0up7A1e5gChuPQOirqWo+cyfV4W9AQ/i2pTTbEi9CjcXAo6b vKdEDKOWX0ZTdqE+q9ckogNauRVDXM+A5JvdJZ9oXJLcG2p4JVeT/ZF59DRf o4jeiWyVFukhcUzX6nC3MtgfRUJPIA5zsePyqShOLYDGDd4pGONZACRZk7SI HZ49iJkvgLR7mbRSUmoI7wx0g7TebpNbT3Uoau0l9p1IvP2LidfWmP4ZpNpm k6kWVau4Uc7PgMjZgi3F8pcmsnebv3cra30/ulbj/g5fAKGzFXz2I13IAky6 ifmEpKMdvMJ6O5iGvU9R5K+A+Gp3ixQfzXvs0tbeeRSv+Gi321+BiMe5iOEC InpbGNEz98B6RURaJLxYf6+XQOyTQmwig/b+oJbwtmxtCsTcZow6WTHSC+J6 BcS7nokXpHhoD2zAfa+3NzKG+xOPYxbVlVXyFCbS23J/rpIQ9x67dLqGmOZ2 mXrooIc2/edrIGi591YJarpRSvdqKhqxN0mneel/vQEylZvym2VqnZ6Sdb0P QYJnDQSfgU4by2S7JWVbZrKN9dUzyReSZyqNtc3cbaNFIF5uFyJkdqREINtq i+5y8WtT6GVasWE7ivItaI3yQaGqpuEtWQ1+Aw0q7U7BikTFhp3hirJFKbMH 4Sb1q5BMlvcOGZsEMMIr5AbuST0rnsqoeRlJxQ4/ik7O7bioJSLkbwHv95t5 R4/z2V4Yd1/eyQCpruP1llTo4bQKOWwvVIrao5nKElO5lVQWfUPsP5LsFxO3 DugFO67e3ChqP1DDaf31N+L8rMq5t3FVHB3zHEuzwgLMh0xGlglw1QjtAlRO UwjVTRuZp3I89Zfx+Q1B/vGlcGyovndu2dzpyzb5TPKZ71hpgtVwHBPhPam2 fiyiePybO6I/7c6xWU7su7AKxxJWqJ/0ImM3zUkO4XShCIi7tunI4uQn/VS/ BCgbAUjF/Zr4xYf97AxvZdDc34JWT6X2sDrkPQpbGT5ByWTWO1Fm2peIybJj E4i9iGNk5MuMNVl4p0b3sos0+S/9/6iCr9qY2wU3c4CWzyj0X14SeF+nLHF+ GqHsBc87sWA8szxje5X9b4Q1mnSMX94UfF4KjnZaZVCoHmSJ5WPU85jY/W53 dtv0FD6aFAKe44uxRIx27pKifg8VweTuCjlGQ4A4N3+9T9r2yeVxi1fGbOXJ r/6+lZMl1QmWovH9j2/bHBUj5E0bJ16cvJaEnieN1uLjS7WfHyHbI+GDbWqc +EdKXo3Saf1VOdh1YKNDbOzi5tCgpXIc7Q2F38DCLp7LjIcm0Xg/bFTQOKno hpcXM23GglZuvpA3b8FgKm28fByUrVCzU3zATwjTZ0hjrNvkM+8+eQi2ATYW tXJwWX9NHtpd3Kzm/yZuVjlkXJ2KLXS3crirfSd05md2nPkFxCoDrv66c8E7 FV7DiD4rJ8cfk33WmMeWGKqUzYOdLWdXHHVZOWL/KGc2+wTUWD40dTt1iH// c5r9vOOZfElBud2E72CeIRfrPHSRLhQNeAxJ9tSmejDJsEn10RTV7Khcbibz EILHLoKlqWDDO5uihtofk3pik0rbjFaaMaWYRKn/NUmtylimLk8nZFTPJOv6 lePo1ePA6vzuEGrvE0hh6iaTQnMQwvSbY9YWqvGqvNRCvFNHMhDFqbznRLxY p7hb5cU6m1wF5B1dNfENKy8A2gDxQvUlQ2tcY+pdSmlBoaUGikwn+crNo/Iy Jt5tMv1Kqiyr8j4oOf4dsyvUl2yhlFUpvgDFp97opbIqxd/nu8nCtf8D9+EA 2g==\ \>"],ImageRangeCache->{{{0., 172.}, {198.75, 0.}} -> {-1., -2., 0., \ 0.}},ExpressionUUID->"b6018255-1a10-4e69-9609-f8b858ce2801"], Cell[BoxData[ TagBox[ RowBox[{"\[SkeletonIndicator]", "Graphics", "\[SkeletonIndicator]"}], False, Editable->False]], "Output",ExpressionUUID->"a45dc658-d7ba-41d4-81de-\ ae0ab54d45f8"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Negative ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "5da157d8-b5c8-499e-ad3e-89965145a1be"], "-intercept" }], "Subsection",ExpressionUUID->"419614f5-ceaf-4711-b2ad-192751ba47f5"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"deltoid", "/.", RowBox[{"t", "\[Rule]", "\[Pi]"}]}]], "Input",ExpressionUUID->"3cb9a3a0-\ 192c-4a70-96fe-4e53b042eecc"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"-", FractionBox["1", "3"]}], ",", "0"}], "}"}]], "Output",ExpressionUUID->\ "30382caf-87f1-4e90-a75a-e701fa2a5293"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"pedal", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", RowBox[{ RowBox[{"-", "1"}], "/", "3"}]}], ",", RowBox[{"y", "\[Rule]", "0"}]}], "}"}]}], "]"}]], "Input",ExpressionUUID-\ >"8f5777b2-d5ed-4f4d-8582-0a4d80ed9b86"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", FractionBox["1", "3"]}], " ", SuperscriptBox[ RowBox[{"Cos", "[", "t", "]"}], "2"]}], ",", RowBox[{ FractionBox["1", "3"], " ", RowBox[{"(", RowBox[{"1", "+", RowBox[{"Cos", "[", "t", "]"}]}], ")"}], " ", RowBox[{"Sin", "[", "t", "]"}]}]}], "}"}]], "Output",ExpressionUUID->\ "fcd36fc3-1616-42ef-9b69-1e8b99457a13"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PedalCurvePlot", "[", RowBox[{"deltoid", ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", "Pi", ",", ".2"}], "}"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "3"}], ",", "3.2"}], "}"}], ",", "Automatic"}], "}"}]}]}], "]"}]], "Input",ExpressionUUID->"e7b0c491-d770-43ec-b225-\ 4dfe4edf5d5b"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .87999 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.483871 0.16129 0.439996 0.16129 [ [ 0 0 0 0 ] [ 1 .87999 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 .44 m 1 .44 L s .48387 0 m .48387 .87999 L s 0 0 m 1 0 L 1 .87999 L 0 .87999 L closepath clip newpath 1 0 0 r .5 Mabswid .32258 .44 m .32258 .44 L .96774 .44 L s .32258 .44 m .32895 .50344 L .94858 .44127 L s .32258 .44 m .34704 .56066 L .89336 .44991 L s .32258 .44 m .374 .60623 L .80855 .47181 L s .32258 .44 m .40558 .63631 L .70391 .51018 L s .32258 .44 m .43679 .64905 L .59104 .56478 L s .32258 .44 m .46269 .6448 L .48183 .63171 L s .32258 .44 m .47921 .62595 L .38673 .70385 L s .32258 .44 m .48373 .59651 L .31344 .77185 L s .32258 .44 m .47555 .56138 L .26594 .82551 L s .32258 .44 m .45594 .52562 L .2442 .85538 L s .32258 .44 m .42801 .49366 L .24446 .85429 L s .32258 .44 m .39617 .46861 L .26011 .81856 L s .32258 .44 m .36544 .45189 L .28302 .74878 L s .32258 .44 m .34068 .44312 L .30502 .64987 L s .32258 .44 m .32579 .44022 L .31938 .53059 L s 0 g [ 1 0 ] 0 setdash .32258 .44 m .32259 .4425 L .32262 .44477 L .32267 .4474 L .32273 .44988 L .3229 .45429 L .32315 .45907 L .3235 .46432 L .32397 .46987 L .32505 .47977 L .32648 .48988 L .32812 .49924 L .33283 .52002 L .33901 .54025 L .34576 .55772 L .35482 .57661 L .36421 .59251 L .38292 .61669 L .3934 .62691 L .40475 .63575 L .41564 .64219 L .42169 .64493 L .42717 .64688 L .43187 .64815 L .43449 .64868 L .4369 .64906 L .4382 .64922 L .43941 .64934 L .4405 .64942 L .44166 .64948 L .44226 .6495 L .44292 .64951 L .4436 .64952 L .44424 .64951 L .44537 .64948 L .446 .64945 L .44658 .64942 L .44772 .64933 L .44878 .64921 L .45074 .64893 L .45281 .64853 L .45499 .64797 L .45876 .64668 L .46204 .64515 L .4654 .64313 L .46882 .64049 L .47205 .6373 L .47474 .63393 L .47703 .63033 L .47917 .62605 L Mistroke .48083 .62169 L .48205 .6174 L .48256 .61512 L .48296 .61293 L .48326 .61089 L .48351 .6087 L .48362 .60754 L .48371 .60626 L .48378 .60508 L .48383 .60398 L .48386 .60266 L .48387 .60145 L .48386 .60004 L .48383 .5987 L .48378 .59746 L .48371 .59611 L .48353 .59366 L .48322 .5907 L .48285 .58796 L .48183 .58222 L .48039 .57608 L .47714 .56564 L .47247 .55414 L .45955 .53092 L .44165 .50767 L .42039 .4868 L .39907 .47053 L .37815 .45802 L .3604 .44976 L .34493 .44431 L .33768 .44236 L .33208 .44117 L .32974 .44076 L .32754 .44043 L .32593 .44024 L .32456 .44011 L .32363 .44004 L .32323 .44002 L .32294 .44 L .32284 .44 L .32275 .44 L .32264 .44 L .32258 .44 L .32258 .44 L .3226 .44 L .32264 .43999 L .32271 .43999 L .32279 .43999 L .3229 .43999 L .32315 .43998 L Mistroke .32347 .43996 L .32441 .4399 L .32578 .43977 L .32924 .43931 L .33344 .43856 L .34565 .43546 L .35421 .43261 L .36304 .42913 L .38145 .4202 L .40282 .40688 L .42534 .38881 L .44496 .36851 L .46148 .34609 L .46842 .3341 L .47455 .32108 L .47898 .30881 L .4806 .30307 L .48182 .29783 L .48275 .29265 L .48313 .28995 L .48345 .28713 L .48356 .28596 L .48366 .2847 L .48372 .28363 L .48379 .28246 L .48383 .28117 L .48386 .27982 L .48387 .27859 L .48386 .27744 L .48383 .2761 L .4838 .27543 L .48377 .27469 L .48369 .2734 L .4836 .2722 L .48332 .26955 L .48291 .26676 L .48242 .26421 L .48187 .26186 L .4806 .25761 L .47907 .25372 L .4774 .25033 L .47276 .24351 L .46986 .24044 L .46638 .23755 L .4628 .23525 L .45926 .23352 L .45586 .23228 L .45385 .23171 L .45194 .23128 L .44986 .23092 L Mistroke .44793 .23069 L .44685 .23059 L .44566 .23052 L .44452 .23048 L .44346 .23047 L .44231 .23049 L .4412 .23053 L .4402 .23059 L .4391 .23068 L .4379 .2308 L .4366 .23097 L .43425 .23135 L .42953 .23242 L .42516 .23376 L .41989 .23581 L .41411 .23859 L .40349 .24511 L .39341 .25307 L .38252 .26373 L .37145 .27709 L .36193 .29105 L .34567 .32249 L .33827 .34191 L .33245 .36142 L .32816 .38053 L .32644 .39041 L .3249 .40146 L .32389 .41095 L .32345 .41631 L .32313 .42125 L .3229 .4256 L .3228 .4281 L .32272 .43037 L .32266 .43278 L .32261 .43535 L .32259 .43774 L .32258 .44 L Mfstroke [ .05 .02 ] 0 setdash .96774 .44 m .96771 .44 L .96764 .44 L .96749 .44 L .96729 .44 L .96679 .44001 L .96605 .44003 L .96498 .44007 L .96357 .44012 L .96032 .4403 L .95601 .44061 L .95109 .44103 L .93681 .44262 L .91802 .44538 L .89731 .44912 L .83977 .46272 L .77969 .4812 L .70676 .50898 L .55571 .58481 L .42197 .67535 L .31482 .77044 L .25685 .83732 L .24524 .85379 L .24243 .85822 L .24212 .85873 L .24196 .85899 L .24194 .85903 L .24194 .85903 L .24197 .85898 L .24203 .85888 L .24353 .8561 L .24696 .84923 L .25855 .82254 L .27289 .78253 L .287 .73395 L .30119 .67089 L .30765 .63395 L .31341 .59274 L .31747 .55437 L .31926 .53237 L .32056 .51214 L .32149 .49307 L .32189 .48226 L .32218 .47227 L .32237 .46351 L .32245 .45853 L .32251 .45386 L .32255 .44895 L .32257 .44445 L .32258 .43927 L Mistroke .32257 .43434 L .32253 .42903 L .32251 .42603 L .32247 .42328 L .3224 .41822 L .3223 .41289 L .32207 .40364 L .32174 .39353 L .32074 .37122 L .31947 .3506 L .31638 .31413 L .31202 .27629 L .30627 .23749 L .29897 .19791 L .28389 .13435 L .26849 .0843 L .25408 .04658 L .24483 .02643 L .24268 .02229 L .2421 .02124 L .24196 .02099 L .24194 .02095 L .24194 .02097 L .24198 .02103 L .24206 .02117 L .2423 .02156 L .24311 .02287 L .24658 .02823 L .26218 .04969 L .32524 .12006 L .42611 .20785 L .56527 .30075 L .70979 .37228 L .78412 .40029 L .8469 .41919 L .89845 .43109 L .93459 .43708 L .94846 .43871 L .95816 .43955 L .96222 .4398 L .96497 .43993 L .96604 .43996 L .96691 .43998 L .96724 .43999 L .96738 .43999 L .96749 .43999 L .96758 .43999 L .96765 .44 L .96773 .44 L .96774 .44 L Mistroke Mfstroke .05 w .32258 .44 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", CellDingbat->None, Evaluatable->False, AspectRatioFixed->True, ImageSize->{228, 200.625}, ImageMargins->{{34, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}},ImageCache->GraphicsData["CompressedBitmap", "\<\ eJy9XAtzEzcQFrbPdhKTV0NCgIJ5NJRXeU0pAYa0AQItw6u09EUpgXYapqVQ muE39Kf1h/R/uJZ0Wq103+ruEpebsX2Wdj99Wq2klazznfXNjV9erm++eL7e v/lm/fXGi+d/9ddevRkmNXcopf4dvv7pK30/GN7mby/VYDBQTftleKdfm+Zd vcusT0DWeVn8os1SPOvcaMUbOqMGBX0lMJcJs0wowjwri18izDKhCPNMmXgr MNXwOi1rXAbG/X/FmyDtmAxxBYg3tg+BWHwoQ1wD4kdl8etAfEkWvwHEP5DF rwLx1vYhMpB2pB5EG6QdliFWKkIcqgdxYmTinwLxrn43/bZfT3EcpL1fD2IC pO2rB9Ej+gnFz4DiTpC2WA9iEqTtliEeAvEFWfwrID4vi29tqlsFhcySSefq Kc6BtJl6ELtA2nQ9iHmQNiVDoKEQQUzWg1iQXa4qxG7J3atDLIK0iXoQe0Da uAxRc6pA4u+T/3XqKR4AaS0Z4lcg3pTFn+mPts4x3BKSP+kPk5PRXRjF6JS3 VhhNlrvIAsbSjYEtMBcx3xLa75E291mVB9kJxdkKbl8CMQPSpqkOCcVpkU8r MLGr2p8Wak2mXLRJ1FooZvpDhkUTBhogW8RxBGAofPhNhu1JzhTBdohjAgzN zmgCGgNyG9uHRfGF7bLqJshCLBKz6wMvTtmJ6f8BQEfik2TYn63ifaC4ByhO k+Kzeop+tn4aGCcQQrGwATPdcp4gnmwVYjdBPJabCEFERjSfaMr5ToZFa4U9 APbt9iH2UzUfyWBoXLFgwVCm7/uEZ6M9dQvoouWhgHeI8L7keFmgm9clD5iz eHg1348Q0P1yoL3AXRogzYPeHR3oErDObQ5f2D5AyInZXEBBfREFG8ep0hYo J1e+akBgJ/W7Ub0hg6FRLwNpp4nZNRkMjcxtkHaWwOxKSX1REQzFeOcJbAV3 C32hanZB2seAyBUZNjHDIe/4nKia6dsMhihI91tuy7Eiik0QxGVq/AsMosJ6 C607rhLYeRkMrbzQOsg0uekhp2WwSbJPYk2IFHukiCx1iypyTIZAax4Edpsq Yjeb7PhuhsgxYoE85B4pHrSKJvLpaJk2KaJejjYF9lsI08XHtHSTINCo8w1Z wObatdFOLYP2RL7X723y2Fx8QsvsB+JPCd0Httac6p4W/Fvf9fTdjiqlf2tK 13Jm2J+JKTT0V7Gqd62c056PDZWRNrK1KWXcauvmMsJaYG/cZtbiOm0QwqK2 v0SwJmRt6jtjsH7sQ6ZyEXbLvlOasi+0GDhDBU0Rf+NzhwueT/CGjfVBSsvv rVMZGxTLQn10icqfs1Ui5opdR0vINKhcVuucTpMMjcrfS62/GJXvMKhFj8sk WqARoobJy/ZpeaPAfaQZssoBYtWkrpLDOOFTMq/MN0VLbqZGyMqIo/G9Q6yO EKuMuvK5MoeJzINZxc0ZMpggBseIQZsBsctO0W46drnmo1H0X06jwVQcMxvL tQO7s97VIF4R4RNEs8tsrtyvawl6Hal7eetN+AKJGAorloiEH0zyGjmxZU6H wvcAuQf8uZHLM7M4k8ANvYPEpEdMVoqWcI5o7d0t1D76jFzJvcRxCIWH+4iZ 8fyW7XEqSa8ZNhT6VAEJTjDqfogniokXiOeUb8vCeLnKSfc4Kd6k0mc+25QY FfVPxHiWGM9EHVx5Izrh7RFHQy5wUcp3yyHnslIVdlIVZplzCFW4zqvQcT0M ViHiBvLGIO/4E63E0FKvS/WYo6bIvA1oNLjFq9CWqgCt7e3KGbaA6yCCaMnc JNLzRLpNKMGyfDymKlnMGjcYznl2MyIpjvusDiZp4LxhgfzE+RPvn3c46a6K iUZuguYjMORJoi3ByXSpJnWRzNoFXO8VuAaFIq4N75QFC3sBYYCxRPEO0nzI 1M+mDwoOi0hWaXXBppYVDhlcJ0AxnCE7FpJ9mCY7YCFFGB+BFvY1aoYZAS3e 5Gii8VPeeEj1a061J1FFPUx/dj0JUmsB/k3voZyWufdxwgTdRbHLo7SH8sq7 lkxPyHEG2STzbUKji8rb1+j0SPQxJ0XjEiJV6D54uEn1G4l55Ja2y1Nbe9My g+rrB/vN7kNn0KAxMvC22KgdqVaysk8zyeS+JmmSueogn4p/3A5zHCOk8rJi c/GWNTSnQprRTPuEu0mhTV1TR14SecAgHxqkgZ6YZjkr7ozeCWJ6+nrK6TWL iGk7ggZO5PmoCv1o52dj/oMWItYCnXQQJHMTg2TfLRJTGIqsgrCGt/Jz7pYC BW8ZbtF4uiwstVhpmC2aAzpURR+YRfuFEWWVpKyoXFWI8LKwGBZuEmiUa+42 ePn+mIJv4G6hR0a9xPBgUyNeyqBwH8bF+npRRgqPXyqODuP4EsV24khOUD4I jtoup7mWGzeyizwDhzsJ4hxZlXM0fFuXIc6mG/+eZIrmBLRqkANkkVCYxrfv AqdkwvrChy/E9kddL1oJlcyKnp0LB1Uo7U35SuaWsmPYutUM56lFu3UufGGF c/vh8zUixVQa7wdsDMviYpkvBJbmbcL9JehHry3fG9W9s4LHRpT9HdM16lq+ eLiJKlev+GB2ofxoX9ZjgB1SrjwCWuKYrFTVmcw0mzcnH6hDgoHpSwLNZAdB 0UlqHzufRNzlSHJ2cdNX8n6BJ7kNG9SCH4GUwhHWZhVCqWmpSKjQsh3byxTe +MkZrBV19X1WseiKjjZwYyaa/PFZ0EHFoXEL1Iohnb7QqYMiNVKtNd1tk1ri 5IKlhhdeFWZIMU3Y7kocO67LpKphBCaJM9QpJvXqj9J8E7lBRr+V0glPDg+q 20ROa3Em+r30dHt1m2zVvfWVOOp/zdvCqYzGFuHyrz6DUdnAlZg4HcMY8GZI bXuU2cdBzcqlrtatdzkTVwo6g1Ol1FQJeEdueKGD0OGDD7ygCnXBeYlDx6ug mK3uZyUeNELFpGojxXeqQjHSfkdZwaZF0BGf8Hkrl7WlOih81rO0BHFvLwIv feIsNWhKm8K8c5fTxyWkf3EKSih93I5O5wJ3qlRC6TOBqTrA/Vot3y9DLYwc JYAHZcAV89GBNEO6YZ8rfc5T4Ig3Xqs9eKq/Sp3F5Gmh0sdg2zGxqIEHfBCv 9lBtRCxchA2vxENipnrgh0pikXh0+SrpBsenuGSiZK8d/k6uRUuf0O4E1Y60 8xNqCfU2VB9eiQd8L3tNZKiTlTRdmVy7VDNa4ui7U7JO9D8A1i/0+0dlOi3O UF+Jx/XrayxjjcRfFLB/SSB3TPwBQqV/XrgIMMvFgz45uv+fqPlnGBdsj1Hu UB8Uepd/5qF2/AcIGrzx\ \>"],ImageRangeCache->{{{0., 227.}, {199.625, 0.}} -> {-3., -2., 0., \ 0.}},ExpressionUUID->"7e7d4f41-6256-45f8-a6c3-9212098ad406"], Cell[BoxData[ TagBox[ RowBox[{"\[SkeletonIndicator]", "Graphics", "\[SkeletonIndicator]"}], False, Editable->False]], "Output",ExpressionUUID->"c5249a7d-0792-4832-b21b-\ 9f64017beae6"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Origin", "Subsection",ExpressionUUID->"d0099207-5142-493f-b500-44e8e5548b01"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"FullSimplify", "[", RowBox[{"pedal", "/.", RowBox[{"{", RowBox[{ RowBox[{"x", "\[Rule]", "0"}], ",", RowBox[{"y", "\[Rule]", "0"}]}], "}"}]}], "]"}]], "Input",ExpressionUUID-\ >"6b3705d3-95ac-4cb2-9158-75a27de4d945"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ RowBox[{"Cos", "[", "t", "]"}], "-", RowBox[{"Cos", "[", RowBox[{"2", " ", "t"}], "]"}]}], ")"}]}], ",", RowBox[{ FractionBox["1", "6"], " ", RowBox[{"(", RowBox[{ RowBox[{"Sin", "[", "t", "]"}], "+", RowBox[{"Sin", "[", RowBox[{"2", " ", "t"}], "]"}]}], ")"}]}]}], "}"}]], "Output",Expressio\ nUUID->"a4401c1e-3cac-46bd-8b87-c79a13d8c6d8"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PedalCurvePlot", "[", RowBox[{"deltoid", ",", RowBox[{"{", RowBox[{"0", ",", "0"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", RowBox[{"2", "Pi"}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", RowBox[{"Pi", "/", "2"}], ",", ".2"}], "}"}]}], "]"}]], "Input",Expressio\ nUUID->"6e08ff9c-8977-44b2-8fc7-10265f2051a9"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.15469 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.341269 0.21164 0.577349 0.21164 [ [ 0 0 0 0 ] [ 1 1.15469 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash 0 .57735 m 1 .57735 L s .34127 0 m .34127 1.15469 L s 0 0 m 1 0 L 1 1.15469 L 0 1.15469 L closepath clip newpath 1 0 0 r .5 Mabswid .34127 .57735 m .34127 .57735 L .97619 .57735 L s .34127 .57735 m .34751 .63958 L .95105 .57903 L s .34127 .57735 m .36501 .69447 L .87859 .59036 L s .34127 .57735 m .39026 .73573 L .76731 .61909 L s .34127 .57735 m .41808 .75903 L .62999 .66944 L s .34127 .57735 m .44248 .76262 L .4819 .74108 L s .34127 .57735 m .45765 .74746 L .33859 .82891 L s .34127 .57735 m .45896 .71708 L .2138 .92357 L s 0 g [ 1 0 ] 0 setdash .34127 .57735 m .34128 .57981 L .3413 .58205 L .34135 .58464 L .34142 .58708 L .34158 .59142 L .34183 .59612 L .34217 .60128 L .34264 .60673 L .3437 .61644 L .3451 .62634 L .3467 .63548 L .35131 .65567 L .35729 .67513 L .36379 .69171 L .37241 .7093 L .38122 .72371 L .39834 .74448 L .40763 .7525 L .41274 .75603 L .41738 .75868 L .42182 .76072 L .42639 .76229 L .4287 .76286 L .43081 .76325 L .432 .76341 L .43307 .76352 L .43366 .76356 L .43429 .76359 L .43488 .76361 L .43543 .76361 L .43631 .7636 L .43725 .76355 L .43813 .76347 L .43892 .76338 L .43979 .76325 L .44072 .76307 L .44255 .7626 L .44428 .76199 L .44606 .76119 L .44904 .75936 L .45046 .75822 L .45187 .75688 L .45416 .75411 L .45625 .75064 L .45725 .74845 L .45807 .74626 L .45872 .74412 L .45924 .74198 L .45961 .73998 L Mistroke .45994 .73772 L .46008 .73639 L .46018 .73517 L .46022 .73448 L .46026 .73373 L .46029 .73301 L .4603 .73235 L .46032 .73114 L .4603 .72979 L .46029 .7291 L .46026 .72836 L .46019 .727 L .4601 .72565 L .45997 .72414 L .4596 .72102 L .45914 .71805 L .4586 .71521 L .45701 .70859 L .45472 .70118 L .44894 .68699 L .44185 .67347 L .42305 .64625 L .39575 .61723 L .36446 .59211 L .32693 .56955 L .3076 .5605 L .28641 .55237 L .26644 .54635 L .24836 .54229 L .23977 .54084 L .23073 .53964 L .22643 .5392 L .22239 .53886 L .21881 .53863 L .21695 .53853 L .21496 .53845 L .21282 .53838 L .21082 .53833 L .20967 .53831 L .20846 .5383 L .20632 .5383 L .20524 .53831 L .2041 .53833 L .20205 .53838 L .20081 .53842 L .19964 .53847 L .19748 .53857 L .19364 .53883 L .19018 .53914 L .18262 .54009 L Mistroke .17615 .54124 L .16959 .54277 L .15883 .54625 L .15302 .54878 L .14805 .55144 L .14371 .55428 L .14025 .55702 L .13731 .55985 L .13466 .56304 L .13246 .56653 L .13167 .56813 L .13095 .5699 L .13043 .57156 L .13005 .57312 L .1299 .57398 L .12979 .57477 L .12969 .57568 L .12964 .57655 L .12963 .57729 L .12964 .57808 L .12968 .57875 L .12974 .57949 L .12983 .58029 L .12997 .58114 L .1303 .58268 L .13079 .58434 L .13146 .58609 L .13304 .5892 L .13554 .5928 L .13839 .59595 L .14195 .59908 L .14645 .60227 L .15127 .60504 L .15628 .6074 L .16715 .61125 L .17356 .6129 L .18091 .61433 L .18476 .61491 L .18908 .61544 L .19327 .61584 L .19532 .61599 L .19726 .61611 L .19919 .61621 L .20129 .6163 L .20322 .61635 L .20502 .61638 L .20698 .6164 L .20915 .61639 L .21025 .61638 L .21143 .61636 L Mistroke .21358 .6163 L .21565 .61622 L .21795 .61612 L .22266 .61581 L .22708 .61543 L .23128 .61499 L .2409 .61369 L .25146 .6118 L .27122 .60705 L .28956 .60124 L .32551 .58586 L .36274 .56378 L .39429 .53881 L .42219 .5095 L .4333 .49472 L .44323 .47881 L .45001 .46538 L .4529 .45846 L .45548 .45124 L .45747 .44436 L .45883 .43833 L .45932 .43551 L .45975 .43251 L .4599 .43117 L .46004 .42976 L .46014 .42854 L .46022 .42727 L .46028 .42587 L .46031 .4246 L .46032 .42325 L .46031 .4225 L .46029 .42181 L .46024 .42056 L .46017 .41945 L .46007 .41819 L .45994 .41702 L .45957 .41444 L .45911 .41217 L .4585 .40979 L .45771 .40741 L .4561 .40377 L .45432 .40082 L .45192 .39787 L .44931 .39554 L .44664 .39381 L .44396 .39258 L .44233 .39204 L .4408 .39165 L .43993 .39148 L .43898 .39133 L Mistroke .43807 .39122 L .43721 .39115 L .43625 .3911 L .43533 .39109 L .4345 .3911 L .43358 .39115 L .43257 .39123 L .43205 .39128 L .43148 .39135 L .4295 .39167 L .42762 .39209 L .42555 .39265 L .42174 .39401 L .41706 .39619 L .41204 .39912 L .40292 .406 L .39409 .4146 L .38487 .42583 L .36948 .45092 L .36193 .46737 L .35486 .48686 L .3494 .50659 L .34555 .52566 L .34371 .53816 L .34245 .55002 L .34193 .55687 L .34159 .56316 L .34145 .56664 L .34139 .56848 L .34134 .57044 L .34131 .57214 L .34129 .57403 L .34127 .57574 L .34127 .57735 L Mfstroke [ .05 .02 ] 0 setdash .97619 .57735 m .97615 .57735 L .97605 .57735 L .97586 .57735 L .97559 .57736 L .97494 .57737 L .97397 .57739 L .97257 .57744 L .97072 .57752 L .96645 .57775 L .96079 .57815 L .95434 .57871 L .93561 .5808 L .91094 .58441 L .88377 .58933 L .80827 .60716 L .72944 .63142 L .63374 .66786 L .43554 .76737 L .26005 .88617 L .11944 1.01094 L .04337 1.09871 L .02814 1.12032 L .02445 1.12613 L .02405 1.1268 L .02385 1.12714 L .02381 1.1272 L .02381 1.1272 L .02385 1.12713 L .02393 1.12699 L .0259 1.12334 L .0304 1.11433 L .04561 1.07931 L .06443 1.02681 L .08294 .96307 L .10155 .88033 L .11004 .83185 L .11759 .77778 L .12292 .72743 L .12527 .69856 L .12697 .67201 L .12819 .64699 L .12872 .63281 L .1291 .6197 L .12935 .60821 L .12945 .60167 L .12953 .59554 L .12959 .58909 L .12962 .5832 L .12963 .57639 L Mistroke .12961 .56993 L .12957 .56295 L .12953 .55903 L .12949 .55542 L .12939 .54878 L .12926 .54178 L .12896 .52965 L .12853 .51638 L .12722 .4871 L .12555 .46005 L .1215 .4122 L .11577 .36253 L .10822 .31163 L .09865 .25969 L .07886 .17629 L .05865 .11062 L .03975 .06112 L .02761 .03468 L .02478 .02925 L .02402 .02787 L .02384 .02754 L .02381 .02749 L .02382 .02751 L .02387 .0276 L .02398 .02778 L .02428 .02829 L .02534 .03002 L .0299 .03704 L .05038 .0652 L .13312 .15754 L .26548 .27273 L .44808 .39464 L .63771 .48849 L .73525 .52525 L .81763 .55005 L .88527 .56567 L .93269 .57352 L .95089 .57566 L .96362 .57676 L .96894 .57709 L .97256 .57726 L .97396 .57731 L .9751 .57733 L .97553 .57734 L .97572 .57734 L .97587 .57735 L .97598 .57735 L .97607 .57735 L .97617 .57735 L .97619 .57735 L Mistroke Mfstroke .05 w .34127 .57735 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{173.188, 199.938}, ImageMargins->{{51, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}},ImageCache->GraphicsData["CompressedBitmap", "\<\ eJy9WwtzFEUQXm73HnmQ1yWp6JHLJQTQJIAERUiJFFCCWmhRPvCJGFKUiYrG yN/yd/iz1puZm57HfrOzHffYqr3bm+3+uqene7Zntu+z/VeHL17uvzo62B88 PNk/Pjw6+Hvw4M+TYVN6JkmSf4bnv4NEXOfDy9HHXpLneZKqH8MrcR7Lz6T2 Wxm4db1e8oa4824UMxndkiQlmE2XXJ67YfIWIL8WJu+AW1ej5ElF8ra0h7hR okLbwxTHO9VVEEdU48zReHhcCXNMYI7LMY6mw6FV24mxucYs0WsSkKeg7VIU Iv8fEPOA/GKYfAGQXwiTTwFyFACbrx0CxdX5MMR0RYgNHsQ2jxx52FoY4mxF iD4PAjnuKg8Cjcg5HgSyTy8MsQrI3wyT9wH5G2HyGUD+No98DrQt8yBQLC+F IWYrQizyINAU0eVBdMOwVSEWw51DEMj6CGKOB1Ey8W4B8rNh8u1wCCDyHUA+ FSZHwy7dXT7smYw9YpzgMaIYbfEgmI+uNVIVZYYljOvys5AiSPaTMNsGyVPP aRL3V5hn07RVIb8A2kAerW9Xzb4z55buxUtFguIVPY9TYvyNx4ge+Ic8iBbJ ZjKiHPkFD+KYRz5Fqh4oRjQToWQFPSd/5kGg58ozHoRUX/mLS/xT2NestITa MzJDNUbfWxw//1FBoMchytnMSqdT4v7f8QyDEkY0RTZIwDfjEWAs+zVPAIpD tAgxwfbleASgoHzME5UBOvSk+8HxHIcceQ5afyHYaTLR5zwBDUCHMoEZEvBo PALmSMCnMQFNP3wkzQwA7RLox4pujwmAlPjodFArpMt9G8CdCwK8Jgm66xgH zDyV8FBu9GHM7AVklGDKHEhy32bjIbc4D9j3qntgCPYSGfSmolsGYOsADAXf FvX5PR4YmouuENhuNTDdjqbmXcB7lQeL9gBvkPUu88DaAOwWdVgtYpKB+GoL mi6AQCn8bYJQSyS1+pf+JSW2cs9bNXMTgN0FfVF5lVriyx1cuVqQfGj/AqX/ D8SnTPH7ts0oCMS3OJdJdfT8TQHyI+p9z0KWND1SOCXYBoB4DEUlOsYUosSR yk+IK5muyI3QeccJHGBJ3hFNT0lLNSEld0Trc3ElQTpqYMpBnkgjCh5pSZPa TIZVkDaYEk3SUNOCTurR9ftmhjcjifdUf/PRVrtkbORkUbKYbajAoDbIH9X9 TN3Oaa33vriYFJcTil5rugrB9YkdUjAofK1xg4h3qFMzSh1xV8KirTwrmjrK /jqalK2UmkXDD0jKgpEiCC5acS6NvGDc1JiWrIMcdolMtUwuMQo763jL8Qqn a+uu/mp0/QFR5pFd6JGYlPxguwp8y5inMHKyA2uWT+tR99B2qgiaVIx6zOWv TcLWM69tn2uV9a8wIn0S1XGd93oVKSPNtJCGGU+7HXnZCsmddOXeiMg1OJYC LV+k/kZd7pLoaRV9xWkhml00bSWMgV2j52beLkwas6TEDCkho+wDJbpCGg1E F0wfkD5B0metGC9KjCozbVuWFAMDEVEoJYXmyRzabWzfv+MolBcVynyFGiY6 A9Zp2k35aP5VBu16ymSOMgnl9Sq1bfmyNbCZx1shp5wjSZlR2LX4vaAs2/jN kJUDgqfIATJS4WHMym0k3fVFLcv2gdRrEyCTJLVldzy3LB1Vp7CaUyMINKiq VcvVauQCw+OTWESgZbQJ2aIuhUGy1CGoBinRJiNJ4cxVvfdA0w6flmmS6bHV Z5uU8WZN5h5GGrVNamuUmHzHGOErnsjM7ZHTUdV9d45I7Ke7SVu/OLVUG93K wArzoRlwI5XZ1yYePU+gbramBt1k7dQ/UaJ5W5DuVJzbU7E9KfvOKAV+yxM4 cgjR5MZ2c/TpmV4Cfq+uSjaUHWtknseL42kMInPC2HR/eDzjdVEv6O2IUBTu hjkTViWYbQfWN9RzHqQfMZ7dxDcTUQVQEyqJ0hYmfGD6k6P0Cw+qUdSU3OUo BlV8qrpT4vD4VWGEXyIRBvmtrcLvMfaWPTXarNGXfIUsSh5/KLZbxnlJwOj1 5U3P+KOj5OXmRpFDfJS8cR24kZdUfu265gUdMVd/SdwHEMz3zKsAAm2Bxd+q OxHEfKvesyD0gXYZSyDUTo1oQVuosTKECm+EXAjn1gqAKKlpYJZa6DofbRzx C+3gl5SJLFkQum2WB7FI9mVWuaCwLoEYU7lODUVDNZQulY0NgBhTGVdJ4Vif Rrmkdo1Z6nYOkJcU0o2pnK+GokJmaWMNBZY1lHT61dLi9F88RSCYdag1VMPW AMEsnWHWHZe84nHaSsrFaijPZhaEn7LsHJVCoMx4KwaBJZfUyfNr8duYI/aX g+KL5xIOptWZf4Fg/h+DSa7WT+JGff8gQX5f319kXueffZIz/wFI0z72\ \>"],ImageRangeCache->{{{0., 172.188}, {198.938, 0.}} -> {-1., -2., 0., \ 0.}},ExpressionUUID->"e88b446a-1b48-425d-a057-dd4ac5a1e7f3"], Cell[BoxData[ TagBox[ RowBox[{"\[SkeletonIndicator]", "Graphics", "\[SkeletonIndicator]"}], False, Editable->False]], "Output",ExpressionUUID->"3ff8a085-4ed5-4b03-b5c7-\ 58f3a22366db"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] }, WindowSize->{640, 750}, WindowMargins->{{460, Automatic}, {Automatic, 4}}, FrontEndVersion->"11.1 for Mac OS X x86 (32-bit, 64-bit Kernel) (June 2, \ 2017)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[580, 22, 91, 0, 92, "Title", "ExpressionUUID" -> \ "3a093c23-21b1-42fe-be59-2e04ec1ed63f"], Cell[CellGroupData[{ Cell[696, 26, 83, 0, 44, "Subsection", "ExpressionUUID" -> \ "8dd131b5-72a0-4860-9944-38ec753a56e6"], Cell[782, 28, 111, 3, 49, "Text", "ExpressionUUID" -> \ "bf5b769f-cb56-4e6d-a2a1-fe2e9ecbc0c6"], Cell[896, 33, 346, 9, 49, "Text", "ExpressionUUID" -> \ "c054bfdf-349a-44f1-b105-2f99155186e4"], Cell[1245, 44, 364, 10, 49, "Text", "ExpressionUUID" -> \ "6af2bf5a-0bfc-4ccf-b1cc-99f48f5ff884"], Cell[1612, 56, 154, 2, 30, "Text", "ExpressionUUID" -> \ "115309c7-3e85-4526-be42-a1cf9e01c38d"] }, Open ]], Cell[CellGroupData[{ Cell[1803, 63, 78, 0, 64, "Section", "ExpressionUUID" -> \ "39775dc4-969e-4e9c-891f-a8271effbe61"], Cell[1884, 65, 639, 21, 32, "Input", "ExpressionUUID" -> \ "4bf43b1a-4359-46d2-aa19-4b1306eeb764"], Cell[CellGroupData[{ Cell[2548, 90, 281, 7, 32, "Input", "ExpressionUUID" -> \ "06fefbe0-d424-4ef0-ab2b-a26cd3b8cffb"], Cell[2832, 99, 920, 27, 88, "Output", "ExpressionUUID" -> \ "feabd876-028d-454d-8c17-55727153d517"] }, Open ]], Cell[CellGroupData[{ Cell[3789, 131, 81, 0, 44, "Subsection", "ExpressionUUID" -> \ "b200d088-c8e3-4e15-8e58-384b386f2b7c"], Cell[CellGroupData[{ Cell[3895, 135, 142, 3, 32, "Input", "ExpressionUUID" -> \ "be14451f-380b-415e-934b-636d8934a683"], Cell[4040, 140, 128, 2, 32, "Output", "ExpressionUUID" -> \ "4de7ccff-1a24-4b43-b37d-e2a673d66acd"] }, Open ]], Cell[CellGroupData[{ Cell[4205, 147, 263, 7, 32, "Input", "ExpressionUUID" -> \ "c087fdd1-dff0-4abe-9b14-0abdd296abd2"], Cell[4471, 156, 607, 20, 49, "Output", "ExpressionUUID" -> \ "8f89d18e-6e84-49f6-84bd-b3aae729d99b"] }, Open ]], Cell[CellGroupData[{ Cell[5115, 181, 364, 10, 32, "Input", "ExpressionUUID" -> \ "b4b26b37-90c9-4b2a-b289-1f919f4bb845"], Cell[5482, 193, 10679, 456, 208, 5512, 367, "GraphicsData", "PostScript", \ "Graphics", "ExpressionUUID" -> "b6018255-1a10-4e69-9609-f8b858ce2801", Evaluatable->False], Cell[16164, 651, 191, 5, 70, "Output", "ExpressionUUID" -> \ "a45dc658-d7ba-41d4-81de-ae0ab54d45f8"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[16404, 662, 224, 6, 70, "Subsection", "ExpressionUUID" -> \ "419614f5-ceaf-4711-b2ad-192751ba47f5"], Cell[CellGroupData[{ Cell[16653, 672, 146, 3, 70, "Input", "ExpressionUUID" -> \ "3cb9a3a0-192c-4a70-96fe-4e53b042eecc"], Cell[16802, 677, 172, 5, 70, "Output", "ExpressionUUID" -> \ "30382caf-87f1-4e90-a75a-e701fa2a5293"] }, Open ]], Cell[CellGroupData[{ Cell[17011, 687, 313, 9, 70, "Input", "ExpressionUUID" -> \ "8f5777b2-d5ed-4f4d-8582-0a4d80ed9b86"], Cell[17327, 698, 431, 14, 70, "Output", "ExpressionUUID" -> \ "fcd36fc3-1616-42ef-9b69-1e8b99457a13"] }, Open ]], Cell[CellGroupData[{ Cell[17795, 717, 579, 18, 70, "Input", "ExpressionUUID" -> \ "e7b0c491-d770-43ec-b225-4dfe4edf5d5b"], Cell[18377, 737, 9764, 475, 70, 6043, 410, "GraphicsData", "PostScript", \ "Graphics", "ExpressionUUID" -> "7e7d4f41-6256-45f8-a6c3-9212098ad406", Evaluatable->False], Cell[28144, 1214, 191, 5, 70, "Output", "ExpressionUUID" -> \ "c5249a7d-0792-4832-b21b-9f64017beae6"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[28384, 1225, 83, 0, 70, "Subsection", "ExpressionUUID" -> \ "d0099207-5142-493f-b500-44e8e5548b01"], Cell[CellGroupData[{ Cell[28492, 1229, 263, 7, 70, "Input", "ExpressionUUID" -> \ "6b3705d3-95ac-4cb2-9158-75a27de4d945"], Cell[28758, 1238, 518, 17, 70, "Output", "ExpressionUUID" -> \ "a4401c1e-3cac-46bd-8b87-c79a13d8c6d8"] }, Open ]], Cell[CellGroupData[{ Cell[29313, 1260, 390, 11, 70, "Input", "ExpressionUUID" -> \ "6e08ff9c-8977-44b2-8fc7-10265f2051a9"], Cell[29706, 1273, 9250, 472, 70, 6528, 426, "GraphicsData", "PostScript", \ "Graphics", "ExpressionUUID" -> "e88b446a-1b48-425d-a057-dd4ac5a1e7f3"], Cell[38959, 1747, 191, 5, 70, "Output", "ExpressionUUID" -> \ "3ff8a085-4ed5-4b03-b5c7-58f3a22366db"] }, Open ]] }, Open ]] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)