(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; showRuler; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] Inversion of Elementary Algebraic Curves with Respect to a Circle :[font = subtitle; inactive; preserveAspect] by Stefan Welke :[font = subsubtitle; inactive; preserveAspect] Mathematica in Education and Research Volume 4 Number 2 Copyright 1995 TELOS/Springer-Verlag :[font = section; inactive; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] Introduction :[font = text; inactive; preserveAspect] This notebook contains the code used in the article. Evaluate the complete notebook or evaluate step by step. :[font = postscript; PostScript; formatAsPostScript; output; inactive; locked; preserveAspect; pictureLeft = 48; pictureWidth = 284; pictureHeight = 142; endGroup] %! %%Creator: Mathematica %%AspectRatio: .5 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.261905 0.238095 0.25 0.238095 [ [(Figure 1)] .5 .5 0 -2 Msboxa [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p [(Figure 1)] .5 .5 0 -2 Mshowa P 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath p .004 w .2619 .25 m .97619 .34524 L s newpath .2619 .25 .2381 0 365.73 arc s p p .015 w .2619 .25 Mdot .97619 .34524 Mdot .33988 .2604 Mdot P P [(M)] .2619 .25 0 1 Mshowa [(P)] .97619 .34524 0 1 Mshowa [(P')] .33988 .2604 0 -1 Mshowa P % End of Graphics MathPictureEnd :[font = section; inactive; Cclosed; pageBreak; preserveAspect; startGroup] Inversion of algebnraic curves :[font = input; pageBreak; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] inversion[{x_, y_}]:={x,y}/(x^2+y^2) ; inversion[{2,0}] :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] parabola:=x^2 - y ; invparabola:=parabola/.Thread[{x,y}->inversion[{x,y}]] ; invparabola :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] invparabola':=invparabola //Together//Numerator ; invparabola' :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; endGroup] <{list1,{"Curve", "Image curve"}} ] :[font = section; inactive; Cclosed; pageBreak; preserveAspect; startGroup] Anticonformality and invariance of the set of generalized circles :[font = input; pageBreak; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] JacobiMatrix[expr_,vars_]:=D[expr,#]& /@ vars ; DInv:=JacobiMatrix[inversion[{x,y}],{x,y}]//Together ; DInv//MatrixForm :[font = input; preserveAspect] M:=DInv/.Thread[{x,y}->r*{Cos[phi],Sin[phi]}] //Simplify ; M //MatrixForm :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] (M.{a,b}).(M.{c,d}) //Simplify :[font = input; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] Det[M]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] -r^(-4) ;[o] -4 -r :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] gc:=a*(x^2+y^2)+b*x+c*y+d :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; endGroup] invgc:=InversionAlgebraic[ gc, {x,y} ] ; invgc :[font = section; inactive; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] Further examples :[font = subsection; inactive; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] The strophoid :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] strophoid:=y^2*(x-a)+x^2*(a+x) InversionAlgebraic[strophoid, {x,y}] :[font = input; preserveAspect; endGroup] a:=1 ; InversionPlot[ strophoid, {x,y}, {-3,3} ] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] The limacon of Pascal :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] InversionAlgebraic[ (x^2+y^2-b*x)^2-c^2*(x^2+y^2), {x,y} ] :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; endGroup; endGroup] <{-3.5,3.5}, Axes->False, PlotStyle->{GrayLevel[.5],Dashing[{.015}],GrayLevel[0]}, AspectRatio->Automatic ]& /@ params ] ; Show[GraphicsArray[plt6], PlotLabel->"Figure 6" ] :[font = section; inactive; Cclosed; pageBreak; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] Cassinian curves :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] cassini[a_,c_]:=(x^2+y^2)^2-2*c^2*(x^2-y^2)-a^4+c^4 ; params3:={3. , 2.35 , 2. , 1.6 , 1.2} ; plt8:=Block[{$DisplayFunction=Identity}, ImplicitPlot[ cassini[#,2]==0,{x,-4,4}, Axes->False,AspectRatio->Automatic ]& /@ params3 ] ; Show[ plt8, PlotLabel->"Figure 8" ] :[font = input; pageBreak; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] cassini:=(x^2+y^2)^2-2*c^2*(x^2-y^2)+d ; invcassini:=InversionAlgebraic[ cassini, {x,y} ] ; invcassini :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] { cassini, invcassini }/.{d->0} :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] invcassini/d //Expand :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] ?InversionPlot :[font = input; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; endGroup] c:=1 ; d:=-1 ; InversionPlot[ cassini, {x,y}, {-1.6 , 1.6} ] :[font = section; inactive; Cclosed; preserveAspect; startGroup] Figures and Tables :[font = text; inactive; preserveAspect] This notebook contains all the code for the figures in the article "Inversion of elementary algebraic curves with respect to a circle". The code is independent of the code presented in the article. I made no efforts to produce elegant code. :[font = subsection; inactive; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] How to use :[font = subsubsection; inactive; preserveAspect; endGroup] (1) Open the notebook, all cells with code are initialization cells (2) Call fig1, fig2, ... , fig9 for the figures and table1 for the table with the expressions for the conic sections. :[font = subsection; inactive; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] General requirements :[font = input; initialization; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] *) Needs["Graphics`ImplicitPlot`"] Needs["Graphics`Graphics`"] Needs["Graphics`Arrow`"] (* :[font = input; initialization; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; endGroup] *) inversion[expr_]:= (expr/.{x->x/(x^2+y^2), y->y/(x^2+y^2)})//Together//Numerator r := (2Cos[#1] + #2)&; (* :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Figure 1 :[font = input; initialization; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] *) M := {0,0}; P := {3.,.4}; P':= P/P.P; Q := {-.4,.2}; Q':= Q/Q.Q; uc := Circle[{0,0}, 1]; lP := Line[{M, P}] ; plist := Point/@{M, P, P'}; txt := Graphics/@{Text[FontForm["M",{"Times-Italic",10}], M-.15], Text[FontForm["P",{"Times-Italic",10}], P-.15], Text[FontForm["P'",{"Times-Italic",10}], P'+.15]}; obj1 := Graphics /@ {lP, uc, {PointSize[.015], plist}} ; Show[{obj1, txt}, AspectRatio->Automatic] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 240; pictureHeight = 120] %! %%Creator: Mathematica %%AspectRatio: .5 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.261905 0.238095 0.25 0.238095 [ [ 0 0 0 0 ] [ 1 .5 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 .5 L 0 .5 L closepath clip newpath p .004 w .2619 .25 m .97619 .34524 L s newpath .2619 .25 .2381 0 365.72958 arc s p p .015 w .2619 .25 Mdot .97619 .34524 Mdot .33988 .2604 Mdot P P p /Times-Italic findfont 10 scalefont setfont [(M)] .22619 .21429 0 0 Mshowa P p /Times-Italic findfont 10 scalefont setfont [(P)] .94048 .30952 0 0 Mshowa P p /Times-Italic findfont 10 scalefont setfont [(P')] .3756 .29611 0 0 Mshowa P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Figure 2 :[font = input; initialization; Cclosed; preserveAspect; startGroup] *) parabola := x^2 - y invparabola := parabola//inversion ImplicitPlot[invparabola==0, {x,-2,2}, AspectRatio -> Automatic] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 240; pictureHeight = 50] %! %%Creator: Mathematica %%AspectRatio: .21193 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.5 0.238095 0.005046 0.238095 [ [(-2)] .02381 .00505 0 2 Msboxa [(-1)] .2619 .00505 0 2 Msboxa [(1)] .7381 .00505 0 2 Msboxa [(2)] .97619 .00505 0 2 Msboxa [(0.2)] .4875 .05266 1 0 Msboxa [(0.4)] .4875 .10028 1 0 Msboxa [(0.6)] .4875 .1479 1 0 Msboxa [(0.8)] .4875 .19552 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .21293 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .00505 m .02381 .0113 L s P [(-2)] .02381 .00505 0 2 Mshowa p .002 w .2619 .00505 m .2619 .0113 L s P [(-1)] .2619 .00505 0 2 Mshowa p .002 w .7381 .00505 m .7381 .0113 L s P [(1)] .7381 .00505 0 2 Mshowa p .002 w .97619 .00505 m .97619 .0113 L s P [(2)] .97619 .00505 0 2 Mshowa p .001 w .07143 .00505 m .07143 .0088 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{"1.","2.","3.","4.","5.","6.","7."} list2 := { -2 + (-2+x)^2 + (1+y)^2, -1 + x^2 + (-1+y)^2 , 1 + 2*x - y , x + 3*y , -x/2 + y^2 , -1 + x^2 - y^2 , -1 + x^2/4 + 4*y^2/9} TableForm[{#,#//inversion}& /@ list2, TableHeadings->{list1,{"Curve", "Image curve"}} ] (* :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup; endGroup] TableForm[{{-2 + (-2 + x)^2 + (1 + y)^2, 1 - 4*x + 3*x^2 + 2*y + 3*y^2}, {-1 + x^2 + (-1 + y)^2, 1 - 2*y}, {1 + 2*x - y, 2*x + x^2 - y + y^2}, {x + 3*y, x + 3*y}, {-x/2 + y^2, -x^3 + 2*y^2 - x*y^2}, {-1 + x^2 - y^2, x^2 - x^4 - y^2 - 2*x^2*y^2 - y^4}, {-1 + x^2/4 + (4*y^2)/9, 9*x^2 - 36*x^4 + 16*y^2 - 72*x^2*y^2 - 36*y^4}}, TableHeadings -> {{"1.", "2.", "3.", "4.", "5.", "6.", "7."}, {"Curve", "Image curve"}}] ;[o] Curve Image curve 2 2 -2 + (-2 + x) + (1 + y) 2 2 1 - 4 x + 3 x + 2 y + 3 y 2 2 -1 + x + (-1 + y) 1 - 2 y 1 + 2 x - y 2 2 2 x + x - y + y x + 3 y x + 3 y -x 2 -- + y 1. 2 2. 3 2 2 -x + 2 y - x y 3. 2 2 -1 + x - y 4. 2 4 2 2 2 4 x - x - y - 2 x y - y 5. 2 2 x 4 y -1 + -- + ---- 6. 4 9 2 4 2 2 2 4 7. 9 x - 36 x + 16 y - 72 x y - 36 y :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Figure 3 :[font = input; initialization; preserveAspect] *) ivPlot[expr_, {a_,b_}, l_:None]:= Module[{}, ImplicitPlot[{x^2+y^2==1, expr==0, inversion[expr]==0}, {x,a,b}, DisplayFunction->Identity, AspectRatio->Automatic, Axes->False, PlotStyle-> {{GrayLevel[.5]}, {Thickness[.007],Dashing[{.015}],GrayLevel[0]}, {Thickness[.007],GrayLevel[0]} }, Epilog->{{GrayLevel[.5],PointSize[.01],Point[{0,0}]}}]] ; (* :[font = input; initialization; preserveAspect] *) ga3:={Table[ivPlot[list2[[k]],{-3,3},k], {k,1,4}], Table[ivPlot[list2[[k]],{-3,3},k], {k,5,7}]} (* :[font = input; initialization; Cclosed; preserveAspect; startGroup] *) Show[GraphicsArray[ga3]] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 240; pictureHeight = 100] %! %%Creator: Mathematica 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.65737 .85218 L .65873 .83152 L s P P p P P p p p [ .015 ] 0 setdash .007 w .02381 .03781 m .02788 .04596 L .04004 .07026 L .06006 .11031 L .08761 .1654 L .12221 .23462 L .16328 .31676 L .21011 .41042 L .2619 .514 L .31777 .62573 L .37675 .7437 L .43784 .86588 L .5 .99019 L .56216 1.1145 L .62325 1.23669 L .68223 1.35465 L .7381 1.46638 L .78989 1.56996 L s .79892 1.58803 m .78989 1.56996 L s s s s s s s s P P P p p P p p .007 w .1638 .91077 m .16532 .88766 L .16985 .86489 L .17731 .84291 L .18758 .82209 L .20048 .80279 L .21578 .78534 L .23324 .77003 L .25254 .75714 L .27336 .74687 L .29534 .73941 L .31811 .73488 L .34127 .73336 L .36443 .73488 L .3872 .73941 L .40918 .74687 L .43 .75714 L .4493 .77003 L .46676 .78534 L .48206 .80279 L .49496 .82209 L .50523 .84291 L .51269 .86489 L .51722 .88766 L .51874 .91077 L s P p .007 w .1638 .91088 m .16532 .93399 L .16985 .95676 L .17731 .97874 L .18758 .99956 L .20048 1.01886 L .21578 1.03631 L .23324 1.05162 L .25254 1.06452 L .27336 1.07478 L .29534 1.08224 L .31811 1.08677 L .34127 1.08829 L .36443 1.08677 L .3872 1.08224 L .40918 1.07478 L .43 1.06452 L .4493 1.05162 L .46676 1.03631 L .48206 1.01886 L .49496 .99956 L .50523 .97874 L .51269 .95676 L .51722 .93399 L .51874 .91088 L s P P p P P P p .5 g .01 w .5 .83146 Mdot P MathSubEnd P % End of sub-graphic % Start of sub-graphic p 0.754707 0.217823 0.97619 0.406823 MathSubStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.5 0.15873 0.166667 0.15873 [ [ 0 0 0 0 ] [ 1 .33333 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash p P 0 0 m 1 0 L 1 .33333 L 0 .33333 L closepath clip newpath p p p P p p .5 g .004 w .34127 .16661 m .34263 .14595 L .34668 .12558 L .35335 .10592 L .36254 .0873 L .37407 .07004 L .38776 .05443 L .40337 .04074 L .42063 .0292 L .43926 .02002 L .45892 .01335 L .47928 .00929 L .5 .00794 L .52072 .00929 L .54108 .01335 L .56074 .02002 L .57937 .0292 L .59663 .04074 L .61224 .05443 L .62593 .07004 L .63746 .0873 L .64665 .10592 L .65332 .12558 L .65737 .14595 L .65873 .16661 L s P p .5 g .004 w .34127 .16672 m .34263 .18739 L .34668 .20775 L .35335 .22741 L .36254 .24603 L .37407 .2633 L .38776 .27891 L .40337 .2926 L .42063 .30413 L .43926 .31331 L .45892 .31999 L .47928 .32404 L .5 .3254 L .52072 .32404 L .54108 .31999 L .56074 .31331 L .57937 .30413 L .59663 .2926 L .61224 .27891 L .62593 .2633 L .63746 .24603 L .64665 .22741 L .65332 .20775 L .65737 .18739 L .65873 .16672 L s P P p P P p p p [ .015 ] 0 setdash .007 w .02381 .3254 m .02788 .32404 L .04004 .31999 L .06006 .31331 L .08761 .30413 L .12221 .2926 L .16328 .27891 L .21011 .2633 L .2619 .24603 L .31777 .22741 L .37675 .20775 L .43784 .18739 L .5 .16667 L .56216 .14595 L .62325 .12558 L .68223 .10592 L .7381 .0873 L .78989 .07004 L .83672 .05443 L .87779 .04074 L .91239 .0292 L .93994 .02002 L .95996 .01335 L .97212 .00929 L .97619 .00794 L s P P P p p p .007 w .02381 .3254 m .02788 .32404 L .04004 .31999 L .06006 .31331 L .08761 .30413 L .12221 .2926 L .16328 .27891 L .21011 .2633 L .2619 .24603 L .31777 .22741 L .37675 .20775 L .43784 .18739 L .5 .16667 L .56216 .14595 L .62325 .12558 L .68223 .10592 L .7381 .0873 L .78989 .07004 L .83672 .05443 L .87779 .04074 L .91239 .0292 L .93994 .02002 L .95996 .01335 L .97212 .00929 L .97619 .00794 L s P P P P p .5 g .01 w .5 .16667 Mdot P MathSubEnd P % End of sub-graphic P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] GraphicsArray["<<>>"] ;[o] -GraphicsArray- :[font = subsection; inactive; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] Figure 4 :[font = input; initialization; preserveAspect] *) P := {Cos[#],Sin[#]}& @ .7 orth := {-#[[2]],#[[1]]}& M2 := .9*P+.6*(P//orth) d[A_,B_] := (A-B).(A-B) //Sqrt c1 := Circle[{0,0}, 1, {-.3,3.4}] c2 := Circle[M2, d[P,M2]] v1 := Arrow[P, P+(P-M2//orth)] v2 := Arrow[P, P-(P-M2//orth)] v3 := Arrow[P, P+(.6*P//orth)] (* :[font = input; initialization; preserveAspect] *) txt4:={Text[FontForm["v",{"Times-Italic",10}],P+(P-M2//orth)+{0,.1}], Text[FontForm["w",{"Times-Italic",10}],P+(.6*P//orth)+{0,.1}], Text[FontForm["v*",{"Times-Italic",10}],P-(P-M2//orth)-{0,.13}], Text[FontForm["P",{"Times-Italic",10}],P+{.1,0}], Text[FontForm["Unit circle",{"Times-Bold",10}],{1.25,0}], Text[FontForm["Invariant circle",{"Times-Bold",10}],1.65*M2]} (* :[font = input; initialization; Cclosed; preserveAspect; startGroup] *) Show[Graphics[{{Dashing[{.015}],c1},c2,v1,v2,v3,txt4}], AspectRatio->Automatic, Axes->False, PlotRange->All ] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 240; pictureHeight = 214] %! %%Creator: Mathematica %%AspectRatio: .89306 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.44709 0.42328 0.146351 0.42328 [ p /Times-Italic findfont 10 scalefont setfont [(v)] .93781 .65735 0 0 Msboxa P p /Times-Italic findfont 10 scalefont setfont [(w)] .60722 .65561 0 0 Msboxa P p /Times-Italic findfont 10 scalefont setfont [(v*)] .60386 .16802 0 0 Msboxa P p /Times-Italic findfont 10 scalefont setfont [(P)] .81316 .41904 0 0 Msboxa P p /Times-Bold findfont 10 scalefont setfont [(Unit circle)] .97619 .14635 0 0 Msboxa P p /Times-Bold findfont 10 scalefont setfont [(Invariant circle)] .65789 .87179 0 0 Msboxa P [ 0 0 0 0 ] [ 1 .89306 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P % Start of user PostScript /mathtops { gsave MBeginOrig moveto MEndOrig currentpoint grestore } bind def /MAtocoords { mathtops 4 2 roll mathtops 4 copy pop pop 3 -1 roll sub /arry exch def exch sub /arrx exch def arrx dup mul arry dup mul add sqrt /arrl exch def translate } bind def /MAarrowhead1 { gsave MAtocoords [ arrx arrl div arry arrl div -1 arry mul arrl div arrx arrl div 0 0 ] concat -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto fill -0.05 0.0125 moveto 0 0 lineto -0.05 -0.0125 lineto -0.05 0.0125 lineto stroke grestore } def % End of user PostScript p p [ .015 ] 0 setdash .004 w newpath .44709 .14635 .42328 -17.18873 194.80565 arc s P .004 w newpath .57485 .58601 .25747 0 365.72958 arc s p .77083 .41904 m .93781 .61502 L s % Start of user PostScript 0.764842 0.644218 1.15933 1.10723 MAarrowhead1 % End of user PostScript P p .77083 .41904 m .60386 .22305 L s % Start of user PostScript 0.764842 0.644218 0.370359 0.181203 MAarrowhead1 % End of user PostScript P p .77083 .41904 m .60722 .61328 L s % Start of user PostScript 0.764842 0.644218 0.378312 1.10312 MAarrowhead1 % End of user PostScript P p p /Times-Italic findfont 10 scalefont setfont [(v)] .93781 .65735 0 0 Mshowa P p /Times-Italic findfont 10 scalefont setfont [(w)] .60722 .65561 0 0 Mshowa P p /Times-Italic findfont 10 scalefont setfont [(v*)] .60386 .16802 0 0 Mshowa P p /Times-Italic findfont 10 scalefont setfont [(P)] .81316 .41904 0 0 Mshowa P p /Times-Bold findfont 10 scalefont setfont [(Unit circle)] .97619 .14635 0 0 Mshowa P p /Times-Bold findfont 10 scalefont setfont [(Invariant circle)] .65789 .87179 0 0 Mshowa P P P 0 0 m 1 0 L 1 .89306 L 0 .89306 L closepath clip newpath % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] Figure 5 :[font = input; initialization; Cclosed; preserveAspect; startGroup] *) strophoid := y^2 (x-1) + x^2 (x+1) Show[ivPlot[strophoid, {-3,3}], DisplayFunction->$DisplayFunction] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 182; pictureHeight = 287] %! %%Creator: Mathematica %%AspectRatio: 1.57439 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.5 0.15873 0.787197 0.15873 [ [ 0 0 0 0 ] [ 1 1.57439 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 1.57439 L 0 1.57439 L closepath clip newpath p p p P p p .5 g .004 w .34127 .78714 m .34263 .76648 L .34668 .74612 L .35335 .72645 L .36254 .70783 L .37407 .69057 L .38776 .67496 L .40337 .66127 L .42063 .64973 L .43926 .64055 L .45892 .63388 L .47928 .62983 L .5 .62847 L .52072 .62983 L .54108 .63388 L .56074 .64055 L .57937 .64973 L .59663 .66127 L .61224 .67496 L .62593 .69057 L .63746 .70783 L .64665 .72645 L .65332 .74612 L .65737 .76648 L .65873 .78714 L s P p .5 g .004 w .34127 .78725 m .34263 .80792 L .34668 .82828 L .35335 .84794 L .36254 .86656 L .37407 .88383 L .38776 .89944 L .40337 .91313 L .42063 .92466 L .43926 .93385 L .45892 .94052 L .47928 .94457 L .5 .94593 L .52072 .94457 L .54108 .94052 L .56074 .93385 L .57937 .92466 L .59663 .91313 L .61224 .89944 L .62593 .88383 L .63746 .86656 L .64665 .84794 L .65332 .82828 L .65737 .80792 L .65873 .78725 L s P P p P P p p P p p [ .015 ] 0 setdash .007 w .34127 .78717 m .34195 .77988 L .34397 .77274 L .34731 .76593 L .3519 .75963 L .35767 .75398 L .36452 .74911 L .37232 .74516 L .38095 .7422 L .39026 .74032 L .40009 .73955 L .41028 .73991 L .42063 .74138 L .43099 .74388 L .44118 .74733 L .45101 .75159 L .46032 .75646 L .46895 .76173 L .47675 .76714 L .4836 .77241 L .48937 .77725 L .49396 .78138 L .4973 .78454 L .49932 .78652 L .5 .7872 L s P p [ .015 ] 0 setdash .007 w .34127 .78722 m .34195 .79451 L .34397 .80166 L .34731 .80846 L .3519 .81477 L .35767 .82042 L .36452 .82528 L .37232 .82924 L .38095 .83219 L .39026 .83408 L .40009 .83484 L .41028 .83448 L .42063 .83302 L .43099 .83051 L .44118 .82706 L .45101 .82281 L .46032 .81794 L .46895 .81267 L .47675 .80726 L .4836 .80198 L .48937 .79714 L .49396 .79301 L .4973 .78986 L .49932 .78787 L .5 .7872 L s P P p p [ .015 ] 0 setdash .007 w .5 .7872 m .50204 .78513 L .50811 .77866 L .51812 .76687 L .5319 .74809 L .5492 .71941 L .56974 .67546 L .59315 .60463 L .61905 .47223 L .64698 .03749 L s P p [ .015 ] 0 setdash .007 w .5 .7872 m .50204 .78926 L .50811 .79574 L .51812 .80752 L .5319 .8263 L .5492 .85499 L .56974 .89893 L .59315 .96976 L .61905 1.10217 L .64698 1.53691 L s P P P p p p .007 w .02381 .39839 m .02517 .39978 L .02922 .40391 L .03589 .41073 L .04508 .42012 L .05661 .43193 L .0703 .44597 L .08591 .46202 L .10317 .47982 L .1218 .49909 L .14146 .51954 L .16182 .54085 L .18254 .56272 L .20326 .58483 L .22362 .60688 L .24328 .62859 L .2619 .64973 L .27917 .67009 L .29478 .68952 L .30847 .70794 L .32 .72533 L .32919 .74177 L .33586 .7574 L .33991 .77245 L .34127 .78716 L s P p .007 w .02381 1.17601 m .02517 1.17462 L .02922 1.17048 L .03589 1.16367 L .04508 1.15428 L .05661 1.14246 L .0703 1.12842 L .08591 1.11238 L .10317 1.09458 L .1218 1.0753 L .14146 1.05486 L .16182 1.03354 L .18254 1.01168 L .20326 .98957 L .22362 .96752 L .24328 .9458 L .2619 .92466 L .27917 .9043 L .29478 .88488 L .30847 .86646 L .32 .84907 L .32919 .83263 L .33586 .81699 L .33991 .80194 L .34127 .78724 L s P P p P p p .007 w .5 .78716 m .50204 .7691 L .50811 .75041 L .51812 .73058 L .5319 .70922 L .5492 .68605 L .56974 .66097 L .59315 .63402 L .61905 .60535 L .64698 .57522 L .67647 .54398 L .70702 .51203 L .7381 .47982 L .76917 .44782 L .79972 .41651 L .82921 .3864 L .85714 .35797 L .88304 .33166 L .90645 .3079 L .92699 .2871 L .94429 .26959 L .95807 .25566 L .96808 .24554 L .97415 .2394 L .97619 .23734 L s P p .007 w .5 .78724 m .50204 .80529 L .50811 .82399 L .51812 .84381 L .5319 .86518 L .5492 .88834 L .56974 .91342 L .59315 .94037 L .61905 .96905 L .64698 .99917 L .67647 1.03041 L .70702 1.06236 L .7381 1.09458 L .76917 1.12658 L .79972 1.15788 L .82921 1.18799 L .85714 1.21643 L .88304 1.24274 L .90645 1.26649 L .92699 1.28729 L .94429 1.30481 L .95807 1.31874 L .96808 1.32886 L .97415 1.335 L .97619 1.33705 L s P P P P p .5 g .01 w .5 .7872 Mdot P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; initialization; Cclosed; preserveAspect; startGroup] Figure 6 :[font = input; initialization; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] *) eqs := {1, #, 1/#}& params := {1.1, 2., 2.7} plt6 = PolarPlot[eqs[r[t,#]]//Evaluate, {t,0,2*Pi}, PlotRange->{-3.5,3.5}, Axes->False, PlotStyle->{GrayLevel[.5],Dashing[{.015}], GrayLevel[0]}, AspectRatio->Automatic, DisplayFunction->Identity]& /@ params (* :[font = message; inactive; preserveAspect] 1 Power::infy: Infinite expression -- encountered. 0. :[font = message; inactive; preserveAspect] 1 Power::infy: Infinite expression -- encountered. 0. :[font = message; inactive; preserveAspect] ParametricPlot::pptr: {<<2>>} does not evaluate to a pair of real numbers at t = 3.14159. :[font = output; output; inactive; preserveAspect; endGroup] {Graphics["<<>>"], Graphics["<<>>"], Graphics["<<>>"]} ;[o] {-Graphics-, -Graphics-, -Graphics-} :[font = input; initialization; Cclosed; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280; startGroup] *) Show[GraphicsArray[plt6]] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 104] %! %%Creator: Mathematica %%AspectRatio: .37226 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.02381 0.31746 0.008863 0.31746 [ [ 0 0 0 0 ] [ 1 .37226 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 .37226 L 0 .37226 L closepath clip newpath p p % Start of sub-graphic p 0.02381 0.008863 0.321429 0.363392 MathSubStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.406148 0.170174 0.595609 0.170174 [ [ 0 0 0 0 ] [ 1 1.19122 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash p P 0 0 m 1 0 L 1 1.19122 L 0 1.19122 L closepath clip newpath p p .5 g p .004 w .57632 .59561 m .57632 .597 L .5763 .59839 L .57627 .59978 L .57623 .60118 L .57618 .60257 L .57612 .60396 L .57596 .60674 L .57575 .60952 L .5755 .61229 L .57487 .61782 L .57405 .62333 L .57305 .62881 L .57052 .63965 L .56729 .65031 L .56337 .66073 L .55352 .6807 L .54116 .6992 L .52648 .71594 L .50974 .73062 L .49123 .74298 L .48141 .74823 L .47127 .75283 L .46085 .75675 L .45019 .75998 L .43935 .76251 L .43387 .76351 L .42836 .76433 L .42283 .76496 L .42005 .76521 L .41728 .76542 L .4145 .76558 L .41311 .76564 L .41172 .76569 L .41032 .76573 L .40893 .76576 L .40754 .76578 L .40615 .76578 L .40476 .76578 L .40336 .76576 L .40197 .76573 L .40058 .76569 L .3978 .76558 L .39502 .76542 L .39224 .76521 L .38947 .76496 L .38394 .76433 L .37843 .76351 L .37295 .76251 L .3621 .75998 L 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sub-graphic P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] GraphicsArray["<<>>"] ;[o] -GraphicsArray- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Figure 7 :[font = input; initialization; preserveAspect] *) params2 := {1.1, 1.4, 1.7, 2., 2.3, 2.6} layer1 := PolarPlot[Evaluate[r[t,#]& /@ params2], {t, 0, 2*Pi}, AspectRatio->Automatic, DisplayFunction->Identity] (* :[font = input; initialization; preserveAspect] *) layer2 := PolarPlot[Evaluate[(1/r[t,#])& /c@ Take[params2,-2]], {t, 0, 2*Pi},AspectRatio->Automatic, DisplayFunction->Identity] (* :[font = input; initialization; preserveAspect] *) epsi1 := .1 layer3 := PolarPlot[Evaluate[(1/r[t,#])& @ params2[[4]] ], {t, -Pi+epsi1, Pi-epsi1}, AspectRatio->Automatic, DisplayFunction->Identity] (* :[font = input; initialization; preserveAspect] *) limacons := ((x^2+y^2-2*x)^2-c^2*(x^2+y^2)) /. {c -> Take[params2,3]} (* :[font = input; initialization; preserveAspect] *) layer4 := ImplicitPlot[(inversion[#]==0)& /@ limacons, {x,-4,5}, AspectRatio->Automatic, DisplayFunction->Identity] (* :[font = input; initialization; preserveAspect] *) layer5 := Graphics[{Dashing[{0.015}], Circle[{0,0}, 1]}, AspectRatio->Automatic] (* :[font = input; initialization; Cclosed; preserveAspect; startGroup] *) Show[{layer1,layer2,layer3,layer4,layer5}, Axes -> False, DisplayFunction -> $DisplayFunction] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 156; pictureHeight = 287] %! %%Creator: Mathematica %%AspectRatio: 1.83712 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.538415 0.087555 0.918562 0.087555 [ [ 0 0 0 0 ] [ 1 1.83712 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 1.83712 L 0 1.83712 L closepath clip newpath p p p p .004 w .80984 .91856 m .80982 .92078 L .80978 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.55634 .924 L .55589 .92402 L .55543 .92403 L .55496 .92403 L .55449 .92402 L .554 .924 L .55351 .92396 L .55249 .92386 L Mistroke .55197 .92379 L .55145 .92371 L .55037 .92351 L .54812 .92296 L .54578 .92219 L .54085 .91996 L .53572 .91676 L .53055 .91252 L .52549 .90723 L .52072 .90087 L .5164 .89346 L .5127 .88505 L .51113 .88048 L .50977 .87569 L .50865 .87069 L .50818 .86811 L .50777 .86548 L .50743 .86281 L .50716 .86009 L .50705 .85872 L .50696 .85733 L .50689 .85594 L .50684 .85453 L .5068 .85311 L .50679 .85169 L .50679 .85026 L .50681 .84881 L .50685 .84736 L .50692 .8459 L .507 .84443 L .5071 .84296 L .50736 .83999 L .50771 .83699 L .50866 .83092 L .50996 .82477 L .51162 .81857 L .51604 .80608 L .52197 .79365 L .52943 .78147 L .53842 .76972 L .5489 .75861 L .56083 .74833 L .57412 .73907 L .58867 .73101 L .60435 .72432 L .61257 .72153 L .62102 .71914 L .62967 .71716 L .6385 .71561 L .64298 .71499 L Mistroke .64749 .71449 L .65204 .71411 L .65432 .71396 L .65661 .71383 L .65891 .71374 L .66122 .71368 L .66353 .71364 L .66585 .71364 L .66817 .71366 L .67049 .71372 L .67282 .7138 L .67516 .71391 L .67984 .71423 L .68452 .71467 L .69392 .7159 L .70331 .71763 L .71268 .71985 L .73122 .72576 L .74931 .73362 L .76672 .74338 L .78324 .75498 L .79865 .76832 L .81276 .78327 L .82539 .79969 L .83638 .81742 L .84557 .83626 L .85284 .85602 L .85573 .86617 L .85811 .87647 L .85997 .88689 L .8607 .89214 L .8613 .8974 L .86177 .90268 L .86195 .90532 L .8621 .90797 L .86222 .91061 L .8623 .91326 L .86235 .91591 L .86237 .91856 L Mfstroke P P p p .004 w .88864 .91856 m .88862 .92143 L .88857 .92429 L .88848 .92716 L .88835 .93002 L .8882 .93288 L .888 .93574 L .88751 .94144 L .88688 .94713 L .88611 .95281 L .88415 .96408 L .88165 .97523 L .87861 .98623 L .87094 1.00766 L .86125 1.02815 L .84966 1.04748 L .83632 1.06547 L .8214 1.08194 L .78756 1.10973 L .76905 1.12083 L .74979 1.12994 L .73 1.13702 L .71998 1.1398 L .70991 1.14206 L .69982 1.14381 L .69478 1.14449 L .68975 1.14505 L .68473 1.14548 L .68222 1.14565 L .67972 1.14579 L .67722 1.1459 L .67472 1.14598 L .67223 1.14602 L .66975 1.14604 L .66727 1.14602 L .6648 1.14598 L .66233 1.1459 L .65987 1.14579 L .65498 1.14549 L .65012 1.14508 L .6453 1.14454 L .64051 1.14389 L .63107 1.14225 L .62183 1.14018 L .6128 1.13768 L .59547 1.13148 L .57924 1.12381 L .56426 1.11483 L .55062 1.10473 L .53842 1.09367 L Mistroke .51854 1.06951 L .51092 1.0568 L .50482 1.04393 L .50234 1.03749 L .50022 1.03108 L .49846 1.02472 L .49705 1.01843 L .49597 1.01224 L .49556 1.00918 L .49522 1.00615 L .49496 1.00316 L .49486 1.00167 L .49478 1.0002 L .49472 .99873 L .49467 .99727 L .49465 .99582 L .49464 .99439 L .49465 .99296 L .49467 .99154 L .49472 .99014 L .49478 .98874 L .49485 .98736 L .49495 .98598 L .49518 .98327 L .49547 .98061 L .49583 .97799 L .49671 .97291 L .4978 .96805 L .49908 .96341 L .50215 .95483 L .50574 .94721 L .50971 .94059 L .51388 .93496 L .5181 .93029 L .5261 .92366 L .52962 .92155 L .53265 .92011 L .53512 .91922 L .53611 .91894 L .53693 .91876 L .53758 .91864 L .53783 .91861 L .53804 .91859 L .5382 .91857 L .53827 .91857 L .53832 .91856 L .53836 .91856 L .53839 .91856 L .53841 .91856 L .53842 .91856 L Mistroke .53841 .91856 L .53839 .91856 L .53836 .91856 L .53832 .91856 L .5382 .91855 L .53804 .91854 L .53783 .91851 L .53758 .91848 L .53693 .91837 L .53611 .91818 L .53512 .91791 L .53265 .91702 L .52962 .91557 L .5261 .91346 L .52222 .91057 L .5181 .90683 L .51388 .90217 L .50971 .89653 L .50574 .88991 L .50215 .8823 L .49908 .87371 L .4978 .86907 L .49671 .86421 L .49583 .85913 L .49547 .85652 L .49518 .85385 L .49495 .85114 L .49485 .84977 L .49478 .84838 L .49472 .84699 L .49467 .84558 L .49465 .84416 L .49464 .84274 L .49465 .8413 L .49467 .83985 L .49472 .83839 L .49478 .83693 L .49496 .83397 L .49522 .83097 L .49556 .82795 L .49597 .82489 L .49705 .81869 L .49846 .8124 L .50022 .80604 L .50482 .7932 L .51092 .78032 L .51854 .76761 L .52771 .75525 L .53842 .74345 L .55062 .7324 L Mistroke .56426 .72229 L .57924 .71331 L .59547 .70564 L .604 .70235 L .6128 .69944 L .62183 .69695 L .63107 .69487 L .64051 .69323 L .6453 .69258 L .65012 .69205 L .65498 .69163 L .65742 .69146 L .65987 .69133 L .66233 .69122 L .6648 .69115 L .66727 .6911 L .66975 .69109 L .67223 .6911 L .67472 .69115 L .67722 .69122 L .67972 .69133 L .68473 .69164 L .68975 .69207 L .69478 .69263 L .69982 .69332 L .70991 .69507 L .71998 .69733 L .73 .7001 L .74979 .70719 L .76905 .7163 L .78756 .72739 L .80507 .74039 L .8214 .75518 L .83632 .77165 L .84966 .78964 L .86125 .80897 L .87094 .82946 L .87861 .85089 L .88165 .86189 L .88415 .87304 L .88611 .88432 L .88688 .88999 L .88751 .89568 L .888 .90139 L .8882 .90425 L .88835 .90711 L .88848 .90997 L .88857 .91283 L .88862 .9157 L .88864 .91856 L Mistroke Mfstroke P P p p .004 w .9149 .91856 m .91488 .92164 L .91483 .92472 L .91474 .9278 L .91461 .93088 L .91444 .93395 L .91424 .93702 L .91372 .94316 L .91306 .94928 L .91225 .95538 L .9102 .96751 L .90757 .97951 L .90437 .99135 L .89631 1.01446 L .88612 1.03659 L .87393 1.05754 L .85988 1.07709 L .84415 1.09507 L .80839 1.12572 L .7888 1.13814 L .76837 1.14851 L .74732 1.15677 L .73664 1.1601 L .7259 1.1629 L .71512 1.16516 L .70973 1.16609 L .70434 1.16689 L .69896 1.16756 L .69359 1.1681 L .6909 1.16832 L .68823 1.1685 L .68555 1.16866 L .68288 1.16878 L .68022 1.16888 L .67756 1.16894 L .6749 1.16896 L .67226 1.16896 L .66961 1.16892 L .66698 1.16886 L .66436 1.16876 L .66174 1.16863 L .65653 1.16829 L .65135 1.16782 L .64112 1.16652 L .63108 1.16476 L .62124 1.16255 L .60227 1.15685 L .58437 1.14957 L .56768 1.14088 L .55233 1.13094 L Mistroke .53842 1.11994 L .51511 1.09555 L .50579 1.08256 L .49803 1.0693 L .49178 1.05595 L .487 1.0427 L .48513 1.03616 L .4836 1.02971 L .4824 1.02336 L .48191 1.02023 L .4815 1.01713 L .48117 1.01407 L .48091 1.01105 L .4808 1.00955 L .48071 1.00806 L .48064 1.00658 L .48059 1.00511 L .48055 1.00365 L .48053 1.0022 L .48052 1.00077 L .48053 .99934 L .48055 .99793 L .48059 .99652 L .48065 .99513 L .48072 .99375 L .4809 .99103 L .48114 .98836 L .48177 .98316 L .48258 .97817 L .48358 .9734 L .486 .96453 L .48887 .95658 L .49535 .94343 L .50183 .93371 L .50728 .9269 L .50935 .92434 L .51089 .92219 L .51144 .92122 L .51165 .92076 L .51183 .9203 L .51197 .91986 L .51207 .91942 L .5121 .91921 L .51213 .91899 L .51214 .91878 L .51215 .91856 L .51214 .91835 L .51213 .91813 L .5121 .91792 L .51207 .9177 L Mistroke .51197 .91726 L .51183 .91682 L .51144 .9159 L .51089 .91494 L .50728 .91022 L .50183 .90341 L .49535 .8937 L .49204 .88757 L .48887 .88054 L .486 .87259 L .48472 .86827 L .48358 .86372 L .48258 .85895 L .48177 .85396 L .48143 .85139 L .48114 .84877 L .4809 .84609 L .4808 .84474 L .48072 .84337 L .48065 .84199 L .48059 .8406 L .48055 .8392 L .48053 .83778 L .48052 .83636 L .48053 .83492 L .48055 .83347 L .48059 .83201 L .48064 .83054 L .48071 .82907 L .48091 .82608 L .48117 .82305 L .4815 .81999 L .4824 .81376 L .4836 .80741 L .487 .79442 L .49178 .78117 L .49803 .76782 L .51511 .74157 L .52599 .72904 L .53842 .71719 L .55233 .70619 L .56768 .69625 L .58437 .68755 L .60227 .68027 L .61163 .67721 L .62124 .67457 L .63108 .67236 L .64112 .6706 L .64622 .6699 L .65135 .66931 L Mistroke .65653 .66884 L .66174 .66849 L .66436 .66836 L .66698 .66827 L .66961 .6682 L .67226 .66816 L .6749 .66816 L .67756 .66819 L .68022 .66825 L .68288 .66834 L .68555 .66846 L .68823 .66862 L .69359 .66903 L .69896 .66956 L .70434 .67023 L .71512 .67196 L .7259 .67423 L .74732 .68035 L .76837 .68861 L .7888 .69898 L .80839 .7114 L .82691 .72579 L .84415 .74205 L .85988 .76003 L .87393 .77959 L .88612 .80053 L .89631 .82266 L .90437 .84577 L .90757 .85761 L .9102 .86962 L .91225 .88174 L .91306 .88784 L .91372 .89396 L .91424 .9001 L .91444 .90317 L .91461 .90625 L .91474 .90932 L .91483 .9124 L .91488 .91548 L .9149 .91856 L Mfstroke P P p p .004 w .94117 .91856 m .94115 .92186 L .94109 .92515 L .94099 .92844 L .94086 .93174 L .94069 .93503 L .94047 .93831 L .93993 .94488 L .93924 .95143 L .93839 .95796 L .93624 .97094 L .93348 .98379 L .93013 .99648 L .92168 1.02126 L .91099 1.04504 L .8982 1.06759 L .88344 1.08871 L .86689 1.10821 L .82923 1.14171 L .80855 1.15546 L .78694 1.16708 L .76464 1.17652 L .7533 1.18041 L .74189 1.18373 L .73042 1.18651 L .71894 1.18873 L .71319 1.18963 L .70746 1.1904 L .70173 1.19103 L .69601 1.19153 L .69032 1.19189 L .68747 1.19203 L .68464 1.19212 L .68181 1.19219 L .67898 1.19222 L .67616 1.19222 L .67335 1.19219 L .67055 1.19213 L .66776 1.19203 L .66497 1.1919 L .66219 1.19174 L .65666 1.19133 L .65117 1.19079 L .64033 1.18934 L .62968 1.18742 L .60906 1.18223 L .58949 1.17534 L .57111 1.16692 L .55405 1.15715 L .53842 1.1462 L Mistroke .51169 1.1216 L .50067 1.10832 L .49123 1.09467 L .48333 1.08082 L .47694 1.06697 L .47199 1.05327 L .47002 1.04653 L .46837 1.03988 L .46703 1.03335 L .46599 1.02695 L .46558 1.0238 L .46523 1.02069 L .46495 1.01762 L .46473 1.01459 L .46464 1.01309 L .46457 1.0116 L .46452 1.01013 L .46447 1.00866 L .46445 1.0072 L .46443 1.00576 L .46443 1.00433 L .46445 1.00291 L .46447 1.0015 L .46451 1.0001 L .46463 .99734 L .4647 .99598 L .46479 .99463 L .465 .99197 L .46555 .98681 L .46625 .98185 L .46803 .97257 L .4702 .96414 L .4726 .95656 L .47757 .94377 L .48191 .9337 L .48359 .92947 L .48428 .9275 L .48485 .92561 L .4853 .92379 L .48547 .9229 L .48562 .92202 L .48573 .92115 L .48582 .92028 L .48585 .91985 L .48587 .91942 L .48588 .91899 L .48588 .91856 L .48588 .91813 L .48587 .9177 L .48585 .91727 L Mistroke .48582 .91684 L .48573 .91597 L .48562 .9151 L .4853 .91333 L .48485 .91151 L .48359 .90766 L .48191 .90342 L .47757 .89336 L .4726 .88057 L .4702 .87298 L .46803 .86455 L .46708 .86002 L .46625 .85527 L .46555 .85031 L .46525 .84776 L .465 .84515 L .46479 .84249 L .46463 .83978 L .46456 .83841 L .46451 .83702 L .46447 .83563 L .46445 .83422 L .46443 .8328 L .46443 .83136 L .46445 .82992 L .46447 .82846 L .46452 .827 L .46457 .82552 L .46464 .82403 L .46473 .82253 L .46495 .8195 L .46523 .81643 L .46599 .81017 L .46703 .80377 L .46837 .79724 L .47199 .78386 L .47694 .77016 L .48333 .7563 L .49123 .74245 L .51169 .71553 L .52428 .70283 L .53842 .69092 L .55405 .67998 L .57111 .67021 L .58949 .66179 L .60906 .6549 L .61925 .65208 L .62968 .6497 L .64033 .64778 L .64573 .647 L Mistroke .65117 .64634 L .65666 .6458 L .65942 .64557 L .66219 .64538 L .66497 .64522 L .66776 .64509 L .67055 .645 L .67335 .64493 L .67616 .6449 L .67898 .6449 L .68181 .64493 L .68464 .645 L .68747 .6451 L .69032 .64523 L .69316 .64539 L .69601 .64559 L .70173 .64609 L .70746 .64672 L .71894 .64839 L .73042 .65061 L .74189 .65339 L .76464 .6606 L .78694 .67004 L .80855 .68166 L .82923 .69541 L .84875 .7112 L .86689 .72892 L .88344 .74842 L .8982 .76954 L .91099 .79209 L .92168 .81587 L .93013 .84064 L .93348 .85334 L .93624 .86619 L .93839 .87917 L .93924 .8857 L .93993 .89224 L .94047 .89881 L .94069 .9021 L .94086 .90539 L .94099 .90868 L .94109 .91197 L .94115 .91527 L .94117 .91856 L Mfstroke P P P p p P P p p .004 w s s s s s s 0 .69945 m .19511 .73975 L s .19511 .73975 m .33708 .77876 L .41203 .80462 L .45635 .82316 L .50385 .84826 L .52733 .86483 L .5405 .87692 L .54849 .8864 L .55355 .89425 L .55681 .90107 L .55795 .90421 L .55884 .90722 L .55949 .91014 L .55975 .91157 L .55995 .91298 L .56011 .91439 L .56017 .91509 L .56022 .91578 L .56025 .91648 L .56027 .91683 L .56028 .91717 L .56029 .91752 L .5603 .91769 L .5603 .91787 L .5603 .91804 L .5603 .91822 L .5603 .91839 L .5603 .91856 L .5603 .91874 L .5603 .91891 L .5603 .91908 L .5603 .91926 L .5603 .91943 L .56029 .9196 L .56028 .91995 L .56027 .9203 L .56025 .92064 L .56022 .92134 L .56017 .92204 L .56011 .92274 L .55995 .92414 L .55975 .92556 L .55949 .92699 L .55884 .9299 L .55795 .93292 L .55681 .93606 L .55355 .94287 L .54849 .95073 L .5405 .9602 L .52733 .97229 L Mistroke .50385 .98887 L .45635 1.01397 L .41203 1.0325 L .33708 1.05836 L .19511 1.09738 L .07092 1.12556 L Mfstroke 0 1.13929 m .07092 1.12556 L s s s s s s s P P p p p p .004 w .1882 .29365 m .18981 .29612 L .19464 .30347 L .2026 .3156 L .21355 .33229 L .2273 .35326 L .24362 .37815 L .26223 .40655 L .28281 .43798 L .30501 .4719 L .32845 .50776 L .35273 .54495 L .37743 .58287 L .40213 .62088 L .4264 .6584 L .44984 .69482 L .47204 .72961 L .49262 .76228 L .51123 .79244 L .52755 .81982 L .54131 .84428 L .55225 .86592 L .56021 .88506 L .56504 .90232 L .56666 .91853 L s P p .004 w .1882 1.54348 m .18981 1.54101 L .19464 1.53365 L .2026 1.52152 L .21355 1.50483 L .2273 1.48387 L .24362 1.45897 L .26223 1.43057 L .28281 1.39915 L .30501 1.36522 L .32845 1.32936 L .35273 1.29217 L .37743 1.25426 L .40213 1.21624 L .4264 1.17873 L .44984 1.1423 L .47204 1.10752 L .49262 1.07484 L .51123 1.04468 L .52755 1.01731 L .54131 .99284 L .55225 .97121 L .56021 .95206 L .56504 .9348 L .56666 .9186 L s P P p P p p .004 w .6357 .91853 m .63716 .90318 L .6415 .88692 L .64866 .86906 L .65851 .84906 L .67088 .82661 L .68556 .80164 L .70231 .77423 L .72082 .74461 L .74079 .71314 L .76188 .68023 L .78372 .64637 L .80594 .61208 L .82817 .57789 L .85001 .54437 L .87109 .51206 L .89107 .4815 L .90958 .4532 L .92633 .42762 L .94101 .4052 L .95338 .38632 L .96323 .3713 L .97039 .36038 L .97473 .35375 L .97619 .35153 L s P p .004 w .6357 .9186 m .63716 .93395 L .6415 .9502 L .64866 .96806 L .65851 .98806 L .67088 1.01051 L .68556 1.03548 L .70231 1.0629 L .72082 1.09251 L .74079 1.12398 L .76188 1.15689 L .78372 1.19075 L .80594 1.22504 L .82817 1.25923 L .85001 1.29275 L .87109 1.32506 L .89107 1.35562 L .90958 1.38393 L .92633 1.4095 L .94101 1.43192 L .95338 1.4508 L .96323 1.46583 L .97039 1.47675 L .97473 1.48337 L .97619 1.48559 L s P P P p p p .004 w .1882 .47794 m .1898 .47959 L .1946 .48454 L .2025 .49268 L .21338 .50389 L .22704 .51799 L .24325 .53473 L .26174 .55384 L .28219 .57501 L .30424 .59789 L .32753 .62211 L .35164 .64728 L .37618 .67301 L .40072 .6989 L .42484 .72457 L .44812 .74965 L .47017 .77381 L .49062 .79679 L .50911 .81835 L .52532 .83838 L .53898 .85683 L .54986 .87381 L .55776 .88951 L .56256 .90428 L .56417 .91853 L s P p .004 w .1882 1.35919 m .1898 1.35753 L .1946 1.35259 L .2025 1.34444 L .21338 1.33323 L .22704 1.31914 L .24325 1.30239 L .26174 1.28328 L .28219 1.26211 L .30424 1.23923 L .32753 1.21501 L .35164 1.18984 L .37618 1.16411 L .40072 1.13822 L .42484 1.11255 L .44812 1.08747 L .47017 1.06331 L .49062 1.04034 L .50911 1.01877 L .52532 .99875 L .53898 .98029 L .54986 .96332 L .55776 .94761 L .56256 .93284 L .56417 .91859 L s P P p P p p .004 w .68434 .91853 m .68559 .906 L .68931 .89311 L .69545 .8796 L .70389 .86524 L .7145 .84987 L .72708 .83343 L .74143 .81593 L .7573 .79745 L .77442 .77816 L .7925 .75824 L .81122 .73794 L .83027 .71753 L .84931 .69729 L .86803 .67753 L .88611 .65854 L .90323 .64063 L .9191 .62407 L .93345 .60913 L .94604 .59605 L .95664 .58505 L .96508 .57629 L 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1.19731 L .24295 1.18687 L .26133 1.17492 L .28167 1.16164 L .3036 1.14722 L .32675 1.13189 L .35074 1.11586 L .37514 1.09933 L .39954 1.08254 L .42352 1.06569 L .44668 1.04895 L .46861 1.03251 L .48894 1.01648 L .50732 1.00097 L .52345 .98603 L .53703 .97167 L .54785 .95785 L .55571 .94448 L .56048 .93144 L .56208 .91859 L s P P p P p p .004 w .83027 .91853 m .83089 .91053 L .83275 .90249 L .83582 .8944 L .84004 .88626 L .84534 .87806 L .85164 .86981 L .85881 .86152 L .86675 .85322 L .87531 .84495 L .88434 .83675 L .8937 .82867 L .90323 .82079 L .91275 .81316 L .92211 .80586 L .93115 .79898 L .93971 .79257 L .94764 .78672 L .95482 .78149 L .96111 .77695 L .96642 .77315 L .97064 .77015 L .9737 .76797 L .97557 .76665 L .97619 .76621 L s P p .004 w .83027 .91859 m .83089 .92659 L .83275 .93464 L .83582 .94273 L .84004 .95087 L .84534 .95906 L .85164 .96731 L .85881 .9756 L .86675 .9839 L .87531 .99217 L .88434 1.00038 L .8937 1.00845 L .90323 1.01634 L .91275 1.02396 L .92211 1.03126 L .93115 1.03815 L .93971 1.04455 L .94764 1.0504 L .95482 1.05563 L .96111 1.06017 L .96642 1.06397 L .97064 1.06698 L .9737 1.06915 L .97557 1.07047 L .97619 1.07091 L s P P P P p [ .015 ] 0 setdash .004 w newpath .53842 .91856 .08756 0 365.72958 arc s P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Figure 8 :[font = input; initialization; preserveAspect; backColorRed = 65280; backColorGreen = 65280; backColorBlue = 65280] *) cassini[a_,c_] := (x^2+y^2)^2-2*c^2*(x^2-y^2)-a^4+c^4 params3 := {3., 2.35, 2., 1.6, 1.2} plt := ImplicitPlot[cassini[#,2]==0, {x,-4,4}, Axes -> False, AspectRatio -> Automatic, DisplayFunction->Identity]& /@ params3 (* :[font = input; Cclosed; preserveAspect; startGroup] Show[plt,DisplayFunction->$DisplayFunction] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174] %! %%Creator: Mathematica %%AspectRatio: .62017 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.5 0.132071 0.310087 0.132071 [ [ 0 0 0 0 ] [ 1 .62017 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 .62017 L 0 .62017 L closepath clip newpath p p p P p p .004 w .02381 .31001 m .02788 .26492 L .04004 .22091 L .06006 .17915 L .08761 .14071 L .12221 .10655 L .16328 .0775 L .21011 .05415 L .2619 .03677 L .31777 .02518 L .37675 .01855 L .43784 .01556 L .5 .01477 L .56216 .01556 L .62325 .01855 L .68223 .02518 L .7381 .03677 L .78989 .05415 L .83672 .0775 L .87779 .10655 L .91239 .14071 L .93994 .17915 L .95996 .22091 L .97212 .26492 L .97619 .31001 L s P p .004 w .02381 .31016 m .02788 .35525 L .04004 .39927 L .06006 .44103 L .08761 .47947 L .12221 .51362 L .16328 .54268 L .21011 .56603 L .2619 .5834 L .31777 .595 L .37675 .60162 L .43784 .60462 L .5 .60541 L .56216 .60462 L .62325 .60162 L .68223 .595 L .7381 .5834 L .78989 .56603 L .83672 .54268 L .87779 .51362 L .91239 .47947 L .93994 .44103 L .95996 .39927 L .97212 .35525 L .97619 .31016 L s P P p P P p p P p p .004 w .09245 .31003 m .09593 .27618 L .10633 .24338 L .12347 .21279 L .14705 .18549 L .17667 .16249 L .21182 .14475 L .2519 .13307 L .29622 .12798 L .34404 .12936 L .39452 .13577 L .4468 .14353 L .5 .14712 L .5532 .14353 L .60548 .13577 L .65596 .12936 L .70378 .12798 L .7481 .13307 L .78818 .14475 L .82333 .16249 L .85295 .18549 L .87653 .21279 L .89367 .24338 L .90407 .27618 L .90755 .31003 L s P p .004 w .09245 .31015 m .09593 .34399 L .10633 .37679 L .12347 .40738 L .14705 .43469 L .17667 .45769 L .21182 .47543 L .2519 .4871 L .29622 .49219 L .34404 .49082 L .39452 .48441 L .4468 .47665 L .5 .47305 L .5532 .47665 L .60548 .48441 L .65596 .49082 L .70378 .49219 L .7481 .4871 L .78818 .47543 L .82333 .45769 L .85295 .43469 L .87653 .40738 L .89367 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.59339 .22666 L .6153 .21235 L .63844 .1999 L .6624 .18993 L .68678 .18284 L .71116 .17888 L .73512 .17813 L .75825 .18058 L .78017 .18613 L .80048 .1946 L .81885 .20575 L .83496 .21929 L .84853 .23489 L .85934 .25217 L .86719 .27075 L .87196 .2902 L .87355 .31004 L s P p .004 w .5 .31009 m .5016 .31168 L .50636 .31645 L .51422 .32426 L .52502 .33489 L .5386 .34788 L .55471 .36258 L .57307 .37812 L .59339 .39351 L .6153 .40783 L .63844 .42027 L .6624 .43024 L .68678 .43733 L .71116 .44129 L .73512 .44204 L .75825 .43959 L .78017 .43404 L .80048 .42557 L .81885 .41442 L .83496 .40088 L .84853 .38529 L .85934 .368 L .86719 .34942 L .87196 .32998 L .87355 .31014 L s P P p P P p p P p p .004 w .16173 .31005 m .1625 .29888 L .16479 .28787 L .16857 .27726 L .17378 .26723 L .18031 .25796 L .18806 .2496 L .1969 .24232 L .20668 .23623 L .21722 .23145 L .22836 .22805 L .23989 .22608 L .25162 .22558 L .26336 .22654 L .27489 .22895 L .28602 .23273 L .29657 .23783 L .30635 .24412 L .31519 .25149 L .32294 .2598 L .32947 .26891 L .33467 .27865 L .33845 .28886 L .34075 .2994 L .34151 .31005 L s P p .004 w .16173 .31013 m .1625 .32129 L .16479 .3323 L .16857 .34291 L .17378 .35295 L .18031 .36222 L .18806 .37057 L .1969 .37785 L .20668 .38394 L .21722 .38872 L .22836 .39213 L .23989 .39409 L .25162 .39459 L .26336 .39363 L .27489 .39123 L .28602 .38744 L .29657 .38235 L .30635 .37606 L .31519 .36869 L .32294 .36037 L .32947 .35127 L .33467 .34153 L .33845 .33131 L .34075 .32078 L .34151 .31013 L s P P p P p p .004 w .65849 .31005 m .65925 .2994 L .66155 .28886 L .66533 .27865 L .67053 .26891 L .67706 .2598 L .68481 .25149 L .69365 .24412 L .70343 .23783 L .71398 .23273 L .72511 .22895 L .73664 .22654 L .74838 .22558 L .76011 .22608 L .77164 .22805 L .78278 .23145 L .79332 .23623 L .8031 .24232 L .81194 .2496 L .81969 .25796 L .82622 .26723 L .83143 .27726 L .83521 .28787 L .8375 .29888 L .83827 .31005 L s P p .004 w .65849 .31013 m .65925 .32078 L .66155 .33131 L .66533 .34153 L .67053 .35127 L .67706 .36037 L .68481 .36869 L .69365 .37606 L .70343 .38235 L .71398 .38744 L .72511 .39123 L .73664 .39363 L .74838 .39459 L .76011 .39409 L .77164 .39213 L .78278 .38872 L .79332 .38394 L .8031 .37785 L .81194 .37057 L .81969 .36222 L .82622 .35295 L .83143 .34291 L .83521 .3323 L .8375 .32129 L .83827 .31013 L s P P p P P p p P p p .004 w .19196 .31006 m .19237 .30386 L .19361 .29775 L .19564 .29185 L .19844 .28626 L .20195 .28108 L .20612 .2764 L .21088 .2723 L .21614 .26885 L .22181 .26611 L .2278 .26413 L .23401 .26293 L .24032 .26254 L .24664 .26297 L .25284 .2642 L .25883 .26622 L .2645 .26898 L .26976 .27245 L .27452 .27655 L .27869 .28123 L .28221 .28639 L .285 .29195 L .28704 .29783 L .28827 .3039 L .28869 .31006 L s P p .004 w .19196 .31012 m .19237 .31631 L .19361 .32242 L .19564 .32833 L .19844 .33392 L .20195 .33909 L .20612 .34377 L .21088 .34787 L .21614 .35132 L .22181 .35406 L .2278 .35605 L .23401 .35724 L .24032 .35763 L .24664 .35721 L .25284 .35597 L .25883 .35396 L .2645 .35119 L .26976 .34773 L .27452 .34362 L .27869 .33895 L .28221 .33378 L .285 .32822 L .28704 .32235 L .28827 .31627 L .28869 .31012 L s P P p P p p .004 w .71131 .31006 m .71173 .3039 L .71296 .29783 L .715 .29195 L .71779 .28639 L .72131 .28123 L .72548 .27655 L .73024 .27245 L .7355 .26898 L .74117 .26622 L .74716 .2642 L .75336 .26297 L .75968 .26254 L .76599 .26293 L .7722 .26413 L .77819 .26611 L .78386 .26885 L .78912 .2723 L .79388 .2764 L .79805 .28108 L .80156 .28626 L .80436 .29185 L .80639 .29775 L .80763 .30386 L .80804 .31006 L s P p .004 w .71131 .31012 m .71173 .31627 L .71296 .32235 L .715 .32822 L .71779 .33378 L .72131 .33895 L .72548 .34362 L .73024 .34773 L .7355 .35119 L .74117 .35396 L .74716 .35597 L .75336 .35721 L .75968 .35763 L .76599 .35724 L .7722 .35605 L .77819 .35406 L .78386 .35132 L .78912 .34787 L .79388 .34377 L .79805 .33909 L .80156 .33392 L .80436 .32833 L .80639 .32242 L .80763 .31631 L .80804 .31012 L s P P p P P P % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Figure 9 :[font = input; initialization; Cclosed; preserveAspect; startGroup] *) cass := (x^2 + y^2)^2 - 2(x^2 - y^2) - 1 Show[ivPlot[cass, {-1.6, 1.6}], DisplayFunction -> $DisplayFunction] (* :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 282] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart %% Graphics /Helvetica findfont 6 scalefont setfont % Scaling calculations 0.5 0.306473 0.5 0.306473 [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p P 0 0 m 1 0 L 1 1 L 0 1 L closepath clip newpath p p p P p p .5 g .004 w .19353 .49992 m .19615 .46 L .20397 .42068 L .21686 .38272 L .23459 .34676 L .25686 .31343 L .28329 .28329 L .31343 .25686 L .34676 .23459 L .38272 .21686 L .42068 .20397 L .46 .19615 L .5 .19353 L .54 .19615 L .57932 .20397 L .61728 .21686 L .65324 .23459 L .68657 .25686 L .71671 .28329 L .74314 .31343 L .76541 .34676 L .78314 .38272 L .79603 .42068 L .80385 .46 L .80647 .49992 L s P p .5 g .004 w .19353 .50008 m .19615 .54 L .20397 .57932 L .21686 .61728 L .23459 .65324 L .25686 .68657 L .28329 .71671 L .31343 .74314 L .34676 .76541 L .38272 .78314 L .42068 .79603 L .46 .80385 L .5 .80647 L .54 .80385 L .57932 .79603 L .61728 .78314 L .65324 .76541 L .68657 .74314 L .71671 .71671 L .74314 .68657 L .76541 .65324 L .78314 .61728 L .79603 .57932 L .80385 .54 L .80647 .50008 L s P P p P P p p P p p [ .015 ] 0 setdash .007 w .02381 .49994 m .02788 .4601 L .04004 .42148 L .06006 .38543 L .08761 .35318 L .12221 .32591 L .16328 .30472 L .21011 .29051 L .2619 .28384 L .31777 .28455 L .37675 .29097 L .43784 .29901 L .5 .30276 L .56216 .29901 L .62325 .29097 L .68223 .28455 L .7381 .28384 L .78989 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