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F/F," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "expand(ParametrizedToIm
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\"\"\"F%*&*$)%\"yG\"\"#\"\"\"F+*$)%\"bG\"\"#F+!\"\"F%*&*$)%\"xGF*F+F+*
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{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%&kappaG6#%*hyperbolaG*&-%$absG6#*&%
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{SECT 1 {PARA 3 "" 0 "Cissoid of Diocles" {TEXT -1 18 "Cissoid of Dioc
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0.800000 70.000000 55.000000 2 }}}}}}{SECT 1 {PARA 3 "" 0 "Tractrix" 
{TEXT -1 40 "Tractrix (equal length tangent segments)" }}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "tractrix = mat(tractrix(a,t));" "6#/%)tractri
xG-%$matG6#-F$6$%\"aG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%)tract
rixG-%'matrixG6#7$7#*&%\"aG\"\"\"-%$sinG6#%\"tGF,7#*&F+\"\"\",&-%$cosG
F/F,-%#lnG6#-%$tanG6#,$F0#F,\"\"#F,F," }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "curveplot2D(tractrix(1,t),t = 0 .. 1,title = \"Tractrix
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\"/%&titleGQ.Tractrix(1,t)6\"" }}{PARA 13 "" 1 "" {GLPLOT2D 614 298 
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{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Curvature" }}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "kappa = kappa2D(lissajous(n,m,a,b,phi,t),t);" "6#/%&k
appaG-%(kappa2DG6$-%*lissajousG6(%\"nG%\"mG%\"aG%\"bG%$phiG%\"tGF0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&kappaG*&-%$absG6#*,%\"aG\"\"\"%\"bG
F+%\"nG\"\"\"%\"mGF.,&*(-%$sinG6#,&*&F/F+%\"tGF+F+%$phiGF+F+-%$cosG6#*
&F-F+F7F.F+F/F.!\"\"*(-F:F4F+-F3F;F+F-F.F+F+F.*$),&*()F*\"\"#F.)F9FFF.
)F-FFF.F+*()F,FFF.)F?FFF.)F/FFF.F+#\"\"$FFF.!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "scalarplot(kappa2D(lissajous(7,5,1,1,.2*Pi,t),t)
,t = 0 .. 2*Pi,labels = [\"t\", \"k\"],title = \"Curvature of Lissajou
s(7,5)\",numpoints = 500);" "6#-%+scalarplotG6'-%(kappa2DG6$-%*lissajo
usG6(\"\"(\"\"&\"\"\"\"\"\"*&$\"\"#!\"\"\"\"\"%#PiGF4%\"tGF6/F6;\"\"!*
&\"\"#F4F5F4/%'labelsG7$Q\"t6\"Q\"kF@/%&titleGQ<Curvature~of~Lissajous
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The Folium of Descartes is easier \+
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]q7$$\"1#)eD)yA'z:F*Fhbp7$7$F^cpFginFc]q7$Fg]q7$F^dp$\"1&GWh0djA\"F*7$
7$FbdpF_\\oFi]q7$F]^q7$F`epF^ep7$7$FdepFg^oF_^q7$Fa^q7$FdfpFbfp7$7$Fhf
pF\\aoFc^q7$Fe^qFg\\q-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%,CONSTRAINE
DG-%'COLOURG6&%$RGBGF6F6$\"*++++\"!\")-%&TITLEG6$Q4Folium~of~Descartes
6\"-%%FONTG6%%&TIMESG%'ITALICG\"#=-%+AXESLABELSG6%%\"xG%\"yG-F\\`q6%F^
`q%&ROMANGF``q-%*AXESTICKSG6%%(DEFAULTGF\\aq-F\\`q6$%*HELVETICAG\"#7" 
1 2 0 1 0 2 6 1 4 1 1.000000 45.000000 45.000000 0 }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 9 "Curvature" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "kappa = simplify(kappa2D(folium(t),t),assume = real);" "6#/%&kappaG
-%)simplifyG6$-%(kappa2DG6$-%'foliumG6#%\"tGF./%'assumeG%%realG" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "eq := fn(rhs(%),t):" "6#>%#eqG-%#fnG6$-
%$rhsG6#%\"%G%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&kappaG,$*&*&)
,&%\"tG\"\"\"F+F+\"\"%\"\"\"),(*$)F*\"\"#F-F+F*!\"\"F+F+F,F-F-*$),.F+F
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2F-!\"\"#F2F9" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "scalarplot(eq(
t),t = -2 .. 2,labels = [\"t\", \"k\"],title = \"Curvature of Folium(t
)\");" "6#-%+scalarplotG6&-%#eqG6#%\"tG/F);,$\"\"#!\"\"\"\"#/%'labelsG
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RP#yK>#F17$$\"1YYY1o-t7F1$\"1c)o)Rp.>>F17$$\"1dccwr,:8F1$\"1JE-45#))o
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\\g8F17$$\"1*)))))o&)QL9F1$\"1R\"\\%3S&fB\"F17$$\"1^]]]#4jZ\"F1$\"18)
\\TNcV7\"F17$$\"1MLLtvW8:F1$\"1m%HEo4Q/\"F17$$\"1FFF(G-hb\"F1$\"1'e;*H
fih'*Fhs7$$\"1\"333'**>'f\"F1$\"1*4v/\\x+0*Fhs7$$\"1baaM.\"fj\"F1$\"1`
YS,]tO&)Fhs7$$\"1===)H_cn\"F1$\"1!HCk>G))4)Fhs7$$\"1YXX&e'=;<F1$\"1z8s
7a(pr(Fhs7$$\"1\"444D_\"f<F1$\"1!z=T(H\\rtFhs7$$\"1zyyeva)z\"F1$\"1SIe
X&)[*4(Fhs7$$\"1EEE'yNu$=F1$\"1&*)e?))\\j'oFhs7$$\"1UTThm+z=F1$\"1.zi&
H.,l'Fhs7$$\"1TSSSq[?>F1$\"1H@jNbAjkFhs7$$\"1&QQQ9=x&>F1$\"1@e$p>_lJ'F
hs7$$\"\"#F*$\"1c<n_\"o2<'Fhs-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%.UNCON
STRAINEDG-%'COLOURG6&%$RGBGF*F*$\"*++++\"!\")-%&TITLEG6$Q7Curvature~of
~Folium(t)6\"-%%FONTG6%%&TIMESG%'ITALICG\"#=-%+AXESLABELSG6%Q\"tFbamQ
\"k6\"-Fdam6%Ffam%&ROMANGFham-%*AXESTICKSG6%%(DEFAULTGFebm-Fdam6$%*HEL
VETICAG\"#7-%%VIEWG6$;F(F[`mFebm" 1 2 0 1 0 2 6 1 2 2 1.000000 
45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F := f
n(folium(t),t):" "6#>%\"FG-%#fnG6$-%'foliumG6#%\"tGF+" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "c1 := curveplot3D([F(t)[1], F(t)[2], eq
(t)],t = -.9 .. 50,numpoints = 1500,thickness = 3):" "6#>%#c1G-%,curve
plot3DG6&7%&-%\"FG6#%\"tG6#\"\"\"&-F+6#F-6#\"\"#-%#eqG6#F-/F-;,$$\"\"*
!\"\"!\"\"\"#]/%*numpointsG\"%+:/%*thicknessG\"\"$" }}}{EXCHG {PARA 0 
"" 0 "" {XPPEDIT 19 1 "c2 := curveplot3D([F(t)[1], F(t)[2], eq(t)],t =
 -50 .. -1.1,numpoints = 500,thickness = 3):" "6#>%#c2G-%,curveplot3DG
6&7%&-%\"FG6#%\"tG6#\"\"\"&-F+6#F-6#\"\"#-%#eqG6#F-/F-;,$\"#]!\"\",$$
\"#6!\"\"F</%*numpointsG\"$+&/%*thicknessG\"\"$" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "pltx := plottools[transform](proc (x, y) options o
perator, arrow; [x, y, 0] end):" "6#>%%pltxG-&%*plottoolsG6#%*transfor
mG6#R6$%\"xG%\"yG7\"6$%)operatorG%&arrowG6\"7%F-F.\"\"!F3F3F3" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "c3 := curveplot2D(F(t),t = -.9 .
. 50,numpoints = 1500,color = black):" "6#>%#c3G-%,curveplot2DG6&-%\"F
G6#%\"tG/F+;,$$\"\"*!\"\"!\"\"\"#]/%*numpointsG\"%+:/%&colorG%&blackG
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "c4 := curveplot2D(F(t),t = \+
-50 .. -1.1,numpoints = 500,color = black):" "6#>%#c4G-%,curveplot2DG6
&-%\"FG6#%\"tG/F+;,$\"#]!\"\",$$\"#6!\"\"F0/%*numpointsG\"$+&/%&colorG
%&blackG" }}}{EXCHG {PARA 0 "" 0 "tmp1" {XPPEDIT 19 1 "plots[display](
c1, c2, pltx(c3), pltx(c4), labels = [\"x\", \"y\", \"k\"], title = \"
Curvature of Folium(t)\", axes = framed, view = [-2 .. 2, -2 .. 2, 0 .
. 4], projection = .8, orientation = [-150, 80]);" "6#-&%&plotsG6#%(di
splayG6,%#c1G%#c2G-%%pltxG6#%#c3G-F,6#%#c4G/%'labelsG7%Q\"x6\"Q\"yF6Q
\"kF6/%&titleGQ7Curvature~of~Folium(t)F6/%%axesG%'framedG/%%viewG7%;,$
\"\"#!\"\"\"\"#;,$\"\"#FE\"\"#;\"\"!\"\"%/%+projectionG$\"\")!\"\"/%,o
rientationG7$,$\"$]\"FE\"#!)" }}{PARA 13 "" 1 "" {GLPLOT3D 529 435 
435 {PLOTDATA 3 "60-%'CURVESG6&7hhp7%$!+J'*4j**!\"*$\"+o'*ym*)F*$\"+x`
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{PARA 3 "" 0 "Cassinian Oval" {TEXT -1 14 "Cassinian Oval" }}{EXCHG 
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+y^2+a^2)^2-4*a^2*x^2-b^4:" "6#>%$tmpG,(*$,(*$%\"xG\"\"#\"\"\"*$%\"yG
\"\"#F+*$%\"aG\"\"#F+\"\"#F+*(\"\"%F+*$F0\"\"#F+F)\"\"#!\"\"*$%\"bG\"
\"%F8" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(tmp,y^2);" "6#
-%(isolateG6$%$tmpG*$%\"yG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$
)%\"yG\"\"#\"\"\",(*$-%%sqrtG6#,&*&)%\"aGF'F()%\"xGF'F(\"\"%*$)%\"bGF4
F(\"\"\"F(F8*$F2F(!\"\"*$F0F(F:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "" "6#%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "" "6#%#%?G
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "cassinian(a,b,t);" "6#-%*cassinianG6%%\"aG%\"bG%\"tG" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$*&,&*&)%\"aG\"\"#\"\"\"
-%$cosG6#,$%\"tGF,\"\"\"F3*$-%%sqrtG6#,&*$)%\"bG\"\"%F-F3*&)F+F<F-)-%$
sinGF0F,F-!\"\"F-F3F3-F/6#F2F3*&F(F--FAFDF3" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "(((a^2*cos(2*t)+sqrt(b^4-a^4*sin(2*t)^2))*cos(t))^2+(
(a^2*cos(2*t)+sqrt(b^4-a^4*sin(2*t)^2))*sin(t))^2+a^2)^2-4*a^2*((a^2*c
os(2*t)+sqrt(b^4-a^4*sin(2*t)^2))*cos(t))^2-b^4:" "6#,(*$,(*$*&,&*&%\"
aG\"\"#-%$cosG6#*&\"\"#\"\"\"%\"tGF1F1F1-%%sqrtG6#,&*$%\"bG\"\"%F1*&F*
\"\"%-%$sinG6#*&\"\"#F1F2F1\"\"#!\"\"F1F1-F-6#F2F1\"\"#F1*$*&,&*&F*\"
\"#-F-6#*&\"\"#F1F2F1F1F1-F46#,&*$F8\"\"%F1*&F*\"\"%-F=6#*&\"\"#F1F2F1
\"\"#FBF1F1-F=6#F2F1\"\"#F1*$F*\"\"#F1\"\"#F1*(\"\"%F1*$F*\"\"#F1*&,&*
&F*\"\"#-F-6#*&\"\"#F1F2F1F1F1-F46#,&*$F8\"\"%F1*&F*\"\"%-F=6#*&\"\"#F
1F2F1\"\"#FBF1F1-F-6#F2F1\"\"#FB*$F8\"\"%FB" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "Collect(simplify(%),[sqrt(b^4-a^4+a^4*cos(2*t)^2), co
s(2*t)],loc,factor);" "6#-%(CollectG6&-%)simplifyG6#%\"%G7$-%%sqrtG6#,
(*$%\"bG\"\"%\"\"\"*$%\"aG\"\"%!\"\"*&F4\"\"%-%$cosG6#*&\"\"#F2%\"tGF2
\"\"#F2-F:6#*&\"\"#F2F>F2%$locG%'factorG" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#,**&,&*&)%\"aG\"\"'\"\"\")-%$cosG6#,$%\"tG\"\"#\"\"$F*\"\")*()F(
F1F*,**$F5F*!\"\"*$)%\"bG\"\"%F*F8*$)F(F<F*\"\"\"*&)-F-6#F0F1F*F5F*F1F
?F,F?!\"%F?-%%sqrtG6#,(F9F?F=F8*&F>F*)F,F1F*F?F*F?*&)F(F3F*)F,F<F*F3*(
F>F*,*F@F1F=F1F9!\"#F7F8F?FJF*FD**,&F;F?F(F8F?,&F;F?F(F?F?,&*$)F;F1F*F
?F7F?F?,,F9F8F?F?F@F<F7FPF=F?F?F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
119 "Sqrt[M/2] \{Cos[t], Sin[t]\}, where M is 2 a^2 Cos[2 t] + 2 Sqrt[
(-a^4+b^4) + a^4 Cos[2 t]^2], 0 < t <= 2 Pi, and a < b. " }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "build_curves();" "6#-%-build_curvesG6\"
" }}{PARA 7 "" 1 "" {TEXT -1 37 "Warning, new definition for cassinian
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7?%\"JG%7ParametrizedToImplicitG%(
astroidG%)binormalG%)cardioidG%*cassinianG%)catenaryG%'circleG%(cissoi
dG%)clothargG%)clothoidG%(cycloidG%(deltoidG%(ellipseG%'foliumG%&helix
G%*hyperbolaG%&kappaG%(kappa2DG%+lemniscateG%*lissajousG%*logspiralG%)
parabolaG%(tangentG%(torsionG%*torusknotG%)tractrixG%)twicubicG%+unitn
ormalG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "curveplot2D(cassinian
(1,1,t),t = 0 .. 2*Pi,scaling = constrained);" "6#-%,curveplot2DG6%-%*
cassinianG6%\"\"\"\"\"\"%\"tG/F+;\"\"!*&\"\"#\"\"\"%#PiGF1/%(scalingG%
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*-%'CURVESG6#7ft7$$\"\"#\"\"!F*7$$\"1[x#H]#\\%*>!#:$\"1L,%>;=Wi'!#<7$$
\"17'G1v>!y>F.$\"1i(z=K#R:8!#;7$$\"1bU6u!z[&>F.$\"1`e1aH[o=F77$$\"1HDB
BiqB>F.$\"1)Rn)*o+7S#F77$$\"19L&GD?$z=F.$\"16Ho3rgoHF77$$\"1#\\\")pOHa
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1d_#zy.O/&F77$$\"1)QWk7MuV\"F.$\"1v'>CqJVC&F77$$\"14$*=A(HiM\"F.$\"1L3
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fjf\"pJ'o$F7$\"1<oUi1`-HF77$$\"1wV-!GHPs#F7$\"1^\\gm;WwAF77$$\"125x,R#
zn\"F7$\"1c)4SQ)o)\\\"F77$$\"1w6&3bN<!oF1$\"1%zQ')>-#*['F17$$\"1@+])fe
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tFet7$$\"1fE.z_;GEFit$\"14#y\\1n'y5FcsFetFetFet7$$\"1L_Qw'G^+#!#L$!1\\
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1I3!**oE%QnF17$$!1(R=>')*****>F.$\"1T#o>hJOK$!#>7$$!1o8JJ?T%*>F.$!1v'H
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7$!1UdUj:p6XF77$$!1h>2_j\"R*eF7$!17`=s!*)R0%F77$$!1gZt(o;*4[F7$!1T3<3g
VNNF77$$!1'o8dX%pYPF7$!1.y#fIS\"RHF77$$!1GzDDm#)eEF7$!1HuALmCJAF77$$!1
$pK3Pj:h\"F7$!1P;y[9hX9F77$$!1BJ#[u<b0(F1$!1(f`e*)p(>nF17$$!1%*4QP:\">
\\\"!#I$!1L\"=!f<R3:F^gl7$$\"1+I4$=))ol&Fcs$\"1#pC#[835hFcs7$$!1V&3Te>
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F1$!1]j\"o_#QP=F17$$\"1<Za\"*pk(4'F1$!1dltAJLXeF17$$\"1$pF:aBQi\"F7$!1
6NeOEXb9F77$$\"1T:,t3H)o#F7$!1Q[7aC\"=D#F77$$\"1.b#)Qu2FPF7$!1'Rfd)*ys
#HF77$$\"1*=$Q_q'\\z%F7$!1\")[bc$ow_$F77$$\"1hlCKDNteF7$!1/[2Tb&\\/%F7
7$$\"1_!y-e%RkpF7$!1`yN0\\v#[%F77$$\"1\\eSBQwg!)F7$!1>)3%*y0&R[F77$$\"
1[MsZMF`\"*F7$!1i\")3Xjw8^F77$$\"1[];pD\\D5F.$!1#*)Qn?9#3`F77$$\"1=9\"
\\K0N8\"F.$!1,0;$RRqT&F77$$\"1s9d\\*yYC\"F.$!1$f`Vv\"RSaF77$$\"1Pi*HkY
9N\"F.$!1H&*f&G%er`F77$$\"1-iq'GOYW\"F.$!1U-)G#4^I_F77$$\"1`-jhKeK:F.$
!1rh3j2N=]F77$$\"1:0)R2#f8;F.$!1Zcyf#pJu%F77$$\"1CIncu@)o\"F.$!1:A')fG
;2WF77$$\"1+Us$>$Hg<F.$!1%f#*)pbR')RF77$$\"1\">N'fT!Q#=F.$!1><j`Uf3NF7
7$$\"1SK]m.4y=F.$!1$>d4()eA)HF77$$\"1W&)yC#HG#>F.$!19Z7JW:9CF77$$\"1\"
y;SeLX&>F.$!1tkt*3!Qv=F77$$\"1yD(p%3,y>F.$!1M!=@QacJ\"F77$$\"1-:0n-\\%
*>F.$!1G%GvDddi'F17$F($\"1\\E7Gu#3k\"!#C-%(SCALINGG6#%,CONSTRAINEDG-%+
AXESLABELSG6%%!GFecm-%%FONTG6%%&TIMESG%&ROMANG\"#=-%*AXESTICKSG6%%(DEF
AULTGF_dm-Fgcm6$%*HELVETICAG\"#7-%*AXESSTYLEG6#%&FRAMEG-%&TITLEG6$Fecm
-Fgcm6%Ficm%'ITALICGF[dm-%'COLOURG6&%$RGBGF*F*$\"*++++\"!\")-%%VIEWG6$
F_dmF_dm" 1 2 0 1 0 2 6 1 3 1 1.000000 45.000000 45.000000 0 }}}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "factormat(cassinian(a,b,t),facto
r);" "6#-%*factormatG6$-%*cassinianG6%%\"aG%\"bG%\"tG%'factorG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&)%\"aG\"\"#\"\"\"-%$cosG6#,$%\"t
GF(\"\"\"F/*$-%%sqrtG6#,$*&,&*&F&F)-%$sinGF,F/F/*$)%\"bGF(F)!\"\"F/,&F
7F/F:F/F/F=F)F/F/-%'vectorG6#7$-F+6#F.-F9FDF/" }}}{EXCHG {PARA 0 "" 0 
"" {XPPEDIT 19 1 "cassinian(a,b,t);" "6#-%*cassinianG6%%\"aG%\"bG%\"tG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7$*&,&*&)%\"aG\"\"#\"
\"\"-%$cosG6#,$%\"tGF,\"\"\"F3*$-%%sqrtG6#,&*$)%\"bG\"\"%F-F3*&)F+F<F-
)-%$sinGF0F,F-!\"\"F-F3F3-F/6#F2F3*&F(F--FAFDF3" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "ParametrizedToImplicit(cassinian(a,b,t),[x, y],t);
" "6#-%7ParametrizedToImplicitG6%-%*cassinianG6%%\"aG%\"bG%\"tG7$%\"xG
%\"yGF+" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, computation interrupt
ed" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(x = cassinian(a,b,t)
[1],y = cassinian(a,b,t)[2],(x^2+y^2+a^2)^2-4*a^2*x^2-b^4);" "6#-%%sub
sG6%/%\"xG&-%*cassinianG6%%\"aG%\"bG%\"tG6#\"\"\"/%\"yG&-F*6%F,F-F.6#
\"\"#,(*$,(*$F'\"\"#\"\"\"*$F2\"\"#F=*$F,\"\"#F=\"\"#F=*(\"\"%F=*$F,\"
\"#F=F'\"\"#!\"\"*$F-\"\"%FH" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*$),
(*&),&*&)%\"aG\"\"#\"\"\"-%$cosG6#,$%\"tGF-\"\"\"F4*$-%%sqrtG6#,&*$)%
\"bG\"\"%F.F4*&)F,F=F.)-%$sinGF1F-F.!\"\"F.F4F-F.)-F06#F3F-F.F4*&F(F.)
-FBFFF-F.F4*$F+F.F4F-F.F4*(F(F.FDF.F+F.!\"%F:FC" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "simplify(%,assume = real);" "6#-%)simplifyG6$%\"%G
/%'assumeG%%realG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,H*&)%\"aG\"\")\"
\"\")-%$cosG6#,$%\"tG\"\"#F/F(!\")*&)F&F/F()%\"bG\"\"%F(F/*(F3F()F&F5F
(F)F(F'*$F7F(\"\"\"*$F3F(!\"\"*$)F4F'F(F9*&F3F(F7F(!\"#*$F%F(F9*$)F&\"
\"'F(F?**)-F+6#F.F/F(F7F(F*F9-%%sqrtG6#,(F:F9F8F;*&F7F(F)F(F9F(F0*(FEF
(FBF(F)F(F0*(FEF(F2F(F3F(!\"%*&FEF(FBF(F5*(FBF()F*\"\"$F(FHF(F'*(F7F(F
*F(FHF(F5**F2F(F*F(FHF(F3F(F5*(FBF(F*F(FHF(FO*&F%F()F*F5F(F'*&FBF(F)F(
F5" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Collect(%,[sqrt, b^4-a^4+
a^4*cos(2*t)^2],factor,loc);" "6#-%(CollectG6&%\"%G7$%%sqrtG,(*$%\"bG
\"\"%\"\"\"*$%\"aG\"\"%!\"\"*&F/\"\"%-%$cosG6#*&\"\"#F-%\"tGF-\"\"#F-%
'factorG%$locG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,@**)%\"aG\"\"#\"\"
\"-%$cosG6#,$%\"tGF'\"\"\",,*&)-F*6#F-F'F(F%F(!\"#*&)F&\"\"%F()F)F'F(F
'*$F%F(F.*$)%\"bGF7F(F.*$F6F(!\"\"F.-%%sqrtG6#,(F:F.F=F>F5F.F(F7*&)F&
\"\")F(F8F(!\")*&F%F(F;F(F'*(F;F(F6F(F8F(FEF=F.F:F>*$)F<FEF(F.*&F;F(F6
F(F4*$FDF(F.*$)F&\"\"'F(F4*&FDF()F)F7F(FE*(F1F(FNF(F8F(FF*(F1F(F%F(F;F
(!\"%*&F1F(FNF(F7*&FNF(F8F(F7" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "" "6#%#%?G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "" "6#%#%?G" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "%?;" "6#%#%?G" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "CassiniPlot := proc (p, q) options operator, arrow; imp
licitplot2D(subs(a = p,b = q,cassinian(a,b,x,y)),x = -2 .. 2,y = -2 ..
 2,scaling = constrained,axes = normal,grid = [100, 100]) end:" "6#>%,
CassiniPlotGR6$%\"pG%\"qG7\"6$%)operatorG%&arrowG6\"-%/implicitplot2DG
6(-%%subsG6%/%\"aGF'/%\"bGF(-%*cassinianG6&F5F7%\"xG%\"yG/F;;,$\"\"#!
\"\"\"\"#/F<;,$\"\"#FA\"\"#/%(scalingG%,constrainedG/%%axesG%'normalG/
%%gridG7$\"$+\"\"$+\"F-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "
plts := [CassiniPlot(1.0,1.0), CassiniPlot(1,1.05), CassiniPlot(1,1.1)
, CassiniPlot(1,1.2), CassiniPlot(1,1.5), CassiniPlot(1,2), CassiniPlo
t(1,2.5), CassiniPlot(1,3)]:" "6#>%%pltsG7*-%,CassiniPlotG6$$\"#5!\"\"
$\"#5!\"\"-F'6$\"\"\"$\"$0\"!\"#-F'6$\"\"\"$\"#6!\"\"-F'6$\"\"\"$\"#7!
\"\"-F'6$\"\"\"$\"#:!\"\"-F'6$\"\"\"\"\"#-F'6$\"\"\"$\"#D!\"\"-F'6$\"
\"\"\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "plots[display](plt
s,axes = normal,title = \"Cassinian Ovals\");" "6#-&%&plotsG6#%(displa
yG6%%%pltsG/%%axesG%'normalG/%&titleGQ0Cassinian~Ovals6\"" }}{PARA 13 
"" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "60-%'CURVESG6acl7$7$$!1$RRR
RRRR\"!#:$!1t/dHs%>O\"!#;7$$!1:j.?)oFS\"F*$!1(*4555555F-7$F.7$$!19!flp
>YS\"F*$!1hy/R)zH.*!#<7$7$$!1(3\"RE4h49F*$!1DfggggggF9F47$F;7$$!1Z:-\\
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