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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 1 " " }{TEXT 257 26 "Four w
ays to plot a circle" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 201 "Plottin
g a circle is quite straightforward, and also unexciting.  However, si
nce it is so familiar an object, it is probably useful to use it to co
mpare four different methods for doing the same thing." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "First, le
t's recall that a circle can be expressed as the set of points (" }
{XPPEDIT 18 0 "(x,y)" "6$%\"xG%\"yG" }{TEXT -1 22 ")  which are distan
ce " }{XPPEDIT 18 0 "R" "I\"RG6\"" }{TEXT -1 18 " from the center (" }
{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 12 ").  That is," }}
{PARA 257 "" 0 "" {XPPEDIT 18 0 "sqrt((x-a)^2+(y-b)^2) = R" "/-%%sqrtG
6#,&*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#F**$,&%\"yGF*%\"bGF,\"\"#F*%\"RG" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "or, since " }{XPPEDIT 
18 0 "r" "I\"rG6\"" }{TEXT -1 75 " is always positive, we can square b
oth sides to obtain the usual equation " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "circleEqn:= (x-a)^2+(y-b)^2=R^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*circleEqnG/,&*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#F**$,&%\"
yGF*%\"bGF,F-F**$%\"RGF-" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "1. A
s a pair of functions" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "To plot t
he circle as a pair of functions, we need to solve for " }{XPPEDIT 18 
0 "y" "I\"yG6\"" }{TEXT -1 51 ".  We can do this easily in our heads, \+
(at least I " }{TEXT 259 4 "hope" }{TEXT -1 23 " you can) or ask maple
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "circfuncs:= \{solve(ci
rcleEqn,y)\};" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*circfuncsG<$,&%\"bG\"\"\"*$,**$%\"xG\"\"#!\"\"*&F,F(
%\"aGF(F-*$F0F-F.*$%\"RGF-F(#F(F-F(,&F'F(F)F." }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 96 "Since we need a specific choice for the center and rad
ius for this to work, let's substitute in " }{XPPEDIT 18 0 "a=2,b=1,R=
3" "6%/%\"aG\"\"#/%\"bG\"\"\"/%\"RG\"\"$" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "mycircfuncs:= subs(a=2,b=1,R=3,circ
funcs);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,mycircfuncsG<$,&\"\"\"F'
*$,(*$%\"xG\"\"#!\"\"F+\"\"%\"\"&F'#F'F,F',&F'F'F(F-" }}}{EXCHG {PARA 
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0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 23 "2. In polar coordinates" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 
"If the circle is centered at the origin, it is, in some sense, most n
aturally expressed in polar coordinates, as " }{XPPEDIT 18 0 "r=R" "/%
\"rG%\"RG" }{TEXT -1 209 ".  If the center is not at the origin, thing
s are a bit messier, but not too bad.  Recall that polar coordinates r
epresent a point in the plane in terms of its distance from the origin
 (usually represented as " }{XPPEDIT 18 0 "r" "I\"rG6\"" }{TEXT -1 16 
") and the angle " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 28 "
 it makes with the positive " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 
80 "-axis (measured counterclockwise).  Thus, from elementary trigonom
etry, we have " }{XPPEDIT 18 0 "x=r*cos(theta" "/%\"xG*&%\"rG\"\"\"-%$
cosG6#%&thetaGF&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=r*sin(theta" "
/%\"yG*&%\"rG\"\"\"-%$sinG6#%&thetaGF&" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Making this change g
ives us the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "s
ubs(x=r*cos(theta),y=r*sin(theta),circleEqn);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&*$,&*&%\"rG\"\"\"-%$cosG6#%&thetaGF)F)%\"aG!\"\"\"\"
#F)*$,&*&F(F)-%$sinGF,F)F)%\"bGF/F0F)*$%\"RGF0" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 79 "which isn't too bad...   Since we usually try to rep
resent a polar function as " }{XPPEDIT 18 0 "r=f(theta)" "/%\"rG-%\"fG
6#%&thetaG" }{TEXT -1 19 ", we can solve for " }{XPPEDIT 18 0 "r" "I\"
rG6\"" }{TEXT -1 25 " to obtain the following." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "polarcirc:=solve(\",r);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%*polarcircG6$,(*&-%$cosG6#%&thetaG\"\"\"%\"aGF,F,*&-%
$sinGF*F,%\"bGF,F,*$,.*&F(\"\"#F-F5F,**F(F,F-F,F/F,F1F,F5*&F/F5F1F5F,*
$F1F5!\"\"*$%\"RGF5F,*$F-F5F9#F,F5F,,(F'F,F.F,F2F9" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 59 "This is really rather dreadful looking, but notic
e that if " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "b" "I\"bG6\"" }{TEXT -1 40 " are both zero, the first s
implifies to " }{XPPEDIT 18 0 "r=R" "/%\"rG%\"RG" }{TEXT -1 25 " and t
he second becomes  " }{XPPEDIT 18 0 " r=-R" "/%\"rG,$%\"RG!\"\"" }
{TEXT -1 89 ".  On closer examination, you can see that if  the circle
 surrounds the origin, (that is " }{XPPEDIT 18 0 "a^2+b^2<R^2" "2,&*$%
\"aG\"\"#\"\"\"*$%\"bG\"\"#F'*$%\"RG\"\"#" }{TEXT -1 139 "), then the \+
second always corresponds to negative values of the radius, and can be
 ignored for our purposes.  Again, since we want to use  " }{XPPEDIT 
18 0 "a=2,b=1,R=3" "6%/%\"aG\"\"#/%\"bG\"\"\"/%\"RG\"\"$" }{TEXT -1 
36 ", we substitute into the first one. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "mypolarcirc:= subs(a=2,b=1,R=3,polarcirc[1]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,mypolarcircG,(-%$cosG6#%&thetaG\"\"
#-%$sinGF(\"\"\"*$,**$F&F*\"\"%*&F&F-F+F-F1*$F+F*F-F1F-#F-F*F-" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "To plot it, we just tell maple th
at we are using polar coordinates.  We need to allow theta to range ov
er the whole circle, of course." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "plot(mypolarcirc,theta=0..2*Pi,coords=polar);" }}{PARA 13 "" 
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\"\"F+F+" 2 245 150 150 2 0 1 0 2 9 0 4 1 1.000000 45.000000 
45.000000 10030 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 
1 0 0 0 114 92 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "I s
hould point out that maple can also do parametric polar plots, where b
oth the radius and angle are functions of a parameter " }{XPPEDIT 18 
0 "t" "I\"tG6\"" }{TEXT -1 60 ", for example, we can make a \"butterfl
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{PARA 0 "" 0 "" {TEXT -1 77 "This, of course, has nothing to do with p
lotting a circle, but is fun anyway." }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 21 "3. In parametric form" }}{PARA 0 "" 0 "" {TEXT -1 130 "Re
presenting the circle parametrically is quite simple.  Recall that for
 a circle centered at the origin, the pair of  equations  " }{XPPEDIT 
18 0 "\{x=R*cos(theta), y=R*sin(theta)\}" "<$/%\"xG*&%\"RG\"\"\"-%$cos
G6#%&thetaGF'/%\"yG*&F&F'-%$sinG6#F+F'" }{TEXT -1 30 "  describe a cir
cle of radius " }{XPPEDIT 18 0 "R" "I\"RG6\"" }{TEXT -1 4 " as " }
{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 18 " varies from 0 to " 
}{XPPEDIT 18 0 "2*Pi" "*&\"\"#\"\"\"%#PiGF$" }{TEXT -1 107 ".  If we w
ant to move the center from the origin, all we have to do is add the a
ppropriate constant to the " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 13 " coordinates." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([2+3*cos(theta),1+3*si
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{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "4. Described implicitly" }}{PARA 
0 "" 0 "" {TEXT -1 212 "Maple can also plot the relation describing th
e circle implicitly.  It has to work much harder in this case, and the
 results aren't always as nice.  But the call is quite easy, although \+
first we must load in the " }{TEXT 260 14 "plots library." }}{EXCHG 
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" 0 "" {MPLTEXT 1 0 60 "implicitplot( (x-2)^2 + (y-1)^2= 3^2, x=-10..1
0, y=-10..10);" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6O7$7$$!1(***
**********R!#;$!1kbbbbbbvF*7$$!1!H9dG9dG'F*F(7$F-7$$!1jbbbbbbvF*$!1BVW
WWWWW!#<7$7$$!1Yr&G9dG9*F*$\"1/++++++SF*F17$7$F9$\"10++++++SF*7$$!1S:Y
Q:YQ&*F*$\"1U:YQ:YQ&*F*7$7$$!1:dG9dG9(*F*$\"1+++++++7!#:FA7$FG7$$!1y6%
HN#)eq)F*$\"1=THN#)eq;FL7$7$$!1.++++++!)F*$\"\"#\"\"!FN7$7$$!1-++++++!
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LF^r7$FbuFet7$F]u7$$\"1i#yM/8Rx$FL$!1h#yM/8RP\"FL7$7$$\"1,++++++SFLFio
Fgu7$F]v7$$\"1F0@%ot%*=%FL$!1h_5Uot%*)*F*7$7$$\"1+++++++WFL$!1^bbbbbbv
F*Fav7$7$FhvF_oF_u7$Fgv7$$\"14EyM/8RXFL$!1%3EyM/8R&F*7$7$$\"1H9dG9dGYF
LF(F_w7$Few7$$\"1_5Uot%*yZFL$\"1)y%*y:j_5#F67$7$$\"1:dG9dG9\\FLF;Fiw7$
F_x7$$\"1MLLLLLL\\FL$\"1smmmmmmmF*7$7$$\"1r&G9dG9(\\FLFJFcx7$7$$\"1s&G
9dG9(\\FLFJ7$$\"13444444\\FL$\"1\"444444\\\"FL7$7$$\"1+++++++[FLFWF`y7
$7$$\"1*************z%FLFWF]w-%'COLOURG6&%$RGBG\"\"\"FYFY-%+AXESLABELS
G6$%\"xG%\"yG" 2 272 221 221 6 0 1 0 2 9 0 4 1 1.000000 45.000000 
45.000000 10010 10061 10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 
1 0 0 0 96 32 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "It i
s worth pointing out a few things here.  First, notice that the circle
 looks a bit ragged.  Also, note that I specified a rather large range
 for " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "y" "I\"yG6\"" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "-10" ",$\"#5!\"\"" }
{TEXT -1 4 " to " }{XPPEDIT 18 0 "10" "\"#5" }{TEXT -1 200 ") but mapl
e chose to only show a much smaller range.  This is because of the way
 implicitplot works.  The range we specify specifies not the desired p
lotting region, but where to search for solutions." }}{PARA 0 "" 0 "" 
{TEXT -1 258 "Maple chops up the region given into a number of subregi
ons (50 by 50, unless you say otherwise) and searches for a solution i
n each of them.  It then plots whatever it found, assuming it is conne
cted \"in a nice way\".  Sometimes it makes mistakes, of course." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "We can greatly improve our picture
 by specifying a better search area." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "implicitplot( (x-2)^2 + (y-1)^2= 3^2, x=-1..5, y=-2..
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\"1++++++XNFjo7$7$F_^l$\"1\"444444c$FjoFi_l7$F^^l7$$\"1)************H&
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$$\"1&************H)F*FablFi`l7$Feal7$$\"15++++vosF*$\"1,+++]78PFjo7$7
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7$7$$\"1++++++S;Fjo$!1+++++]x>FjoFjfl7$Fffl7$$\"1I9dG9d`9Fjo$\"1s&G9dG
k%RFjo7$7$Fagl$\"1+++++]xRFjoFfgl7$F`gl7$$\"1FFFFFFx=Fjo$!1FFFFFF(*>Fj
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$$\"1++++++?@FjoFihlFfil7$Fbil7$$\"1-++++]#)=FjoFcil7$7$F[jlFcilF^jl7$
Fjil7$$\"1p2Bp2BRBFjo$!1p2Bp2Bz>Fjo7$7$$\"1************fBFjoFcglFdjl7$
Fbjl7$$\"1vsssssA@Fjo$\"1FFFFFF(*RFjo7$7$$\"1++++++gBFjoF]hlF^[m7$Fjjl
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\"1-+++++(Q#Fjo$\"1++++++tRFjo7$7$F_\\mFgflFb\\m7$F^\\m7$$\"1,+++++UFF
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\"1mmmmmm;RFjo7$7$Fa]mFaelFd]m7$F`]m7$$\"1,+++]7GHFjo$!1++++]7[=Fjo7$7
$$\"1,+++++!3$FjoF\\blF\\^m7$Fj]m7$$\"1/++++vBIFjo$\"1*********\\i\"QF
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Fjo$!1\"======p\"FjoFh_m7$7$F_`m$\"1\"======p$Fjo7$$\"1'***********pJF
joFacl7$7$$\"1(***********pJFjoFaclF\\_m7$F^`m7$$\"1O#)eqk<JMFjo$!1N#)
eqk<J;Fjo7$7$$\"1,+++++gNFjo$!1!444444c\"FjoF_am7$Fd`m7$$\"11+++++qMFj
o$\"1(***********4OFjo7$7$Ffam$\"1!444444c$FjoF[bm7$Feam7$$\"1,+++++&e
$Fjo$!1***********\\a\"Fjo7$7$$\"1:dG9dGCOFjoFf^lFebm7$F[cm7$$\"1IN#)e
qkFPFjo$!1HN#)eqkZ9Fjo7$7$$\"1-++++++QFjo$!1)**********pR\"FjoF_cm7$7$
Ffcm$\"1)**********pR$Fjo7$$\"17dG9dGCOFjoFg_l7$F^dmFabm7$7$Ffcm$!1***
********pR\"Fjo7$$\"1,+++++lQFjo$!1***********\\M\"Fjo7$7$$\"1+++++DYR
FjoF]\\lFfdm7$F\\em7$$\"1y6%HN#)e*RFjo$!1w6%HN#)eB\"Fjo7$7$$\"1-+++++S
SFjo$!1lmmmmm'>\"FjoF`em7$7$Fgem$\"1lmmmmm'>$Fjo7$$\"1*********\\i%RFj
oF^]l7$F_fmF[dm7$Ffem7$$\"1MLLLLL=TFjo$!1LLLLLL=6Fjo7$7$$\"1mmmmmm'>%F
joFhyFdfm7$Fjfm7$$\"1x6%HN#)eB%Fjo$!1`<THN#)e**F*7$7$$\"1-+++++!G%Fjo$
!1w********\\i%*F*F^gm7$7$Fegm$\"1(********\\i%HFjo7$$\"1lmmmmm'>%FjoF
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jo$!15r&G9dGC'F*F\\im7$7$Fcim$\"16dG9dGCEFjo7$$\"1***********pR%FjoFcx
7$F[jmFjgm7$Fbim7$$\"1,+++++XXFjo$!1')**********\\eF*7$7$$\"1\"444444c
%FjoF[tF`jm7$Ffjm7$$\"1O#)eqk<JYFjo$!1QB)eqk<J%F*7$7$$\"1\"======p%Fjo
$!1-++++++KF*Fjjm7$7$Fa[nFjt7$$\"1++++]78ZFjo$!1%)*******\\7t#F*7$7$$
\"1-+++++gZFjo$!1X***********p\"F*Fg[n7$7$F^\\n$\"1&***********p@Fjo7$
Fa[nFeu7$Ff\\n7$$\"1(***********4YFjo$\"11+++++qCFjo7$7$FgjmFavFh\\n7$
F^]nFhim7$7$F^\\n$!1Y***********p\"F*7$$\"1&HN#)eqky%Fjo$!1EHN#)eqk5F*
7$7$$\"1**********\\(z%FjoF5Fd]n7$Fj]n7$$\"1++++]7[[Fjo$\"1!=++++v=(F2
7$7$$\"1+++++]x[FjoF@F^^n7$Fd^n7$$\"1***********>!\\Fjo$\"1?+++++!e#F*
7$7$$\"1**********\\P\\FjoFLFh^n7$7$$\"1+++++]P\\FjoFL7$$\"1r&G9dGk%\\
Fjo$\"12Vr&G9d`%F*7$7$$\"1+++++]x\\FjoFenFe_n7$7$$\"1**********\\x\\Fj
oFen7$$\"1q2Bp2Bz\\Fjo$\"1KBp2Bp2mF*7$7$$\"1+++++](*\\FjoFaoFb`n7$Fh`n
7$Fi`n$\"1F+++++D))F*7$7$Fi`nF]pF\\an7$F`an7$$\"1FFFFFF(*\\Fjo$\"1wsss
ssA6Fjo7$7$F\\`nFhpFban7$7$F``nFhp7$$\"1++++++t\\Fjo$\"1/+++++(Q\"Fjo7
$7$Fc_nFbqF[bn7$Fabn7$$\"1mmmmmm;\\Fjo$\"1PLLLLL$o\"Fjo7$7$Fe^nF\\rFcb
n7$Fibn7$$\"1)********\\i\"[Fjo$\"1/++++vB?Fjo7$7$F[^nFfrF[cn7$FacnFc
\\n-%'COLOURG6&%$RGBG\"\"\"\"\"!Fhcn-%+AXESLABELSG6$%\"xG%\"yG" 2 272 
221 221 2 0 1 0 2 9 0 4 1 1.000000 45.000000 45.000000 10030 10061 
10056 10074 0 0 0 20030 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 96 32 0 0 0 0 
0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Also, we can specify the n
umber of points to search using the \"grid\" option.  Rather than the \+
circle, let's use a more complicated example to better show what goes \+
wrong.  " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "A trickier example o
f implicitplot." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 " A deltiod has
 3 cusps, making it trickier to get right.  The equation (at least in \+
Cartesian cordinates) is also nastier." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 117 "implicitplot( (x^2+y^2)^2 + 8*x*(3*y^2-x^2) + 18*(x^
2+y^2) = 27,\nx=-10..10,y=-10..10,scaling=constrained,axes=boxed);" }}
{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6E7$7$$!1+++++++7!#:$!1LLLLL$e/
#F*7$$!1<7ON*3G@\"F*$!\"#\"\"!7$F-7$$!1*[\">`v;27F*$!16&3oWKG*>F*7$7$F
($!1\"Hh^kI8*=F*F47$7$F($\"1*Gh^kI8*=F*7$F.$\"\"#F27$FA7$$!1!QTs^lkC\"
F*$\"1!QTs^lk/#F*7$7$F($\"1MLLLL$e/#F*FE7$F'7$$!1!46azeN>\"F*$!15*)e/7
W1?F*7$7$$!1+++++]v6F*F0FO7$FUF:7$7$$!1(*************R!#;$!1M?KfNc(R\"
F*7$$!1E(yHQ1,'*)FgnF(7$Fjn7$$!1ussss(*yrFgn$!1BFFFF-@))Fgn7$7$$!1BoQ&
))f4'*)FgnFenF^o7$7$$!1?oQ&))f4'*)FgnFen7$Feo$\"1!HoQ&))f4'*!#<7$7$$!1
AoQ&))f4'*)Fgn$\"1/++++++SFgnF[p7$7$Feo$\"10++++++SFgn7$$!1+++++DT5F*$
\"1,++++DT5F*7$7$$!1G(yHQ1,'*)Fgn$\"1+++++++7F*Fip7$F_qF>7$7$Fen$\"1L?
KfNc(R\"F*7$FVFB7$FiqFK7$FZ7$$!1?UF]vrgLFgn$!1yD(\\CGRE\"F*7$7$$!1]h%Q
:'458FgnF(F\\r7$Fbr7$$\"10TybZN3XF^p$!10%ybZN3X)Fgn7$7$Fcp$!1X$)oM)o%e
#)FgnFfr7$7$Fcp$\"1U$)oM)o%e#)Fgn7$$!1fh%Q:'458FgnFbq7$FcsFfq7$7$Fgp$!
1Y$)oM)o%e#)Fgn7$$\"1<5[yA3*)fFgn$!155[yA3*)fFgn7$7$Fbq$!1cp%Rn[=f%Fgn
F[t7$F`s7$$\"1>===obF5F*$\"16====VCdFgn7$7$Fbq$\"1fp%Rn[=f%FgnFet7$7$F
bq$!1dp%Rn[=f%Fgn7$$\"1K-Dq)f^B\"F*$!18B]-()f^VFgn7$7$$\"1\"zT;n:J[\"F
*FenFbu7$Fhu7$$\"1!zT;n:J[\"F*$\"1*4#e$GV)o6Fgn7$7$F]vFcpF\\v7$7$$\"1*
yT;n:J[\"F*Fcp7$Fbq$\"1ep%Rn[=f%Fgn-%'COLOURG6&%$RGBG\"\"\"F2F2-%(SCAL
INGG6#%,CONSTRAINEDG-%+AXESLABELSG6$%\"xG%\"yG-%*AXESSTYLEG6#%$BOXG" 
2 398 218 218 2 0 1 0 2 9 0 2 1 1.000000 45.000000 45.000000 10030 
10061 10056 10074 0 0 0 20010 0 12010 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 
0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Certainly some refinin
g is in order.  Note the strange little island down in the lower left \+
corner.  Reducing the area of search will help a bit." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "implicitplot( (x^2+y^2)^2 + 8*x*(3
*y^2-x^2) + 18*(x^2+y^2) = 27,\nx=-1.5..3,y=-2.5..2.5,scaling=constrai
ned,axes=boxed);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6[u7$7$$!1+
+++++?8!#:$!1d/wbm<BBF*7$$!1G3gtZ!=L\"F*$!1+++++++BF*7$F-7$$!1ep:CsyC8
F*$!1Dy/i3o%H#F*7$7$F($!1W\"=&3!Q8B#F*F37$7$F($\"1P\"=&3!Q8B#F*7$$!1H3
gtZ!=L\"F*$\"1*************H#F*7$7$$!1I3gtZ!=L\"F*$\"1+++++++BF*7$$!1(
>ve9*Q(Q\"F*$\"1=-vGo([P#F*7$7$F($\"1f/wbm<BBF*FK7$F'7$$!1h=:bA4;8F*$!
1a,sQ>M/BF*7$7$$!1T9s$)yI38F*F0FU7$7$$!1S9s$)yI38F*F0F97$7$$!1++++++S6
F*$!1o?*G3vM8#F*7$$!1lOc&em$[7F*$!1+++++++@F*7$Fbo7$$!1>&pf['or6F*$!1,
sO$z#zk?F*7$7$$!1G]fq[y97F*$!1+++++++>F*Fho7$F^p7$$!1B\\=C;J#=\"F*$!1v
*QU3()H&=F*7$7$$!17q]V&ez<\"F*$!1+++++++<F*Fdp7$7$$!18q]V&ez<\"F*F]q7$
$!1Os8.Idk6F*$!1\\3=jmps;F*7$7$$!1JE$*z*fZ9\"F*$!1+++++++:F*Fcq7$Fiq7$
$!1O5zgvKV6F*$!1#=K-r-j\\\"F*7$7$F^o$!1F58j\\vs9F*F_r7$7$F^o$\"1F58j\\
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