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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 38 "Making an ellipse look \+
like an ellipse" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 45 "Suppose we have a dog in a yard.  We know, or" }
{TEXT 258 7 " should" }{TEXT -1 204 " know, that if we tie his leash d
own at one point and let him roam freely, he can reach any point which
 lies in a circle whose radius is the length of the leash, and the cen
ter is the point it is tied at." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Now, suppose we tie the leash at \+
" }{TEXT 257 4 "both" }{TEXT -1 159 " ends, but allow the collar to sl
ip along the rope.  What set of points can the dog get to?  Well, if w
e give the points the leash is attached at  coordinates " }{XPPEDIT 
18 0 "[x[1],y[1]" "7$&%\"xG6#\"\"\"&%\"yG6#\"\"\"" }{TEXT -1 4 "and " 
}{XPPEDIT 18 0 "[x[2],y[2]" "7$&%\"xG6#\"\"#&%\"yG6#\"\"#" }{TEXT -1 
105 " and let the leash have length c, then the set of points which ar
e accessible are given by the inequality" }}}{EXCHG {PARA 257 "" 0 "" 
{XPPEDIT 18 0 "dist([x[1],y[1]],[x,y])+dist([x[2],y[2]],[x,y])=c" "/,&
-%%distG6$7$&%\"xG6#\"\"\"&%\"yG6#\"\"\"7$F)F-\"\"\"-F%6$7$&F)6#\"\"#&
F-6#\"\"#7$F)F-F1%\"cG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 126 "However, we need only consider the case of equality.  Fu
rthermore, using the distance formula, this becomes the set of points \+
" }{XPPEDIT 18 0 "[x,y]" "7$%\"xG%\"yG" }{TEXT -1 14 " which satisfy" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "dogEqn:= sqrt( (x-x[1])^2
 + (y-y[1])^2) + sqrt( (x-x[2])^2 + (y-y[2])^2) = c:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 62 "If you don't already know that this is an ellip
se, we can use " }{TEXT 260 12 "implicitplot" }{TEXT -1 99 " to see th
at.  Let us take our points to be (-1,0) and (1,0), and c=4.  (The poi
nts are called the " }{TEXT 259 4 "foci" }{TEXT -1 17 " of the ellipse
)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "myDogEqn := subs(x[1]
=-1,y[1]=0,x[2]=1,y[2]=0,c=3,dogEqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%)myDogEqnG/,&*$,**$%\"xG\"\"#\"\"\"F*F+F,F,*$%\"yGF+F,#F,F+F,*$,*
F)F,F*!\"#F,F,F-F,F/F,\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "implici
tplot(myDogEqn, x=-2..2, y=-2..2, scaling=constrained,axes=boxed);" }}
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 187 "That surely looks like an ellipse
 to me.  But how do we know it is an ellipse?  From high school, I see
m to remember (and maybe you do too) that the equation of an ellipse i
s of the form " }{XPPEDIT 18 0 "x^2/a^2+y^2/b^2=1" "/,&*&%\"xG\"\"#*$%
\"aG\"\"#!\"\"\"\"\"*&%\"yG\"\"#*$%\"bG\"\"#F*F+\"\"\"" }{TEXT -1 77 "
, and this doesn't look like that.  So now we get to make it look like
 that. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "But how?  By algebrai
c messing around.  This is, in fact, a little easier to do by hand on \+
paper, but we will coerce maple into doing it anyway. " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 261 7 "WARNING" }{TEXT -1 3 ":  " }{TEXT 262 
148 "most of this would be much easier to do on paper, with maple's as
sistance. but we are doing this for pedagogical reasons, and out of sh
eer meanness." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 186 "First, so that we can keep track of what is going on,
 let's replace the complicated stuff inside the square roots with some
thing simpler.  Then after we are done, we will put it back in." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dogEqn1 := sqrt(e)+sqrt(f)=c
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dogEqn1G/,&*$%\"eG#\"\"\"\"\"#
F**$%\"fGF)F*%\"cG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 184 "We need to
 manipulate the left and right sides of the equation independantly, so
 lets give them names.  The maple functions rhs and lhs give us the ri
ght and left sides of an equation:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "dogEqn1r := rhs(dogEqn1); dogEqn1l := lhs(dogEqn1);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)dogEqn1rG%\"cG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%)dogEqn1lG,&*$%\"eG#\"\"\"\"\"#F)*$%\"fGF(F)" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Now we square both sides, in hopes
 that the square roots will go away (they won't)." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "expand((dogEqn1l)^2) = (dogEqn1r)^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\"eG\"\"\"*&F%#F&\"\"#%\"fGF(F)F*F
&*$%\"cGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "Doesn't look much \+
better, huh?  But notice that if we move the terms without a square ro
ot over to the other side, and then square both sides again, it will g
et rid of the roots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "exp
and((dogEqn1l)^2-(e+f)) = (dogEqn1r)^2 - (e+f);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,$*&%\"eG#\"\"\"\"\"#%\"fGF'F),(*$%\"cGF)F(F&!\"\"F*F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "expand(((dogEqn1l)^2-(e
+f)))^2 = ((dogEqn1r)^2 - (e+f))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/,$*&%\"eG\"\"\"%\"fGF'\"\"%*$,(*$%\"cG\"\"#F'F&!\"\"F(F/F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "The roots are gone.  So that is wh
at we want.  Lets put everything on one side:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 70 "dogFunc := expand( ((dogEqn1l)^2-(e+f))^2 - ((do
gEqn1r)^2 - (e+f))^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(dogFuncG,
.*&%\"eG\"\"\"%\"fGF(\"\"#*$%\"cG\"\"%!\"\"*&F,F*F'F(F**&F,F*F)F(F**$F
'F*F.*$F)F*F." }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 46 "Finally, we can substitute the real values of " }
{XPPEDIT 18 0 "e" "I\"eG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f" "I
\"fG6\"" }{TEXT -1 9 " back in." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 89 "dogMess := expand(\n subs(e=(x-x[1])^2 + (y-y[1])^2, f=(x-x[2]
)^2 + (y-y[2])^2, dogFunc));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(dog
MessG,jp*(%\"xG\"\"\"&F'6#F(F(&%\"yG6#\"\"#F.!\"%*(F)F.F,F(F+F(F/*(&F,
F*F.F,F(F+F(F/*(%\"cGF.F'F(&F'F-F(F/*(F,F(F2F(F5F.F/*(F)F.F'F(F5F(F/*(
F2F.F'F(F5F(F/*(F,F(F2F(F+F.F/*(F,F.F2F(F+F(\"\")*(F'F(F)F(F5F.F/*(F4F
.F'F(F)F(F/*(F'F.F)F(F5F(F;**F'F(F)F(F,F(F+F(F;*$F4\"\"%!\"\"*(F4F.F,F
(F2F(F/*(F4F.F,F(F+F(F/*(F'F(F)F(F2F.FA*(F)F.F,F(F2F(FA*(F'F(F5F(F+F.F
A*(F5F.F,F(F+F(FA*&F4F.F'F.FA*&F4F.F,F.FA*$F)FAFB*$F2FAFB*$F5FAFB*$F+F
AFB*&F'F.F5F.F/*&F)F.F'F.F/*&F)F.F5F.F.*&F)F.F+F.F.*&F,F.F+F.F/*&F2F.F
5F.F.*&F2F.F,F.F/*&F2F.F+F.F.*&F4F.F)F.F.*&F4F.F2F.F.*&F4F.F5F.F.*&F4F
.F+F.F.*&F'F(F)\"\"$FA*&F)F.F2F.!\"#*&F,F(F2FfnFA*&F'F(F5FfnFA*&F5F.F+
F.Fhn*&F,F(F+FfnFA**F,F(F2F(F'F(F5F(F;**F'F(F5F(F,F(F+F(!\")**F'F(F)F(
F,F(F2F(F_o" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Well, that looks a
wful.  Lets see what we get when we put in our specific values of c, \+
" }{XPPEDIT 18 0 "[x[1],y[1]" "7$&%\"xG6#\"\"\"&%\"yG6#\"\"\"" }{TEXT 
-1 4 "and " }{XPPEDIT 18 0 "[x[2],y[2]]" "7$&%\"xG6#\"\"#&%\"yG6#\"\"#
" }{TEXT -1 9 ", though:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 
"myDogMess := subs(x[1]=-1,y[1]=0,x[2]=1,y[2]=0,c=3,dogMess);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*myDogMessG,(*$%\"xG\"\"#\"#?!#X\"\"
\"*$%\"yGF(\"#O" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Not bad.  Look
s just like we'd expect an ellipse to look like. Let's plot it:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "implicitplot(myDogMess,x=-2.
.2, y=-2..2, scaling=constrained,axes=boxed);" }}{PARA 13 "" 1 "" 
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es, that is the same thing, and it is clearly the expected form.  But,
 after that work, we have more: we have the general form of an ellips \+
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\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "[x[2],y[2]" "7$&%\"xG6#\"\"#
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>7$7$$\"12r$**Q#=49F*FhsFgam7$F]bm7$$\"1'o>D%)f7Q\"F*$\"179.[d,ueFjs7$
7$$\"1()>q%z!)yP\"F*F\\vFabm7$FgbmF[`m-%'COLOURG6&%$RGBG\"\"\"\"\"!F`c
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "But let's see if we can neaten up \+
this apparent mess a bit, collecting terms in x and y." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "collect(dogMess,[x,y],distributed);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,J*$%\"cG\"\"%!\"\"*&F%\"\"#&%\"xG
6#\"\"\"F)F)*&F%F)&%\"yGF,F)F)*&F%F)&F+6#F)F)F)*&F%F)&F0F3F)F)*$F*F&F'
*&F*F)F/F)!\"#*&F*F)F2F)F)*&F*F)F5F)F)*$F/F&F'*&F/F)F2F)F)*&F/F)F5F)F)
*$F2F&F'*&F2F)F5F)F8*$F5F&F'*&,**$F%F)F&*$F*F)!\"%*&F*F-F2F-\"\")*$F2F
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-F0F-F-*&,*FCF&*$F/F)FE*&F/F-F5F-FG*$F5F)FEF-F0F)F-*&,6*&F%F)F*F-FE*&F
%F)F2F-FE*$F*\"\"$F&*&F*F)F2F-FE*&F*F-F/F)F&*&F*F-F2F)FE*&F*F-F5F)FE*&
F/F)F2F-FE*$F2FZF&*&F2F-F5F)F&F-F+F-F-*&,6*&F%F)F/F-FE*&F%F)F5F-FE*&F*
F)F/F-F&*&F*F)F5F-FE*$F/FZF&*&F/F)F5F-FE*&F/F-F2F)FE*&F/F-F5F)FE*&F2F)
F5F-F&*$F5FZF&F-F0F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 174 "Still \+
pretty dreadful.  Oh well.  But we can make a general ellipse function
 out of it quite easily.  Note that we will specify the two foci as A \+
and B, so we need to have A=" }{XPPEDIT 18 0 "[x[1],y[1]]" "7$&%\"xG6#
\"\"\"&%\"yG6#\"\"\"" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "B=[x[2],y[2
]]" "/%\"BG7$&%\"xG6#\"\"#&%\"yG6#\"\"#" }{TEXT -1 71 ", so it looks l
ike I made a mistake in the subs command.  But I didn't." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "GenEllipse := (A,B,k) ->\n   simpl
ify(subs(x[1]=A[1],y[1]=A[2], x[2]=B[1], y[2]=B[2], c=k, dogMess));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+GenEllipseG:6%%\"AG%\"BG%\"kG6\"6$
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