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{SECT 0 {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 205 13 "with(linalg):" }
}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 62 "______________________________
________________________________" }}{PARA 201 "" 0 "" {TEXT 206 10 "Th
e point " }{XPPEDIT 18 0 "X=(1-t^2)/(1+t^2)" "6#/%\"XG*&,&\"\"\"F'*$%
\"tG\"\"#!\"\"F',&F'F'*$F)F*F'F+" }{TEXT 206 2 ", " }{XPPEDIT 18 0 "Y=
2*t/(1+t^2)" "6#/%\"YG*(\"\"#\"\"\"%\"tGF',&F'F'*$F(F&F'!\"\"" }{TEXT 
206 20 " lies on the circle " }{XPPEDIT 18 0 "X^2+Y^2 = 1;" "6#/,&*$%
\"XG\"\"#\"\"\"*$%\"YGF'F(F(" }{TEXT 206 24 ". Homogenise by putting \+
" }{XPPEDIT 18 0 "X=x/z" "6#/%\"XG*&%\"xG\"\"\"%\"zG!\"\"" }{TEXT 206 
2 ", " }{XPPEDIT 18 0 "Y=y/z" "6#/%\"YG*&%\"yG\"\"\"%\"zG!\"\"" }
{TEXT 206 30 ", so the point with parameter " }{XPPEDIT 18 0 "t" "6#%
\"tG" }{TEXT 206 29 " has homogeneous coordinates " }{XPPEDIT 18 0 "V(
t)" "6#-%\"VG6#%\"tG" }{TEXT 206 7 ", where" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 205 32 "V:=t->vector(
[1-t^2,2*t,1+t^2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "__________
____________________________________________________" }}{PARA 0 "" 0 "
" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "ch(t,u);" "6#-%#chG6$%\"tG%\"uG" }
{TEXT -1 8 " be the " }{TEXT 206 33 "line-coords of the chord joining \+
" }{XPPEDIT 18 0 "V(t);" "6#-%\"VG6#%\"tG" }{TEXT 206 4 " to " }
{XPPEDIT 18 0 "V(u);" "6#-%\"VG6#%\"uG" }{TEXT 206 19 ". (We are assum
ing " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 206 2 ", " }{XPPEDIT 18 0 "
u;" "6#%\"uG" }{TEXT 206 38 " distinct so we can cancel the factor " }
{XPPEDIT 18 0 "2(t-u);" "6#-\"\"#6#,&%\"tG\"\"\"%\"uG!\"\"" }{TEXT 
206 2 ".)" }{TEXT -1 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 
205 73 "ch:=(t,u)->simplify(scalarmul(crossprod(V(t),V(u)),1/(2*(u-t))
));ch(t,u);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT -1 62 "________________
______________________________________________" }}{PARA 201 "" 0 "" 
{TEXT 206 38 "We are going to work with the circles " }{XPPEDIT 18 0 "
X^2+Y^2 = 1;" "6#/,&*$%\"XG\"\"#\"\"\"*$%\"YGF'F(F(" }{TEXT 206 5 " an
d " }{XPPEDIT 18 0 "(X-d)^2+Y^2 = r^2;" "6#/,&*$,&%\"XG\"\"\"%\"dG!\"
\"\"\"#F(*$%\"YGF+F(*$%\"rGF+" }{TEXT 206 74 ". We need the matrix of \+
the second circle, and of the dual conic envelope:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 79 "expand((x-d)^2+y^2-r^2);\nA:=matrix(3,3,[1,
0,-d,0,1,0,-d,0,d^2-r^2]);\nM:=adj(A);" }}}{EXCHG {PARA 201 "" 0 "" 
{TEXT -1 62 "_________________________________________________________
_____" }}{PARA 201 "" 0 "" {XPPEDIT 18 0 "p(t, u) = 0;" "6#/-%\"pG6$%
\"tG%\"uG\"\"!" }{TEXT 206 15 " iff the chord " }{XPPEDIT 18 0 "ch(t,u
);" "6#-%#chG6$%\"tG%\"uG" }{TEXT 206 71 " touches the second circle, \+
that is, iff the chord belongs to the dual:" }}}{EXCHG {PARA 200 "> " 
0 "" {MPLTEXT 1 205 63 "p:=(t,u)->expand(evalm(transpose(ch(t,u))&*M&*
ch(t,u)));p(t,u);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "____________
__________________________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 34 "Rearrange this as a polynomial in " }{XPPEDIT 18 0 "u;" "
6#%\"uG" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"collect(%,u);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "_______________
_______________________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Given " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 26 ", ther
e are two zeros for " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 16 ", wh
ich we call " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 53 ". Their sum and product can
 be expressed in terms of " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 1 
":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "uplusv:=coeff(-p(t,u)
,u,1)/coeff(p(t,u),u,2);\nutimesv:=coeff(p(t,u),u,0)/coeff(p(t,u),u,2)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "(These are just stored for \+
future use.)\n________________________________________________________
______" }}{PARA 0 "" 0 "" {TEXT -1 38 "Next, we need the point of cont
act of " }{XPPEDIT 18 0 "ch(t,u);" "6#-%#chG6$%\"tG%\"uG" }{TEXT -1 
39 " with the second circle, which we call " }{XPPEDIT 18 0 "contact(t
,u);" "6#-%(contactG6$%\"tG%\"uG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 47 "contact:=(t,u)->evalm(ch(t,u)&*M);contact(t,u
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "___________________________
___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 29 "No
w we find the line joining " }{XPPEDIT 18 0 "contact(t,u);" "6#-%(cont
actG6$%\"tG%\"uG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "V(v);" "6#-%\"VG6
#%\"vG" }{TEXT -1 16 ", which we call " }{XPPEDIT 18 0 "join(t,u,v);" 
"6#-%%joinG6%%\"tG%\"uG%\"vG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 66 "join:=(t,u,v)->simplify(crossprod(contact(t,u),V
(v)));join(t,u,v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "___________
___________________________________________________" }}{PARA 0 "" 0 "
" {TEXT -1 9 "Finally, " }{XPPEDIT 18 0 "meet(t,u,v);" "6#-%%meetG6%%
\"tG%\"uG%\"vG" }{TEXT -1 10 " is where " }{XPPEDIT 18 0 "join(t,u,v);
" "6#-%%joinG6%%\"tG%\"uG%\"vG" }{TEXT -1 7 " meets " }{XPPEDIT 18 0 "
join(t,v,u);" "6#-%%joinG6%%\"tG%\"vG%\"uG" }{TEXT -1 8 ". Since " }
{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v;" "6#%
\"vG" }{TEXT -1 40 " are distinct, we can cancel the factor " }
{XPPEDIT 18 0 "u-v;" "6#,&%\"uG\"\"\"%\"vG!\"\"" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "meet:=(t,u,v)->simplify(scal
armul(crossprod(join(t,u,v),join(t,v,u)),1/(u-v)));meet(t,u,v);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "__________________________________
____________________________" }}{PARA 0 "" 0 "" {TEXT -1 33 "We now us
e the fact that we know " }{XPPEDIT 18 0 "u+v;" "6#,&%\"uG\"\"\"%\"vGF
%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "uv;" "6#%#uvG" }{TEXT -1 13 " i
n terms of " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 19 ". Temporarily
, put " }{XPPEDIT 18 0 "u+v = s;" "6#/,&%\"uG\"\"\"%\"vGF&%\"sG" }
{TEXT -1 5 " (the" }{TEXT 257 4 " sum" }{TEXT -1 6 ") and " }{XPPEDIT 
18 0 "uv = p;" "6#/%#uvG%\"pG" }{TEXT -1 6 " (the " }{TEXT 256 7 "prod
uct" }{TEXT -1 0 "" }{TEXT -1 6 "). So " }{XPPEDIT 18 0 "v = s-u;" "6#
/%\"vG,&%\"sG\"\"\"%\"uG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u^2
 = su-p;" "6#/*$%\"uG\"\"#,&%#suG\"\"\"%\"pG!\"\"" }{TEXT -1 1 ":" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "P:=simplify(subs(v=s-u,meet(
t,u,v)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x:=P[1];y:=P[2
];z:=P[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x:=algsubs(u^
2=s*u-p,x);y:=algsubs(u^2=s*u-p,y);z:=algsubs(u^2=s*u-p,z);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x:=algsubs(u^2=s*u-p,x);y:=a
lgsubs(u^2=s*u-p,y);z:=algsubs(u^2=s*u-p,z);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 62 "_______________________________________________________
_______" }}{PARA 0 "" 0 "" {TEXT -1 23 "We have now got rid of " }
{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v;" "
6#%\"vG" }{TEXT -1 80 ", and so can finish the job by substituting the
 values we previously stored for " }{XPPEDIT 18 0 "s;" "6#%\"sG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 13 " in ter
ms of " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 120 "x:=factor(subs(\{s=uplusv,p=utimesv\},x)
);y:=factor(subs(\{s=uplusv,p=utimesv\},y));z:=factor(subs(\{s=uplusv,
p=utimesv\},z));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "_____________
_________________________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 112 "Multiply through by the common denominator to get rid of
 the fractions, and divide through by the common factor " }{XPPEDIT 
18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 52 "x:=x*denom(z)/r^2;y:=y*denom(z)/r^2;z:=numer(z)/r^
2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "___________________________
___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 12 "We
 now have " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "y;" "6#%\"yG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z;" "6#%\"zG" }
{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 
37 ". Reverting to cartesian coordinates " }{XPPEDIT 18 0 "X;" "6#%\"X
G" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Y;" "6#%\"YG" }{TEXT -1 10 ", we h
ave " }{XPPEDIT 18 0 "X = x/z;" "6#/%\"XG*&%\"xG\"\"\"%\"zG!\"\"" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "Y = y/z;" "6#/%\"YG*&%\"yG\"\"\"%\"zG!
\"\"" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "zX-x = 0;" "6#/,&%#zX
G\"\"\"%\"xG!\"\"\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "zY-y = 0;
" "6#/,&%#zYG\"\"\"%\"yG!\"\"\"\"!" }{TEXT -1 19 ". Substituting for \+
" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y;" "
6#%\"yG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "z;" "6#%\"zG" }{TEXT -1 17 "
 and eliminating " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 46 ", we ge
t the cartesian equation of our curve: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "res:=resultant(z*X-x,z*Y-y,t);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 62 "____________________________________________________
__________" }}{PARA 0 "" 0 "" {TEXT -1 42 "Let's check on the degree o
f this monster:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "degree(r
es,\{X,Y\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The number of bra
ckets here is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nops(res)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The number of terms in that \+
last bracket is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nops(op
(8,res));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "____________________
__________________________________________" }}{PARA 0 "" 0 "" {TEXT 
-1 75 "Now put in the \"Poncelet condition\", that is, Euler's triangl
e formula. So " }{XPPEDIT 18 0 "R^2-2*rR-d^2 = 0;" "6#/,(*$%\"RG\"\"#
\"\"\"*&F'F(%#rRGF(!\"\"*$%\"dGF'F+\"\"!" }{TEXT -1 7 "; here " }
{XPPEDIT 18 0 "R = 1;" "6#/%\"RG\"\"\"" }{TEXT -1 5 ", so " }{XPPEDIT 
18 0 "r = (1-d^2)/2;" "6#/%\"rG*&,&\"\"\"F'*$%\"dG\"\"#!\"\"F'F*F+" }
{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "res2:=subs
(r=(1-d^2)/2,op(8,res));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "_____
_________________________________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 21 "Let's factorise this:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "factor(res2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
62 "______________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 58 "The interesting part is the last bracket \+
in the numerator:" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "op(1,op(6,%));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "q1:=sort(collect(%,\{X,Y\}),[X,Y]);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 18 "This is a circle! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "It is the
 same as this one:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "q2:=(
-d^2+9)*((X-4*d*(3-d^2)/(9-d^2))^2+Y^2-(d^2*(1-d^2)/(9-d^2))^2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "simplify(q1-q2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "c1:=X^2+Y^2-1;c2:=(X-d)^2+Y^2-((1-d
^2)/2)^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "3*(d^2-1)*c1+4
*(3-d^2)*c2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(%-
q1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 
0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }