6.1 Cassini $B$N\t7A(B

2$BDjE@$+$i$N5wN%$N@Q$,0lDj$G$"$kE@$N(B $B50@W$O(B Cassini $B$N\t7A(B (Cassini $B$NMq7A@~(B, Cassini oval, oval of Cassini) $B$G$"$k!#(B Giovanni Domenico Cassini (1625-1712) $B$K$A$J$s$GL>IU$1$i$l$?!#(B

$B$A$g$C$H@0M}(B

$BJ?LL>e$N(B $2$ $BDjE@$+$i$N5wN%$NOB!":9!"@Q!"(B $B>&$,0lDj$G$"$kE@$N50@W$r5a$a$kLdBj$OM-L>$G$"$k!#(B

• $BOB$,0lDj(B $B!D(B $BBJ1_(B
• $B:9$,0lDj(B $B!D(B $BAP6J@~(B
• $B@Q$,0lDj(B $B!D(B Cassini $B$N\t7A(B
• $B>&(B ($BHf(B) $B$,0lDj(B $B!D(B Apollonius¹ $B$N(B $B1_(B ($BFCJL$J>l9g$H$7$F?bD>FsEyJ,@~(B)

$BFsDjE@$r(B $(\pm a,0)$ $B$H$7$F!"5wN%$N@Q$r(B $b^2$ $B$H$9$k$H!"(B

$$\left[(x-a)^2+y^2\right]\left[(x+a)^2+y^2\right]=b^4$$

$B$H$$$&J}Dx<0$,F@$i$l$k!#(B

$$(x^2+y^2+a^2)^2=4a^2x^2+b^4.$$

$B6K7A<0$G$O(B

$$r^4-2a^2r^2\cos2\theta+a^4=b^4.$$

$b$ $B$,>.$5$/!"(B$b<a$ $B$J$i$PFs$D$NO"[email protected],$+$i$J$j!"(B $a\le b$ $B$J$i$PO"7k$J6J@~$G$"$k!#(B

$b=a$ $B$J$i$P!"(B $(x^2+y^2)^2=2a^2(x^2-y^2)$ $B$H$J$k$N$G!"(B $a':=\sqrt{2}a$ $B$H$*$/$H(B

$$(x^2+y^2)^2=a'^2(x^2-y^2)$$

$B$H$J$k!#(B $B$3$l$O%Y%k%L!<%$$N%l%`%K%9%1!<%H(B (lemniscate of Bernoulli) $B$H8F$P$l!"(B 8$B$N;z7A$r$7$F$$$k!#(B

$b>a$ $B$J$i$P@5B'$JJD6J@~$G$"$k!#(B $BFC$K(B $a<b<\sqrt{2}a$ $B$J$i$P1z$_$N$"$kJD6J@~!"(B $b\ge \sqrt{2}a$ $B$J$i$PFLJD6J@~$G$"$k!#(B

$B?^(B 3: Cassini $B$N(B oval ($a=2$, $b=2.1$)

    

$B?^(B 4: Cassini $B$N(B oval ($a=2$, $b=3$)

2$BJQ?t4X?t$N%l%Y%k%;%C%H$rIA$-$?$$>l9g!"(B Mathematica $B$G$O!"(BImplicitPlot[] $B$,MxMQ$G$-$k!#(B

Decartes $B$NMU@~(B, $B%l%`%K%9%1!<%H(B, Cassini $B$N\t7A$rIA$/(B

  Needs["Graphics`ImplicitPlot`"]  ($B:G=i$K$3$N%*%^%8%J%$$,I,MW(B)

  decartes[a_]:=ImplicitPlot[x^3+y^3-3 a x y==0,{x,-4,4},{y,-4,4}]
  lemniscate[a_]:=ImplicitPlot[(x^2+y^2)^2==a^2(x^2-y^2),{x,-4,4},{y,-4,4}]
  limason[a_,b_]:=ImplicitPlot[(x^2+y^2-a x)^2==b^2(x^2+y^2),{x,-4,4},{y,-4,4}]
  cassini[a_,b_]:=ImplicitPlot[(x^2+y^2+a^2)^2==4a^2x^2+b^4,{x,-4,4},{y,-4,4}]

  g=lemniscate[4]                  (a=4 $B$N%l%`%K%9%1!<%H$rIA$/(B)
  SetDirectory[$HomeDirectory <> "\My Documents"]]
  Export["lemniscate.eps", g]

$B$J$*!"$3$3$G$O(B $x$, $y$ $BN>J}$NHO0O$r;XDj$7$?$,!"(B $B$3$&$9$k$HFbIt$G(B ContourPlot[] $B$rMxMQ$7$FEy9b@~$rIA2h$9$k!#(B $B$3$l$KBP$7$F!"(B $x$ $B$^$?$O(B $y$ $B$N0lJ}$N$_$NHO0O$r;XDj$9$k$H!"(B Solve[] $B$rMxMQ$7$F6J@~$rIA2h$9$k!#(B $B$3$NJ}$,=hM}$O=E$$$,!"7k2L$O$-$l$$$K$J$k(B ($B$=$&$G$"$k(B)$B!#(B