恰好分成2條曲線

hbghlyj

\[
1(x^{4}+y^{4})-2(x^{2}+y^{2})+3x^{2}y^{2}+k=0
\]
當$k\approx0.8$時，曲線能恰好分成2條曲線
$k$的精确值是？

k=0.8
wolframalpha.com/input?i=contourplot+x%5E4+%2B+3+x%5E2+y%5E2+%2B ... 2B+y%5E2%29%2B.8%3D0
unitsize(4cm);
import contour; // load the contour module
real f(real x, real y) { // define the function
  return (x^4+y^4)-2(x^2+y^2)+3*x^2*y^2;
}
real[] levels = {-.8}; // define the contour levels
draw(contour(f,(-2,-2),(2,2),levels,200)); // generate the contour guides
k=0.7999
wolframalpha.com/input?i=contourplot+x%5E4+%2B+3+x%5E2+y%5E2+%2B ... y%5E2%29%2B.7999%3D0
unitsize(4cm);
import contour; // load the contour module
real f(real x, real y) { // define the function
  return (x^4+y^4)-2(x^2+y^2)+3*x^2*y^2;
}
real[] levels = {-.7999}; // define the contour levels
draw(contour(f,(-2,-2),(2,2),levels,200)); // generate the contour guides

hbghlyj

懷疑就是$k=0.8$，WolframAlpha、Asmptote畫的圖不凖，0.8時還有缺口。
代入$y=x$，$-4 x^2 + 5 x^4$ 的最小值$=-0.8$.

如何使Asmptote畫出更精確的圖？在contour参數中加上nx=200

kuing

k=0.8 时原式分解为
\[\left(x^2+\frac{3+\sqrt5}2y^2-1-\frac1{\sqrt5}\right)\left(x^2+\frac{3-\sqrt5}2y^2-1+\frac1{\sqrt5}\right)=0\]

hbghlyj

kuing 发表于 2024-2-10 14:10
k=0.8 时原式分解为
\[\left(x^2+\frac{3+\sqrt5}2y^2-1-\frac1{\sqrt5}\right)\left(x^2+\frac{3-\sqrt5}2 ...

怎麼看出的呢