如何画伯努利双纽线？

利用伯努利双纽线的极坐标表达式，弄了一个简单例子。

\documentclass{standalone}

\usepackage{tikz}

\begin{document}
  \begin{tikzpicture}% 注意定义域
    \draw[->] (-3,0) -- (3,0);
    \draw[->] (0,-2) -- (0,2);
    \draw[domain=-pi/4:pi/4,samples=400] plot ({deg(\x)}:{2*sqrt(cos(2*deg(\x)))});
    \draw[domain=-pi/4:pi/4,samples=400] plot ({deg(\x)}:{-2*sqrt(cos(2*deg(\x)))});
  \end{tikzpicture}
\end{document}


\begin{tikzpicture}
  \coordinate (O) at (0,0);
  \draw[->] (-3,0) -- (3,0);
  \draw[->] (0,-2) -- (0,2);
  \draw[domain=-pi/4:pi/4,samples=400]
    plot ({1.5*sqrt(2*cos(2*deg(\x)))*cos(deg(\x))},
          {1.5*sqrt(2*cos(2*deg(\x)))*sin(deg(\x))});
  \draw[domain=-pi/4:pi/4,samples=400]
    plot ({-1.5*sqrt(2*cos(2*deg(\x)))*cos(deg(\x))},
          {-1.5*sqrt(2*cos(2*deg(\x)))*sin(deg(\x))});
\end{tikzpicture}%%% 双纽线参数方程，这个画出来有一些问题

画曲线的代码有两行，感觉不太完美，主要是为了避免给负数开方。

伯努利双纽线的极坐标方程很奇怪，感觉有某些约定。

分左右两段去画即可（抄自https://tex.stackexchange.com/questions/299218/cassini-oval-curve-in-tikz/299225#299225）

\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[
        x=2cm,
        y=2cm,
    ]
    \def\samples{100}
    \def\c{1}
    \pgfmathsetmacro\cc{\c*\c}
    \pgfmathsetmacro\cccc{\cc*\cc}
    \def\a{1.01}
    \draw (0, 0) --
    plot[
            variable=\t,
            domain=-45+1/\samples:45-1/\samples,
            samples=\samples,
            smooth,
        ]
    (\t:{\cc*sqrt(\cc*cos(2*\t) + \cc*cos(2*\t))})
    -- cycle;
    \draw (0, 0) --
    plot[
            variable=\t,
            domain=180-45+1/\samples:180+45-1/\samples,
            samples=\samples,
            smooth,
        ]
    (\t:{\cc*sqrt(\cc*cos(2*\t) + \cc*cos(2*\t))})
    -- cycle;
    %\path plot[mark=x] coordinates { (-\c, 0) (\c, 0) };
\end{tikzpicture}
\end{document}

在 向老师的 tikz 绘图笔记二 里看到了方案, 也是分段解决, 代码如下

\documentclass[margin=5pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}[samples=200,scale=1.2]
\draw[-Stealth] (-2,0)--(2,0) node[below] {$x$};
\draw[-Stealth] (0,-0.6)--(0,0.8) node[left] {$y$};
\draw[domain=-pi/4:pi/4] plot ({1.414*sqrt(cos(2*\x r))*cos(\x r)},{1.414*sqrt(cos(2*\x r))*sin(\x r)})--(0,0);
\draw[domain=-pi/4:pi/4,rotate=180] plot ({1.414*sqrt(cos(2*\x r))*cos(\x r)},{1.414*sqrt(cos(2*\x r))*sin(\x r)})--(0,0);
\node at (-4pt,-9pt) {$o$};
\node[below] at (-1.65,0) {$-a$};
\node[below] at (1.55,0) {$a$};
\node[align=center] at (0,-1.1) {$(x^2+y^2)^2=a^2(x^2-y^2)$\\
$\rho^2=a^2\cos2\theta$};
\end{tikzpicture}
\end{document}

\tikz{%r^2=a^2*cos(2*i)
  \def\a{2}
  \def\tempsave{}
  \foreach \i in {-45,-44.5,...,45,135,135.5,...,225}
  {
    \pgfmathparse{sqrt(\a^2*cos(2*\i))}
    \xdef\tempsave{\tempsave(\i:\pgfmathresult)}
  }
  \draw plot[smooth] coordinates {\tempsave};
}