在tikz中绘制带渐近线的双曲线的一个分支

Question

我正在尝试使用 tikz 绘制双曲线的单个分支，并且我在此论坛的此帖子中发现了一个精彩的脚本（@DouglasMencken）：这个帖子

\documentclass[tikz, margin=10]{standalone}

\usepackage{bm}

\usepackage{tikz}
\usetikzlibrary{shapes.geometric} % for shape=ellipse
\usetikzlibrary{calc}

\begin{document}

\def\tikzscale{0.8}
\begin{tikzpicture}[scale=\tikzscale]

\tikzset{
    elli/.style args={#1:#2and#3}{
        draw,
        shape=ellipse,
        rotate=#1,
        minimum width=2*#2,
        minimum height=2*#3,
        outer sep=0pt,
    }
}

%
% #1 optional parameters for \draw
% #2 angle of rotation in degrees
% #3 offset of center as (pointx, pointy) or (name-o-coordinate)
% #4 length of plus (semi)axis, that is axis which hyperbola crosses
% #5 length of minus (semi)axis
% #6 how much of hyperbola to draw in degrees, with 90 you’d reach infinity
%
\newcommand\tikzhyperbola[6][thick]{%
    \draw [#1, rotate around={#2: (0, 0)}, shift=#3]
        plot [variable = \t, samples=1000, domain=-#6:#6] ({#4 / cos( \t )}, {#5 * tan( \t )});

}

\def\angle{90}
\def\bigaxis{1.5cm}
\def\smallaxis{1.5cm}

\coordinate (center) at (0, 0);

\node [scale=\tikzscale, elli=\angle:\bigaxis and \smallaxis, line width = 1.2pt, color=black, dotted] at (center) (e) {};

\draw [-{stealth}, line width = 1.2pt, color = orange] ([shift={(\angle:-12)}] e.center) -- ([shift={(\angle:12)}] e.center) node [above right] {$\bm{a}_1$};
\draw [-{stealth}, line width = 1.2pt, color = orange] ([shift={(90+\angle:-8)}] e.center) -- ([shift={(90+\angle:8)}] e.center) node [above left]  {$\bm{a}_2$};

\tikzhyperbola[line width = 1.2pt, color=blue!80!black]{\angle}{(center)}{\bigaxis}{\smallaxis}{77}

\pgfmathsetmacro\axisratio{\smallaxis / \bigaxis}

% asymptotes
\def\lengthofasymptote{15}
\draw [color=black!40, line width = 0.4pt, rotate around={\angle + atan( \axisratio ): (center)}]
    ($ (-\lengthofasymptote, 0) + (center) $) -- ++(2*\lengthofasymptote, 0) ;
\draw [color=black!40, line width = 0.4pt, rotate around={\angle - atan( \axisratio ): (center)}]
    ($ (-\lengthofasymptote, 0) + (center) $) -- ++(2*\lengthofasymptote, 0) ;


\end{tikzpicture}

\end{document}

但是，正如您所看到的：我只对绘制图形的上部感兴趣，它只有一个分支：我不知道如何获得它，因为我最终得到了一个非常大的轴（我不理解构建轴 a_1 和 a_2 的机制）。例如，如果我只想要渐近线的上部并将 lengthasymptote = 0 作为第一个坐标：整个图形将向右移动。

任何帮助都将不胜感激，以理解这个脚本，以获得期望的结果，因为它看起来非常优雅和干净，

谢谢。