# Hyperbolic Orbit \documentclass[border=2pt]{standalone}

% Drawing
\usepackage{tikz}

% Tikz Library
\usetikzlibrary{decorations.markings, calc}

% Notation
\usepackage{physics}

% Newcommand
%% Midline Label
\newcommand{\midlinelabel}{
\node[inner sep = 1.3pt] (midlabel) at ($(#1)!.5!(#2)$) {#3};
\draw[thick, line cap = round] (#1) -- (midlabel);
\draw[-latex, thick, line cap = round] (midlabel) -- (#2);
}

% Styles
%% Arrows
\tikzset{arrow/.style = {postaction=decorate, decoration={markings,mark=between positions 0.1 and 0.9 step 3cm with \arrow{stealth}}}}

\begin{document}

% Macros
\def\a{5}
\def\b{3}
\def\Theta{atan(\b/\a)}
\def\r{1}
\def\sm{0.17}

\begin{tikzpicture}
%Grid
% \draw[thin, dotted] (-1,-1) grid (10,10);
% \foreach \i in {1,...,8}
% {
% \node at (\i,-2ex) {\i};
% }
% \foreach \i in {1,...,8}
% {
% \node at (-2ex,\i) {\i};
% }
% \node at (-2ex,-2ex) {0};

% Rotate the Hyperbola
\begin{scope}[rotate=-\Theta]
% Dashed Axis
\draw[domain=-2:8, dashed] plot (\x,{\b/\a*\x});
\draw[domain=1.5:-4, dashed] plot (\x,{-\b/\a*\x});
% Plot
\draw[domain=40:-22, samples=100, thick, blue, -stealth, arrow] plot ({\sm*\x},{\sm*\b*sqrt(1+(\x)^(2)/(\a)^2)});
% r_o
\midlinelabel{$(0,0)+(90:-1.17)$}{0,\sm*\b}{$r_o$}
\end{scope}

% Angles
\draw[-latex, thick] (-\r,0) arc (180:(90+\Theta):\r) node[pos=0.6, left] {$\theta$};

% Particle
\draw[-latex, thick, red] (10,0) -- +(-1.2,0) node [above, shift={(0.1,0.1)}] {$\vb{v}$};
\draw[fill=black] (10,0) circle [radius=2pt] node [above, shift={(0,0.2)}] {$m$};

% Node
\node[blue] at (-0.8,3) {Orbit};

% Vectors
\midlinelabel{-0.6,-1}{10,0}{$\vb{r}_\infty$}
\midlinelabel{-0.6,-1}{-0.6,0}{$b$}

% Point
\draw[fill=black] (-0.6,-1) circle [radius=1.5pt] node [below, shift={(0,-0.1)}] {$\mathrm{O}$};
\end{tikzpicture}

\end{document}

Hyperbolic-Orbit.tex