\documentclass[tikz,border=3mm]{standalone}
\newif\iftikzepitrochoidhasxis
\tikzepitrochoidhasxistrue
\tikzset{pics/epitrochoid/.style={code={%
  \tikzset{epitrochoid/.cd,#1}
  \def\pv##1{\pgfkeysvalueof{/tikz/epitrochoid/##1}}
  \pgfmathsetmacro{\mymax}{2*max(abs(\pv{r}),abs(\pv{d}))+abs(\pv{R})+0.5}
  \iftikzepitrochoidhasxis
   \draw[-stealth] (-\mymax,0) -- (\mymax,0) node[below left]{$x$};
   \draw[-stealth] (0,-\mymax) -- (0,\mymax) node[below left]{$y$};
  \else 
   \path (-\mymax,-\mymax) rectangle (\mymax,\mymax);
  \fi 
  \draw[blue,semithick] circle[radius=\pv{R}];
  \draw[semithick] ({deg(\pv{t})}:{abs(\pv{r})+abs(\pv{R})}) coordinate (c) circle[radius=\pv{r}];
  \pgfmathtruncatemacro\itest{(\pv{t}>0?1:0)}
  \ifnum\itest=1
   \pgfmathsetmacro{\isamples}{5+2*(\pv{R}+\pv{r})*int(\pv{t})}
   \draw[red,thick] plot[smooth,samples=\isamples,domain=0:\pv{t},variable=\t,smooth]
    ({(\pv{R}+\pv{r})*cos(deg(\t))-\pv{d}*cos((\pv{R}+\pv{r})*deg(\t)/\pv{r})},
     {(\pv{R}+\pv{r})*sin(deg(\t))-\pv{d}*sin((\pv{R}+\pv{r})*deg(\t)/\pv{r})})
     node[circle,inner sep=1.2pt,fill](e){};
    \draw  (c) node[circle,inner sep=1.2pt,fill]{}  -- (e); 
   \fi
}},
epitrochoid/.cd,R/.initial=3,r/.initial=1,d/.initial=0.5,t/.initial=1,
    axis/.is if=tikzepitrochoidhasxis,axis/.default=true}
\begin{document}
\foreach \Time in {0,0.1,...,6.2}
{\begin{tikzpicture}
 \pic{epitrochoid={t=\Time}};
\end{tikzpicture}}
\end{document}

LaTeX 绘制的 limacon 图，Happy LaTeXing