我是不懂这个代码的，both 数学原理 and LaTeX Tikz实现
我只是个轮子的搬运工
参考(copy)自下面的链接John Kormylo‘s brilliant solution on TSE
\documentclass[tikz,border=2pt]{standalone}
\usetikzlibrary{calc,intersections}
\newlength{\bigenough}
\newcount\total
% locates both points on a closed shape tangent to a point outside the shape.
% \tangent{pathname}{center}{point}{first}{second}
\newcommand{\tangent}[5]% #1=path name for shape, #2=coordinate name for center, #3=cordinate name for outside point
{\begingroup% #4=coordinate name for first tangent point, #5=coordinate name for second coordinate point
\setlength{\bigenough}{1cm}
\loop% loop until big enough
\path[name path=temp] ($(#2)!\bigenough!-90:(#3)$)--($(#2)!\bigenough!90:(#3)$);
\path[name intersections={of = #1 and temp, total=\t}]%
\pgfextra{\global\total=\t};%
\ifnum\total<2 \global\bigenough=2\bigenough\repeat%
\endgroup
\coordinate (#4) at (intersection-1);% initial guess
\coordinate (#5) at (intersection-2);%
\tangentsearch{#1}{#2}{#3}{#4}%
\tangentsearch{#1}{#2}{#3}{#5}}
% find tangent using binary search
\newcommand{\tangentsearch}[4]% #1=path name for shape, #2=coordinate name for center, #3=cordinate name for outside point
{\begingroup% #4=coordinate name for tangent point (initail guess -> final)
\loop% loop until only 1 intersection
\path[name path=temp] (#3)--($(#4)!-\bigenough!(#3)$);
\path[name intersections={of = #1 and temp, total=\t}]%
\pgfextra{\global\total=\t};%
\ifnum\total=2 \coordinate (#4) at ($(intersection-1)!0.5!(intersection-2)$);
%\draw[pink] (intersection-1)--(intersection-2);% included only for debugging purposes
\path[name path=temp] (#4)--($(#4)!-\bigenough!(#2)$);
\path[name intersections={of = #1 and temp}];%
\coordinate (#4) at (intersection-1);%
\repeat%
\endgroup}
\begin{document}
\begin{tikzpicture}
\draw [gray!70] (-4,-4) grid (4,4);
\draw [->] (-4,0) -- (4,0) node [below left] {$x$};
\draw [->] (0,-4) -- (0,4) node [below left] {$y$};
\draw[name path=ellipse,thick,fill=gray!30,opacity=0.5] (0,0) ellipse (2 and 1);
% \draw [thick] (2,0) arc (0:360:2 and 1);
%\coordinate (O) at (0,0);
%\node at (O) [below left] {$O$};
%这里可以缩行
\coordinate[label={below left:$O$}] (O);
% \coordinate (P) at (1,2);
% \fill (P) at (1,2) circle (1pt) node [above right] {$P$};
%这里也可以缩行
\node[label={above right:$P$},circle, fill, inner sep=1pt] (P) at (1,2) {};
\tangent{ellipse}{O}{P}{X}{Y}
\draw (X)--(P)--(Y);
\end{tikzpicture}
\end{document}
问 过标准椭圆外一定点，引椭圆的两条切线，绘制切线