Update
With [bend right=70] I need to adapt the nodes used at the end but the result is correct
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\coordinate (n_3) at (1,1);
\coordinate (h_1) at (1.5,3);
\fill[blue] (n_3) circle (2pt);
\fill[blue] (h_1) circle (2pt);
\draw (n_3) to [bend right=70]
node[pos=0,sloped,minimum width=.8cm] (n_3) {}
node[pos=1,sloped,minimum width=.8cm] (h_1) {}
(h_1) ;
\draw (n_3.center) -- (n_3.west);
\draw (h_1.center) -- (h_1.west);
\end{tikzpicture}
\end{document}

Actually with shorten and [bend right=70] we get

In other cases, with simple curves, I prefer Jake's answer but my first try with shorten was not efficient because I kept \documentclass{standalone} so I searched another way.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\coordinate (n_3) at (1,1);
\coordinate (h_1) at (1.5,3);
\fill[blue] (n_3) circle (2pt);
\fill[blue] (h_1) circle (2pt);
\draw (n_3) to [bend right=5]
node[pos=0,sloped,minimum width=.8cm] (n_3) {}
node[pos=1,sloped,minimum width=.8cm] (h_1) {}
(h_1) ;
\draw (n_3.center) -- (n_3.west);
\draw (h_1.center) -- (h_1.east);
\end{tikzpicture}
\end{document}

[border=5mm]the picture is wrong. I think the bounding box is wrong. – Alain Matthes Jun 05 '12 at 06:32shortendoesn't affect the bounding box. I'm not sure if this is a bug or a feature... – Jake Jun 05 '12 at 06:36[bend right=70]the points are not on the curve. I have also a problem in this case because I need to change the nodes. – Alain Matthes Jun 05 '12 at 07:20shortendoes not extend the curve with straight line segments tangential to the ends of your curve but I think the curve is drawn between to others points but I don't know how the point are choiced. – Alain Matthes Jun 05 '12 at 07:27shorten >=-1cmby now? I used a dirty fix for the bounding box issue:\draw [shorten >=-1cm] (A) -- (B) node [inner sep=1cm] {};or placingphantom{blabla}inside the node also has the effect of extending the bounding box, but it's less elegant. – PatrickT Mar 30 '21 at 01:38