Drawing lines in tikz - just like a graph, but with extended edges (?)

Question

I am familiar with graphs in tikz and I feel like I can do some work already. But how would one reproduce extended edges? I know how to create a simple graph with edges between its nodes, but how would one create a line that goes through some nodes?

Below I present what I am trying to achieve:

Answer 1

Something like this perhaps:

\documentclass[tikz, border=10pt]{standalone}
\usetikzlibrary{calc, intersections}
\begin{document}
\begin{tikzpicture}[
        coordinate/.style={
            circle, fill, inner sep=2pt
        },
        blue
    ]

\coordinate[coordinate, label={60:$p_1$}] (P1) at (-2.5,0.75);
\coordinate[coordinate, label={60:$p_2$}] (P2) at (-1.5,-0.75);
\coordinate[coordinate, label={120:$q_1$}] (Q1) at (2.5,0.75);
\coordinate[coordinate, label={120:$q_2$}] (Q2) at (1.5,-0.75);

\draw[name path=L1] ($(P1)!-0.5!(P2)$) node[above] {$l$} -- ($(P1)!3!(P2)$); 
\draw[name path=L2] ($(Q1)!-0.5!(Q2)$) node[above] {$l'$} -- ($(Q1)!3!(Q2)$); 

\coordinate[coordinate, gray, label={[gray]0:$z$}] (Z1) at ($(P1)!0.5!(P2)$);
\coordinate[coordinate, gray, label={[gray]180:$z$}] (Z2) at ($(Q1)!0.5!(Q2)$);

\node[name intersections={of=L1 and L2, by=Y}, coordinate, label={270:$y$}] at (Y) {};

\draw ($(P1)!-0.25!(Q1)$) -- ($(P1)!1.25!(Q1)$) node[above] {$\overline{p_{1}q_{1}}$}
      ($(P2)!-0.5!(Q2)$) -- ($(P2)!1.5!(Q2)$) node[below] {$\overline{p_{2}q_{2}}$};
\draw[gray] ($(P1)!1.25!(Q1)$) to[out=0, in=0] (Z2) to[out=0, in=0] ($(P2)!1.5!(Q2)$)
      ($(P2)!-0.5!(Q2)$) to[out=180, in=180] (Z1) to[out=180, in=180] ($(P1)!-0.25!(Q1)$);

\end{tikzpicture}
\end{document}

Answer 2

Here is an alternative done with Metapost for comparison.

This was produced by compiling the following source with lualatex. I have included comments to try to explain the approach.

\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
    % start with the two lines
    path ell, ell';
    ell = (20 down -- 144 up) rotated 32;
    ell' = ell reflectedabout(up, down);

    % draw them and label the intersection at the origin
    draw ell; draw ell';
    dotlabel.rt("$y$", origin);
    label.top("$\ell$", point 1 of ell);
    label.top("$\ell'$", point 1 of ell');

    % now define two sets of points along each line
    pair p[], q[];
    p1 = point 3/4 of ell;
    q1 = point 3/4 of ell';
    p2 = point 3/8 of ell;
    q2 = point 3/8 of ell';

    % p3, q3 are the midpoints
    p3 = 1/2[p1, p2];
    q3 = 1/2[q1, q2];

    % and now some points shifted off to the sides..
    p11 = p1 shifted 30 left;
    p12 = (xpart p11, ypart p2);
    p13 = (xpart p11, ypart p3);
    q11 = q1 shifted 40 right;
    q12 = (xpart q11, ypart q2);
    q13 = (xpart q11, ypart q3);

    % draw the loop through all these points using the "Hobby" syntax
    draw q12 {right} .. {left} q13 -- q3 -- q13 {right} .. {left} q11 --
         p11 {left} .. {right} p13 -- p3 -- p13 {left} .. {right} p12 --
         cycle;

    % note that drawing `q11 -- p11` goes through q1 and p1 
    % because of the way they were defined above, 

    % add the remaining labels
    label.ulft("$\overline{p_1 q_1}$", q11);
    label.ulft("$\overline{p_2 q_2}$", q12);
    dotlabel.llft("$p_1$", p1);
    dotlabel.llft("$p_2$", p2);
    dotlabel.lrt("$q_1$", q1);
    dotlabel.lrt("$q_2$", q2);
    dotlabel.rt("$z$", p3);
    dotlabel.lft("$z$", q3);

endfig;
\end{mplibcode}
\end{document}