let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=\lbrace\ell\_{ij}\rbrace := \lbrace \lbrace \alpha p+ (1-\alpha q)\rbrace\,|\,0\le\alpha\le 1;\, p,q\in P\rbrace$ the set of line segments connecting pairs of points.


How are the numbers $\operatorname{card}(\lbrace \ell\_{hk}| \lbrace h,k\rbrace\subseteq P\,\setminus\lbrace p,q\rbrace\rbrace)$ of line-segments that intersect $\ell\_{pq}$ distributed?



Distribution of line segment intersections in random pointsets
Let $p\_1,\dots,p\_n$ be iid random points uniformly distributed in a region $R$. For a given (straight) line segment $[a,b]$ connecting points $a$ and $b$, let
$$h\_{a,b}(p,q):=1([p,q]\cap[a,b]\ne\emptyset).$$
Then the cardinality of the random set of all line segments connecting pairs of the random points $p\_1,\dots,p\_n$ that intersect $[a,b]$ is
$$N\_{n;a,b}=n(n-1)U\_{n;a,b},$$
where
$$U\_{n;a,b}:=\frac1{n(n-1)}\sum\_{i\in[n]}\sum\_{j\in[n]\setminus\{i\}}h\_{a,b}(p\_i,p\_j),$$
which is a so-called U-statistic (of order $m=2$) with (symmetric) kernel $h\_{a,b}$. Here, $[n]:=\{1,\dots,n\}$.


By [Hoeffding's Theorem 7.1](https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730196), for $\mu\_{a,b}:=Eh\_{a,b}(p\_1,p\_2)$, the distribution of
$$\sqrt n\,\Big(\frac{N\_{n;a,b}}{n(n-1)}-\mu\_{a,b}\Big)=\sqrt n\,(U\_{n;a,b}-\mu\_{a,b})$$
converges (as $n\to\infty$) to the centered normal distribution with variance $4\zeta\_1$, with $\zeta\_1(=\zeta\_{a,b;1})$ given by [formulas (5.9), (5.6), (5.1)](https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730196) in Hoeffding's paper (and depending on $a,b$). So, the distribution of $N\_{n;a,b}$ is asymptotically normal, with an asymptotic standard deviation asymptotically proportional to $n(n-1)/\sqrt n\sim n^{3/2}$.
For a specific simple region like a square or a disk, it should be straightforward to compute, at least numerically, the values of $\mu\_{a,b}$ and $\zeta\_{a,b;1}$.


If now $[a,b]$ is a random segment independent of $p\_1,\dots,p\_n$, then, accordingly, the limit distribution of the cardinality of the random set of all line segments connecting pairs of the random points $p\_1,\dots,p\_n$ that intersect $[a,b]$ will be a mixture of normal distributions with asymptotic standard deviations asymptotically proportional to $n^{3/2}$.


