  We discuss a unified approach to a class of geometric combinatorics incidence
problems in $2D$, of the Erd\"os distance type. The goal is obtaining the
second moment estimate, that is given a finite point set $S$ and a function $f$
on $S\times S$, an upper bound on the number of solutions of $$
  f(p,p') = f(q,q')\neq 0,\qquad (p,p',q,q')\in S\times S\times S\times S.
\qquad(*) $$ E.g., $f$ is the Euclidean distance in the plane, sphere, or a
sheet of the two-sheeted hyperboloid.
  Our tool is the Guth-Katz incidence theorem for lines in $\mathbb{RP}^3$, but
we focus on how the original $2D$ problem is made amenable to it. This
procedure was initiated by Elekes and Sharir, based on symmetry considerations.
However, symmetry considerations can be bypassed or made implicit. The
classical Pl\"ucker-Klein formalism for line geometry enables one to directly
interpret a solution of $(*)$ as intersection of two lines in $\mathbb{RP}^3$.
This allows for a very brief argument extending the Euclidean plane distance
argument to the spherical and hyperbolic distances. We also find instances of
the question $(*)$ without underlying symmetry group.
  The space of lines in the three-space, the Klein quadric $\mathcal K$, is
four-dimensional. We start out with an injective map $\mathfrak F:\,S\times
S\to\mathcal K$, from a pair of points in $2D$ to a line in $3D$ and seek a
combinatorial problem in the form $(*)$, which can be solved by applying the
Guth-Katz theorem to the set of lines in question. We identify a few new such
problems and generalise the existing ones.
