  The group $PGL(2,q)$ has an embedding into $PGL(3,q)$ such that it acts as
the group fixing a nonsingular conic in $PG(2,q)$. This action affords a
coherent configuration $R(q)$ on the set $L(q)$ of non-tangent lines of the
conic. We show that the relations can be described by using the cross-ratio.
Our results imply that the restrictions $R_{+}(q)$ and $R_{-}(q)$ to the sets
$L_{+}(q)$ of secant lines and to the set $L_{-}(q)$ of exterior lines,
respectively, are both association schemes; moreover, we show that the elliptic
scheme $R_{-}(q)$ is pseudocyclic.
  We further show that the coherent configuration $R(q^2)$ with $q$ even allow
certain fusions. These provide a 4-class fusion of the hyperbolic scheme
$R_{+}(q^2)$, and 3-class fusions and 2-class fusions (strongly regular graphs)
of both schemes $R_{+}(q^2)$ and $R_{-}(q^2). The fusion results for the
hyperbolic case are known, but our approach here as well as our results in the
elliptic case are new.
