  We introduce the notion of delineation. A graph class $\mathcal C$ is said
delineated if for every hereditary closure $\mathcal D$ of a subclass of
$\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if
$\mathcal D$ is monadically dependent. An effective strengthening of
delineation for a class $\mathcal C$ implies that tractable FO model checking
on $\mathcal C$ is perfectly understood: On hereditary closures $\mathcal D$ of
subclasses of $\mathcal C$, FO model checking is fixed-parameter tractable
(FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs
[BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively
delineated, while subcubic graphs are not. On the one hand, we prove that
interval graphs, and even, rooted directed path graphs are delineated. On the
other hand, we show that segment graphs, directed path graphs, and visibility
graphs of simple polygons are not delineated. In an effort to draw the
delineation frontier between interval graphs (that are delineated) and
axis-parallel two-lengthed segment graphs (that are not), we investigate the
twin-width of restricted segment intersection classes. It was known that
(triangle-free) pure axis-parallel unit segment graphs have unbounded
twin-width [BGKTW, SODA '21]. We show that $K_{t,t}$-free segment graphs, and
axis-parallel $H_t$-free unit segment graphs have bounded twin-width, where
$H_t$ is the half-graph or ladder of height $t$. In contrast, axis-parallel
$H_4$-free two-lengthed segment graphs have unbounded twin-width. Our new
results, combined with the known FPT algorithm for FO model checking on graphs
given with $O(1)$-sequences, lead to win-win arguments. For instance, we derive
FPT algorithms for $k$-Ladder on visibility graphs of 1.5D terrains, and
$k$-Independent Set on visibility graphs of simple polygons.
