$f(A) = \cup_{a \in A}f(a)$, for any map $f$. Take an essential, non-annular subsurface $Y \subset S$. We define a map $$\phi_Y: {\mathcal{C}}(S) {\longrightarrow}{\mathscr{P}}({\mathcal{A}}{\mathcal{C}}(Y))$$ such that $\phi_Y(a)$ is the set of arcs and curves obtained from $a \cap Y$ when $\partial Y$ and $a$ are in minimal position. Define another map $$\psi_Y : {\mathscr{P}}({\mathcal{A}}{\mathcal{C}}(Y)) {\longrightarrow}{\mathscr{P}}({\mathcal{C}}(Y))$$ such that if $a$ is a curve, then $\psi_Y(a) = a$, and if $b$ is an arc, then $\psi_Y(b)$ is the union of the non-trivial components of the regular neighborhood of $(b\cap Y) \cup \partial Y$ (see Figure \[nbhd\]).

Composing these two maps we define the map $$\begin{aligned}
\pi_Y: {\mathcal{C}}(S) &{\longrightarrow}{\mathscr{P}}({\mathcal{C}}(Y)) \\
c &\longmapsto \psi_Y(\phi_Y(c))\end{aligned}$$ We use this map to define distances in a subsurface: for any two sets $A$ and $B$ in ${\mathcal{C}}(S)$, $$d_Y(A, B) = d_Y(\pi_Y(A), \pi_Y(B)).$$ We often refer to this as the distance in the subsurface $Y$.

The relationship between hierarchies and these maps give rise to some useful properties including the Bounded Geodesic Image Theorem which was originally proven by Masur-Minsky [@MMII].

\[bounded geodesic image\] Let $Y$ be a subsurface of $S$ with $\xi(Y) \neq 3$ and let $g$ be a geodesic segment, ray, or biinfinite line in ${\mathcal{C}}(S)$, such 