which are illustrated in Fig. \[fig:events\], passages open in the free space. The critical values are the distances corresponding to these events.

Locally correct [Fréchet matching]{}s {#sec:lcm}
=====================================

We introduce *locally correct* [Fréchet matching]{}s, for which the matching between any two matched subcurves is a [Fréchet matching]{}.

\[def:locallycorrect\] Given two polygonal curves $P$ and $Q$, a matching $\mu = (\sigma,\theta)$ is *locally correct* if for all $a,b$ with $0 \leq a \leq b \leq 1$ $$d_\mu[a,b] = \delta_\text{F}(P_\sigma[a,b], Q_\theta[a,b]).$$

Note that not every [Fréchet matching]{} is locally correct. See for example Fig. \[fig:speedcounter\]. The question arises whether a locally correct matching always exists and if so, how to compute it. We resolve the first question in the following theorem.

\[thm:main\] For any two polygonal curves $P$ and $Q$, there exists a locally correct [Fréchet matching]{}.

#### Existence

We prove Theorem \[thm:main\] by induction on the number of edges in the curves. First, we present the lemmata for the two base cases: one of the two curves is a point, and both curves are line segments. In the following, $n$ and $m$ again denote the number of edges of $P$ and $Q$, respectively.

\[lem:pointcurve\] For two polygonal curves $P$ and $Q$ 