\lambda < 1$. As a reparametrization $\sigma : [0,1] \rightarrow [0,m]$ of a curve $P$, we allow any continuous, non-decreasing function such that $\sigma(0) = 0$ and $\sigma(1) = m$. We denote by $P_\sigma(t)$ the actual location according to reparametrization $\sigma$: $P_\sigma(t) = P(\sigma(t))$. By $P_\sigma[a,b]$ we denote the subcurve of $P$ in between $P_\sigma(a)$ and $P_\sigma(b)$. In the following we are always given two polygonal curves $P$ and $Q$, where $Q$ is defined by its vertices $q_0, \ldots, q_n$ and is reparametrized by $\theta  : [0,1] \rightarrow [0,n]$. The reparametrized curve is denoted by $Q_\theta$.

#### [Fréchet matching]{}s

We are given two polygonal curves $P$ and $Q$ with $m$ and $n$ edges. A (monotonous) *matching* $\mu$ between $P$ and $Q$ is a pair of reparametrizations $(\sigma,\theta)$, such that $P_\sigma(t)$ matches to $Q_\theta(t)$. The Euclidean distance between two matched points is denoted by $d_\mu(t) = |P_\sigma(t) - Q_\theta(t)|$. The maximum distance over a range is denoted by $d_\mu[a,b] = \max_{a \leq t \leq b} d_\mu(t)$. The *[Fréchet distance]{}* between two curves is defined as $\delta_\text{F}(P,Q) = \inf_{\mu} d_\mu[0,1]$. A *[Fréchet matching]{}* is a matching $\mu$ that realizes the [Fréchet distance]{}: $d_\mu[0,1] = \delta_\text{F}(P,Q)$ holds.

#### Free space diagrams

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