Natural pseudo-distances between closed curves

Let us consider two closed curves $mathcal{M}$, $mathcal{N}$ of class $C^1$ and two functions $varphi:{mathcal{M}}rightarrow R$, $psi:mathcal{N}rightarrow R$ of class $C^1$, called measuring functions. The natural pseudo-distance ${d}$ between the pairs $({mathcal{M}},varphi)$, $({mathcal{N}},psi)$ is defined as the infimum of $Theta(f)stackrel{def}{=}max_{Pin mathcal{M}}|varphi(P)-psi(f(P))|$, as $f$ varies in the set of all homeomorphisms from $mathcal{M}$ onto $mathcal{N}$. The problem of finding the possible values for $d$ naturally arises. In this paper we prove that under appropriate hypotheses the natural pseudo-distance equals either $|c_1-c_2|$ or $frac{1}{2}|c_1-c_2|$, where $c_1$ and $c_2$ are two suitable critical values of the measuring functions. This equality shows that the relations between the natural pseudo-distance and the critical values of the measuring functions previously obtained in higher dimensions can be made stronger in the particular case of closed curves. Moreover, the examples we give in this paper show that our result cannot be further improved, and therefore it completely solves the problem of determining the possible values for $d$ in the $1$-dimensional case.