A path g in the plane $R^2$ is the sequence of the points $(t_0, t_1, ldots, t_n)$, with coordinates in $Z^2$. The point $t_0$ is the starting point and the point $t_n$ is the arriving point. An elementary step of g is a couple $(t_i, t_{i+1}), 0 leq i leq n - 1$. We denote the length of the path g by $mid$g$mid$ = n.