A periodic curve The graph of the curve $f(x,y)=0$ is formed by infinitely many equal closed curves  as shown in the figure below. Knowing that the distance between two contiguous, vertically and horizontally, of these curves are both equal to $\pi$, is it sufficient   to determine univocally the expression of the function $f$? If yes, what is this expression?


Too long for a comment, I fear I do not get the question.
I must be missing something obvious and it is maybe time to go to bed, but I would say the answer to the question as posed ("determine univocally") is trivially negative (which is what worries me).
For example, let us select among all the points $f(x,y) = 0$, one of the closed curves. 
We then select the point with highest $y$-coordinate, with lowest, and the same for the $x$-coordinate. 
Next  we define a new closed curve $\gamma$, going through these "extreme" (distance defining) points and such that the distance to the lateral and vertical neighbours is unaffected, which is certainly possible (just stay within a square aligned with the axes, going through the "extreme" points).
We then just substitute everywhere the old curves with the new ones, and define the set $\Gamma$ as the union of all periodically translated copies of $\gamma$.
Then we define $g$ as $$
g (x,y) =
\left\{
 \begin{array}{ll}
  0  & \mbox{if } (x,y) \in \Gamma \\
  1 & \mbox{if } (x,y) \notin \Gamma
 \end{array}
\right.
$$
and a new function $g(x,y)$ complies with the same requirements and $g(x,y)=0$ defines different curves. It seems to me that the only requirement of periodicity is far too little.
So what am I missing? Something on how an implicit function is defined?
