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?