  Digital circles not only play an important role in various technological
settings, but also provide a lively playground for more fundamental
number-theoretical questions. In this paper, we present a new recursive
algorithm for the construction of digital circles on the integer lattice
$\mathbb{Z}^2$, which makes sole use of the signum function. By briefly
elaborating on the nature of discretization of circular paths, we then find
that this algorithm recovers, in a space endowed with $\ell^1$-norm, the
defining constant $\pi$ of a circle in $\mathbb{R}^2$.
