On a recursive construction of circular paths and the search for $\pi$ on the integer lattice $\mathbb{Z}^2$

Abstract

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 $\mathbb{R}^2$.