Digital images unveil geometric structures in pairs of relatively prime numbers

We present a transformation, based on the B'ezout’s identity, which maps the set of pairs of relatively prime numbers $(p,q)$ with fixed $p$ and $0<q<p$, to pairs of relatively prime numbers in the $p\times p$ square in ${\mathbb{R}}^2$, in such a way that intriguing quadratic arcs show up. We exhibit parametrizations of quadratic curves which fit such quadratic arcs and we also justify algebraically the ensuing geometry.