  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.
