B\'ezout coefficients of coprime numbers approximate quadratic B\'ezier curves

Abstract

Given a point $(p,q)$ with nonnegative integer coordinates and $p\not=q$, we prove that the quadratic B\'ezier curve relative to the points $(p,q)$, $(0,0)$ and $(q,p)$ is approximately the envelope of a family of segments whose endpoints are the B\'ezout coefficients of coprime numbers belonging to neighborhoods of $(p,q)$ and $(q,p)$, respectively.