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

TL;DR

This paper demonstrates that a quadratic Bézier curve defined by specific points can be approximated by an envelope of segments formed from Bézout coefficients of nearby coprime numbers, linking algebraic number theory with geometric approximation.

Contribution

It introduces a novel geometric interpretation of Bézier curves using Bézout coefficients of coprime integers, connecting algebraic number theory with curve approximation.

Findings

Quadratic Bézier curve approximated by Bézout coefficient segments

Envelope of segments closely matches the Bézier curve

Provides a new perspective on curve approximation using number theory

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.