A parameterization of the Fermat curves satisfying x^(2N)+y^(2N)=1

TL;DR

This paper introduces a natural parameterization for the Fermat curves defined by x^(2N)+y^(2N)=1, and extends the approach to a broader class of similar equations, revealing their geometric behavior as N increases.

Contribution

The paper provides a new parameterization of Fermat curves and generalizes it to a wider class of equations, enhancing understanding of their geometric properties.

Findings

Fermat curves approach the boundary of [-1,1]^2 as N increases

A natural parameterization for these curves is established

The parameterization is extended to more general equations

Abstract

Note that the family of closed curves C_N={(x,y)\in R^2;x^(2N)+y^(2N)=1} for N=1,2,3,... approaches the boundary of [-1,1]^2 as N \to \infty. In this paper we exhibit a natural parameterization of these curves and generalize to a larger class of equations.