Approximations of periodic functions to R^n by curvatures of closed curves

TL;DR

This paper demonstrates that any set of positive periodic functions can be approximated by the curvatures of a closed curve in Euclidean space, but not in hyperbolic space or for parametric families.

Contribution

It establishes the existence of closed curves in R^{n+1} whose curvatures approximate given periodic functions, expanding understanding of curvature approximation in Euclidean geometry.

Findings

Approximation of periodic functions by curvatures in R^{n+1}

Non-existence of similar approximation in hyperbolic space

Limitations for parametric families of curves

Abstract

We show that for any n real periodic functions f_1,..., f_n with the same period, such that f_i>0 for i<n, and a real number e >0, there is a closed curve in R^{n+1} with curvatures k_1, ..., k_n such that |k_i(t)-f_i(t)| < e for all i and t. This neither holds for closed curves in the hyperbolic space H^{n+1}, nor for parametric families of closed curves in R^{n+1}.