Spherical friezes

TL;DR

This paper introduces algorithms for reconstructing points on a sphere from measurements, utilizing novel frieze patterns with glide symmetry and Laurent properties, advancing spherical distance geometry.

Contribution

It presents a new recursive algorithm that produces frieze patterns with unique symmetry and algebraic properties for spherical point recovery.

Findings

Algorithms successfully reconstruct points on the sphere

Frieze patterns exhibit glide symmetry

Patterns demonstrate Laurent phenomenon

Abstract

A fundamental problem in spherical distance geometry aims to recover an $n$-tuple of points on a 2-sphere in $\mathbb{R}^3$, viewed up to oriented isometry, from $O(n)$ input measurements. We solve this problem using algorithms that employ only the four arithmetic operations. Each algorithm recursively produces output data that we arrange into a new type of frieze pattern. These frieze patterns exhibit glide symmetry and a version of the Laurent phenomenon.