Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions

TL;DR

This paper investigates the complexity of approximating points on the unit sphere with rational points of low bit size, providing new bounds, explicit constructions, and practical algorithms for applications like geospatial data.

Contribution

It revises lower bounds on rational sphere approximations, introduces explicit constructions with bounded denominators, and demonstrates practical algorithms with open-source implementation.

Findings

Floating-point solutions are trivial in 2D and 3D.

Constructed rational points with denominators bounded by 10(d-1)/ε².

Practical algorithms enable efficient approximation for large geospatial datasets.

Abstract

Each non-zero point in $\mathbb{R}^d$ identifies a closest point $x$ on the unit sphere $\mathbb{S}^{d-1}$. We are interested in computing an $\epsilon$-approximation $y \in \mathbb{Q}^d$ for $x$, that is exactly on $\mathbb{S}^{d-1}$ and has low bit size. We revise lower bounds on rational approximations and provide explicit, spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in $\mathbb{R}^2$ and $\mathbb{R}^3$. Moreover, we show how to construct a rational point with denominators of at most $10(d-1)/\varepsilon^2$ for any given $\epsilon \in \left(0,\tfrac 1 8\right]$, improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation. Our open-source implementation and experiments demonstrate the practicality of our approach in the context of massive data sets Geo-referenced by latitude and longitude values.