Buffon's Problem determines Gaussian Curvature in three Geometries

TL;DR

This paper extends Buffon's problem to Riemannian surfaces with constant Gaussian curvature, revealing a relationship between Buffon deficits and Gaussian curvature akin to classical geometric theorems.

Contribution

It introduces a generalized Buffon problem in curved geometries and establishes a new connection between Buffon deficits and Gaussian curvature.

Findings

Buffon probability defines a Buffon deficit in curved geometries.

The relationship mirrors the Bertrand-Diguet-Puiseux Theorem.

Gaussian curvature can be determined from Buffon deficits.

Abstract

A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand-Diguet-Puiseux Theorem establishes between Gaussian curvature and both circumference and area deficits.