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Buffon's Problem determines Gaussian Curvature in three Geometries

Aizelle Abelgas, Bryan Carrillo, John Palacios, David Weisbart, Adam Yassine

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.

Buffon's Problem determines Gaussian Curvature in three Geometries

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.

Paper Structure

This paper contains 14 sections, 9 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. A General Framework
  3. Isometries and gratings
  4. Dropping the needle
  5. Consequences of homogeneity
  6. Determination of the Buffon Probabilities
  7. The needle in the plane
  8. The needle in the sphere
  9. The geometry of the Poincaré disk
  10. The needle in the Poincaré disk
  11. Limiting Behavior
  12. Order estimates for spheres
  13. Order estimates for the Poincaré disk
  14. Buffon deficits and Gaussian curvature

Key Result

Theorem \oldthetheorem

For any Riemannian surface $M$ with constant Gaussian curvature,

Theorems & Definitions (18)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 8 more