# Random Triangles and Polygons in the Plane

## TL;DR

This paper explores the probability of a random triangle being obtuse, introduces a geometric correspondence for polygons, and applies Grassmannian geometry to quadrilaterals, with implications for shape analysis.

## Contribution

It establishes a natural probability measure on plane polygons via Grassmannian correspondence and solves classical problems like the probability of obtuse triangles and Sylvester's four-point problem.

## Key findings

- Probability of a random triangle being obtuse is determined.
- Explicit description of the moduli space of quadrilaterals.
- Connection between polygon geometry and Grassmannian manifolds.

## Abstract

We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This correspondence defines a natural probability measure on plane polygons. In these terms, we answer Caroll's question. We then explore the Grassmannian geometry of planar quadrilaterals, providing an answer to Sylvester's four-point problem, and describing explicitly the moduli space of unordered quadrilaterals. All of this provides a concrete introduction to a family of metrics used in shape classification and computer vision.