Random Triangles and Polygons in the Plane

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.

arXiv comment24 pages, 4 figures

References in corpus (5)

• Probability Theory of Random Polygons from the Quaternionic Viewpoint
• The symplectic geometry of closed equilateral random walks in 3-space
• The Toric Geometry of Triangulated Polygons in Euclidean Space
• A Fast Direct Sampling Algorithm for Equilateral Closed Polygons
• The Expected Total Curvature of Random Polygons

Cited by in corpus (3)

• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• A dynamical system in the space of convex quadrangles
• Stiefel manifolds and polygons