Random Triangles and Polygons in the Plane

• Published in 2017
• Added on 2017-02-06

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.

Links

• http://arxiv.org/abs/1702.01027v1
• http://arxiv.org/pdf/1702.01027v1

Other information

BibTeX entry

@article{RandomTrianglesandPolygonsinthePlane,
	key = {RandomTrianglesandPolygonsinthePlane},
	type = {article},
	title = {Random Triangles and Polygons in the Plane},
	author = {Jason Cantarella and Tom Needham and Clayton Shonkwiler and Gavin Stewart},
	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.},
	comment = {},
	date_added = {2017-02-06},
	date_published = {2017-10-09},
	urls = {http://arxiv.org/abs/1702.01027v1,http://arxiv.org/pdf/1702.01027v1},
	collections = {Probability and statistics,Geometry},
	url = {http://arxiv.org/abs/1702.01027v1 http://arxiv.org/pdf/1702.01027v1},
	urldate = {2017-02-06},
	archivePrefix = {arXiv},
	eprint = {1702.01027},
	primaryClass = {math.MG},
	year = 2017
}