# Random Triangles and Polygons in the Plane

- Published in 2017
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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.

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### BibTeX entry

@article{RandomTrianglesandPolygonsinthePlane, title = {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.}, url = {http://arxiv.org/abs/1702.01027v1 http://arxiv.org/pdf/1702.01027v1}, author = {Jason Cantarella and Tom Needham and Clayton Shonkwiler and Gavin Stewart}, comment = {}, urldate = {2017-02-06}, archivePrefix = {arXiv}, eprint = {1702.01027}, primaryClass = {math.MG}, collections = {Probability and statistics,Geometry}, year = 2017 }