Random Triangles and Polygons in the Plane

Abstract

In 1997, Jean-Claude Hausmann and Allen Knutson introduced a natural and beautiful correspondence between planar n-gons and the Grassmann manifold of 2-planes in real n-space. This construction leads to a natural probability distribution and a natural metric on polygons which has been used in shape classification and computer vision. In this paper, we provide an accessible introduction to this circle of ideas by explaining the Grassmannian geometry of triangles. We use this to find the probability that a random triangle is obtuse, which was a question raised by Lewis Carroll. 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.

MSC:

• Primary 53A04, Secondary 60D05
• 52A22
• 14M15

Acknowledgments

The authors wish to thank the MAA for the invitation to present some of this paper as an invited address at the 2017 Joint Mathematics Meetings and the Simons Foundation for their support of Cantarella and Shonkwiler. In addition, we’d like to thank the many colleagues who have helped us understand Grassmannians and the triangle problem, including Harrison Chapman, Rebecca Goldin, Ben Howard, Johannes Koelman, Chris Manon, Sébastien Martineau, John McCleary, Chris Peterson, Stu Whittington, and Seth Zimmerman. We are deeply grateful to the anonymous referees for carefully reading this paper and making numerous thoughtful suggestions for improving it.

Notes

1 Why not perimeter 1? The theory is the same either way, but if we make that choice, there will be many messy denominators to keep track of later on.

2 We will deal with rotations shortly.

3 If one of the z_i is zero, then $z_i=-z_i$ and the order of the cover is a lower power of 2, so this cover is actually branched over the points where some z_i = 0, which correspond to the polygons with ith edge of length 0.

4 The scalar is determined by $\det A^{T}A=1/2\sum {\Delta }_{\mathit{ij}}^2$, where A is the $n\times 2$ matrix with columns $\overrightarrow{u},\overrightarrow{v}$.

5 Note that there is a typo in the original statement of Sylvester’s problem: he used the word “convex” where he meant to say “reentrant.”

6 So called because $\text{ln}\hspace{0.17em}{\Delta }_{\mathit{ij}}(t)$ interpolates linearly between $\text{ln}\hspace{0.17em}{\Delta }_{\mathit{ij}}^0$ and $\text{ln}\hspace{0.17em}{\Delta }_{\mathit{ij}}^1$.

Additional information

Notes on contributors

Jason Cantarella

JASON CANTARELLA received his B.A. from Vassar College and his Ph.D. from the University of Pennsylvania. He held a visiting position at the University of Massachusetts, Amherst before joining the faculty at the University of Georgia, where he is the Associate Head of the Math Department. He enjoys building robots and finding unexpected applications for differential geometry.

Tom Needham

TOM NEEDHAM recently graduated from the University of Georgia. He is currently a postdoctoral researcher at The Ohio State University. In his spare time, he enjoys playing guitar and table tennis.

Clayton Shonkwiler

CLAYTON SHONKWILER was an undergraduate at Sewanee and did his graduate work at the University of Pennsylvania. He held visiting positions at Haverford College and the University of Georgia before coming to Colorado State University. He is well known for his mathematical animations, which he shares on various social media under the username @shonk.

Gavin Stewart

GAVIN STEWART graduated from Colorado State University in May 2016. He is currently a doctoral student at the Courant Institute at New York University.