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Random triangles and polygons in the plane. (English) Zbl 1410.52004

Summary: 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:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
53A04 Curves in Euclidean and related spaces
60D05 Geometric probability and stochastic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
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[1] Arkani-Hamed, N.; Bourjaily, J.; Cachazo, F.; Goncharov, A.; Postnikov, A.; Trnka, J., Grassmannian Geometry of Scattering Amplitudes (2016), Cambridge, UK: Cambridge Univ. Press, Cambridge, UK · Zbl 1365.81004
[2] Avishalom, D., The perimetric bisection of triangles, Math. Mag, 36, 1, 60-62 (1963) · doi:10.2307/2688140
[3] Blaschke, W., Über affine Geometrie XI: Lösung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten, Leipz. Ber, 69, 436-453 (1917) · JFM 46.1113.04
[4] Bonk, M.; Schramm, O., Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal, 10, 2, 266-306 (2000) · Zbl 0972.53021 · doi:10.1007/s000390050009
[5] Cantarella, J.; Deguchi, T.; Shonkwiler, C., Probability theory of random polygons from the quaternionic viewpoint, Commun. Pure Appl. Math, 67, 10, 1658-1699 (2014) · Zbl 1300.60026 · doi:10.1002/cpa.21480
[6] Cantarella, J.; Duplantier, B.; Shonkwiler, C.; Uehara, E., A fast direct sampling algorithm for equilateral closed polygons, J. Phys. A: Math. Theory, 49, 27, 275202 (2016) · Zbl 1342.82063 · doi:10.1088/1751-8113/49/27/275202
[7] Cantarella, J.; Grosberg, A. Y.; Kusner, R.; Shonkwiler, C., The expected total curvature of random polygons, Amer. J. Math, 137, 2, 411-438 (2015) · Zbl 1357.60017 · doi:10.1353/ajm.2015.0015
[8] Cantarella, J.; Shonkwiler, C., The symplectic geometry of closed equilateral random walks in 3-space, Ann. Appl. Probab, 26, 1, 549-596 (2016) · Zbl 1408.53109 · doi:10.1214/15-AAP1100
[9] De Morgan, A., On infinity; and on the sign of equality, Trans. Camb. Philos. Soc, 11, 145-189 (1871)
[10] Draper, B.; Kirby, M.; Marks, J.; Marrinan, T.; Peterson, C., A flag representation for finite collections of subspaces of mixed dimensions, Linear Algebra Appl, 451, 15-32 (2014) · Zbl 1326.14118 · doi:10.1016/j.laa.2014.03.022
[11] Edelman, A.; Strang, G., Random triangle theory with geometry and applications, Found. Comput. Math, 15, 3, 681-713 (2015) · Zbl 1321.51016 · doi:10.1007/s10208-015-9250-3
[12] Ghys, E.; De La Harpe, P., Sur les Groupes Hyperboliques d’après Mikhael Gromov, 83 (1990), Boston, MA: Birkhäuser, Boston, MA · Zbl 0731.20025 · doi:10.1007/978-1-4684-9167-8
[13] Gromov, M.; Gersten, S. M., Essays in Group Theory, Hyperbolic groups, 75-263 (1987), New York: Springer, New York · doi:10.1007/978-1-4613-9586-7_3
[14] Guy, R. K., There are three times as many obtuse-angled triangles as there are acute-angled ones, Math. Mag, 66, 3, 175-179 (1993) · Zbl 0808.51017 · doi:10.2307/2690963
[15] Hausmann, J.-C.; Knutson, A., Polygon spaces and Grassmannians, L’Enseign. Math. (2), 43, 1-2, 173-198 (1997) · Zbl 0888.58007 · doi:10.5169/seals-63276
[16] Howard, B.; Manon, C.; Millson, J. J., The toric geometry of triangulated polygons in Euclidean space, Canad. J. Math, 63, 4, 878-937 (2011) · Zbl 1230.14067 · doi:10.4153/CJM-2011-021-0
[17] Ingleby, C. M., Correction of an inaccuracy in Dr. Ingleby’s note on the four-point problem, Math. Quest. Solut. Educ. Times, 5, 108-109 (1866)
[18] Kapovich, M.; Millson, J. J., The symplectic geometry of polygons in Euclidean space, J. Differential Geom, 44, 3, 479-513 (1996) · Zbl 0889.58017 · doi:10.4310/jdg/1214459218
[19] Kendall, D. G., Exact distributions for shapes of random triangles in convex sets, Adv. Appl. Probab, 17, 2, 308-329 (1985) · Zbl 0566.60007 · doi:10.2307/1427143
[20] Kleiman, S. L.; Laksov, D., Schubert calculus, Amer. Math. Monthly, 79, 10, 1061-1082 (1972) · Zbl 0272.14016 · doi:10.2307/2317421
[21] Mackay, J. S., Formulae connected with the radii of the incircle and the excircles of a triangle, Proc. Edinb. Math. Soc, 12, 86-105 (1893) · JFM 25.0913.06 · doi:10.1017/S0013091500001711
[22] Marrinan, T.; Beveridge, J. R.; Draper, B.; Kirby, M.; Peterson, C., Finding the subspace mean or median to fit your need, 2014 IEEE Conference on Computer Vision and Pattern Recognition, 1082-1089 (2014) · doi:10.1109/CVPR.2014.142
[23] Needham, T., Grassmannian geometry of framed curve spaces (2016)
[24] Needham, T., Kähler structures on spaces of framed curves, Ann. Glob. Anal. Geom, 54, 1, 123-153 (2018) · Zbl 1396.58012 · doi:10.1007/s10455-018-9595-3
[25] Pfiefer, R. E., The historical development of J. J. Sylvester’s four point problem, Math. Mag, 62, 5, 309-317 (1989) · Zbl 0705.52005 · doi:10.2307/2689482
[26] Portnoy, S., A Lewis Carroll pillow problem: probability of an obtuse triangle, Statist. Sci, 9, 2, 279-284 (1994) · Zbl 0955.60502 · doi:10.1214/ss/1177010497
[27] Suzuki, T.; Yamamoto, T.; Tezuka, Y., Constructing a macromolecular \(K_{3,3}\) graph through electrostatic self-assembly and covalent fixation with a dendritic polymer precursor, J. Am. Chem. Soc, 136, 28, 10148-10155 (2014) · doi:10.1021/ja504891x
[28] Sylvester, J. J., Algebraical researches, containing a disquisition on Newton’s rule for the discovery of imaginary roots, and an allied rule applicable to a particular class of equations, together with a complete invariantive determination of the character of the roots of the general equation of the fifth degree, &c, Philos. Trans. R. Soc. London, 154, 579-666 (1864) · doi:10.1098/rstl.1864.0017
[29] Sylvester, J. J., Mathematical question 1491, Educ. Times J. Coll. Precept, 17, 20 (1864)
[30] Sylvester, J. J., On a special class of questions on the theory of probabilities, Rep. Br. Assoc. Advmt. Sci, 35, 8-9 (1866)
[31] Watson, S., Answer to mathematical question 1987, The Lady’s and Gentleman’s Diary, 159, 66-68 (1862)
[32] Wilson, J. M., On the four-point problem and similar geometrical chance problems, Math. Quest. Solut. Educ. Times, 5, 81 (1866)
[33] Woolhouse, W. S. B., Mathematical question 1987, The Lady’s and Gentleman’s Diary, 158, 76 (1861)
[34] Woolhouse, W. S. B., Mathematical question 1835, The Educational Times and Journal of the College of Preceptors, 18, 189 (1865)
[35] Younes, L.; Michor, P. W.; Shah, J.; Mumford, D., A metric on shape space with explicit geodesics, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur, 19, 1, 25-57 (2008) · Zbl 1142.58013 · doi:10.4171/RLM/506
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