Random Triangles Research Articles

Probability distributions arising from isoperimetric random triangles

We analyze the family of triangles whose sides come from a random subdivision of a given line segment into three segments. The usual geometric measurements on these random triangles (heights, bisectors, medians, angles, area, radii of the incircle, excircles, circumcircle) become random variables for which we determine the distribution function, the probability density, the expectation, the variance and higher order moments. This work can serve as a basis for activities at the college or university level. It is located at the crossroads between probability, geometry, integral calculus, special functions and computer algebra.

Lightweight Curvature Estimation on Point Clouds with Randomized Corrected Curvature Measures

AbstractThe estimation of differential quantities on oriented point cloud is a classical step for many geometry processing tasks in computer graphics and vision. Even if many solutions exist to estimate such quantities, they usually fail at satisfying both a stable estimation with theoretical guarantee, and the efficiency of the associated algorithm. Relying on the notion of corrected curvature measures [LRT22, LRTC20] designed for surfaces, the method introduced in this paper meets both requirements. Given a point of interest and a few nearest neighbours, our method estimates the whole curvature tensor information by generating random triangles within these neighbours and normalising the corrected curvature measures by the corrected area measure. We provide a stability theorem showing that our pointwise curvatures are accurate and convergent, provided the noise in position and normal information has a variance smaller than the radius of neighbourhood. Experiments and comparisons with the state‐of‐the‐art confirm that our approach is more accurate and much faster than alternatives. The method is fully parallelizable, requires only one nearest neighbour request per point of computation, and is trivial to implement.

Random triangles on flat tori

Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this computation involves reducing the problem to a new invariant of measurable sets in the plane that is unchanged under area-preserving affine transformations. Our result show that this probability is minimized at all rectangular tori and maximized at the regular hexagonal torus.

Random triangles in random graphs

AbstractIn a recent paper, Oliver Riordan shows that for and p up to and slightly larger than the threshold for a Kr‐factor, the hypergraph formed by the copies of Kr in G(n, p) contains a copy of the binomial random hypergraph with . For r = 3, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem for all . With Riordan's result, this immediately implies an asymptotically sharp bound for the threshold of a Kr‐factor in G(n, p) for . In this note, we resolve the missing case r = 3 by modifying Riordan's argument. This means that Kahn's result also implies a sharp bound for triangle factors in G(n, p).

Random Triangles in a Metric Space of Sequences

We investigate probabilistic properties of random triangles in the space of finite sequences with the Hamming metrics. As a triangle is understood any triple of points with distances between them. Probability measure is given by the classical way. In particular, it is shown that randomly chosen triangle is approximately equilateral with high probability. We also introduce a quantity that characterizes degree of “equilaterality” of triangles in the metric space in average.

Hall’s Conjecture on Extremal Sets for Random Triangles

In this paper we partially resolve Hall's conjecture about the distribution of random triangles. We consider the probability that three points chosen uniformly at random, in a bounded convex region of the plane, form an acute triangle. Hall's conjecture is the "isoprobabilistic inequality" which states that this probability should be maximized by the disk. We first prove that the disk is a weak local maximum for planar regions and that the ball is a weak local maximum in three dimensional regions. We then prove a local $C^{2,1/2}$-type estimate on the probability in the Hausdorff topology. This enables us to prove that the disk is a local maximum in the Gromov-Hausdorff topology (modulo congruences). Finally, we give an explicit upper bound on the isoperimetric ratio of the regions which maximize the probability and show how this reduces verifying the full conjecture to a finite, though currently intractable, calculation. An interesting aspect of our work is the use of tools from outside of geometric probability. We use an autocorrelation integral to provide the appropriate framework for the problem. When we study the problem in $\mathbb{R}^3$, we need non-Abelian harmonic analysis. To set up the proof that the disk is a strong local maximum, we use the Borsak-Ulam theorem.

Random Triangles and Polygons in the Plane

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.

Random triangles in planar regions

In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue measurable set of finite measure) contains a given fixed point O. These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point O. The formulae provide another way to approach the Sylvester’s four-point problem. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: \(\frac{2}{27}+20\frac{\ln 2}{81}\) (Halasz and Kleitman in Stud Appl Math 53:225–237, 1974; Prekopa in Period Math Hung 2:259–282, 1972). We compute this probability in the case of a regular polygon and its center of mass for the point O. Other families of regions are studied. For the family of Limacons \(r=a+\cos t, a>1\), and O the origin of the polar coordinates, the probability is \(\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3\pi ^2}\).

Tracking rates of random walks

We show that simple random walks on (non-trivial) relatively hyperbolic groups stay O(log(n))-close to geodesics, where n is the number of steps of the walk. Using similar techniques we show that simple random walks in mapping class groups stay $$O\left( {\sqrt {n\log \left( n \right)} } \right)$$ -close to geodesics and hierarchy paths. Along the way, we also prove a refinement of the result that mapping class groups have quadratic divergence. An application of our theorem for relatively hyperbolic groups is that random triangles in non-trivial relatively hyperbolic groups are O(log(n))-thin, random points have O(log(n))-small Gromov product and that in many cases the average Dehn function is subasymptotic to the Dehn function.

Random Triangle Theory with Geometry and Applications

What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis, random matrix theory, the Hopf fibration, and much more. One of the best distributions of random triangles takes all six vertex coordinates as independent standard Gaussians. Six can be reduced to four by translation of the center to \((0,0)\) or reformulation as a \(2\times 2\) random matrix problem. In this note, we develop shape theory in its historical context for a wide audience. We hope to encourage others to look again (and differently) at triangles. We provide a new constructive proof, using the geometry of parallelians, of a central result of shape theory: triangle shapes naturally fall on a hemisphere. We give several proofs of the key random result: that triangles are uniformly distributed when the normal distribution is transferred to the hemisphere. A new proof connects to the distribution of random condition numbers. Generalizing to higher dimensions, we obtain the “square root ellipticity statistic” of random matrix theory. Another proof connects the Hopf map to the SVD of \(2\times 2\) matrices. A new theorem describes three similar triangles hidden in the hemisphere. Many triangle properties are reformulated as matrix theorems, providing insight into both. This paper argues for a shift of viewpoint to the modern approaches of random matrix theory. As one example, we propose that the smallest singular value is an effective test for uniformity. New software is developed, and applications are proposed.

Size distributions in random triangles

The random triangles discussed in this paper are defined by having the directions of their sides independent and uniformly distributed on (0, π). To fix the scale, one side chosen arbitrarily is assigned unit length; let a and b denote the lengths of the other sides. We find the density functions of a / b, max{a, b}, min{a, b}, and of the area of the triangle, the first three explicitly and the last as an elliptic integral. The first two density functions, with supports in (0, ∞) and (½, ∞), respectively, are unusual in having an infinite spike at 1 which is interior to their ranges (the triangle is then isosceles).

Reconciling Representations

One student's unexpected solution led us to make sense of the problem about random triangles and persevere to a solution.

Size distributions in random triangles

The random triangles discussed in this paper are defined by having the directions of their sides independent and uniformly distributed on (0, π). To fix the scale, one side chosen arbitrarily is assigned unit length; let a and b denote the lengths of the other sides. We find the density functions of a / b, max{a, b}, min{a, b}, and of the area of the triangle, the first three explicitly and the last as an elliptic integral. The first two density functions, with supports in (0, ∞) and (½, ∞), respectively, are unusual in having an infinite spike at 1 which is interior to their ranges (the triangle is then isosceles).

U-max-statistics and limit theorems for perimeters and areas of random polygons

Abstract Recently Lao and Mayer (2008) considered U -max-statistics, where the maximum of kernels over the set of indices is studied instead of the usual sums. Such statistics emerge frequently in stochastic geometry. The examples include the largest distance between random points in a ball, the maximal diameter of a random polygon, the largest scalar product within a sample of points, etc. Their limit distributions are related to the distributions of extreme values. Among the results obtained by Lao and Mayer, the limit theorems for the maximal perimeter and the maximal area of random triangles inscribed in a circumference are of great interest. In the present paper, we generalize these theorems to the case of convex m -polygons, m ≥ 3 , with random vertices on the circumference. In addition, a similar problem for the minimal perimeter and the minimal area of circumscribed m -polygons is solved in this paper. This problem has not been studied in the literature so far.

Statistical Hyperbolicity in Teichmüller Space

In this paper we explore the idea that Teichmuller space is hyperbolic “on average.” Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichmuller space. We consider several different measures on Teichmuller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichmuller space, showing that “most triangles are mostly thin.”

Shooting randomly against a line in Euclidean and non-Euclidean spaces

In this paper we study a class of distributions related to the r.v. , for different distributions of . The problem is related to the hitting point of a randomly oriented ray and generalizes the Cauchy distribution in different directions. We show that the distribution of solves the Laplace equation of order , possesses even moments of order , and has bimodal structure when is uniform. We study also a number of distributional properties of functionals of , including those related to the arcsine law. Finally we study the same problem in the Poincaré half-plane and this leads to the hyperbolic distribution of which the main properties are explored. In particular we study the distribution of hyperbolic functions of , the law of sums of i.i.d. r.v.'s and the distribution of the area of random hyperbolic right triangles.

On areas of random triangles

By using a representation of powers of the area of a triangle as integral of Bessel functions involving its sides, we present some results on the distribution of the area of triangles possibly formed by given three positive independent and identically distributed random variables, taken to be as the sides.

Effects of global shape on angle discrimination

Previous studies have been inconclusive as to whether angle discrimination performance can be predicted by the sensitivity of orientation discrimination mechanisms or by that of mechanisms specialised for angle coding. However, these studies have assumed that angle discrimination is independent of the shape of the object of which the angle is a part. This assumption was tested by measuring angle discrimination using angles that were parts of different triangular shapes. Angle discrimination thresholds were lowest when angles were presented in isosceles triangles (sides forming the angle were of identical length). Performance was significantly poorer when angles were presented in scalene triangles (sides of different lengths) and as much as three times worse when the sides forming the angle varied randomly in length between presentations. Comparing orientation discrimination for single lines with angle discrimination for different stimulus conditions (isosceles, scalene and random triangles) leads to conflicting conclusions as to the mechanisms underlying angle perception: line orientation sensitivity correctly predicts angle discrimination for random triangles, but underestimates angle acuity for isosceles triangles. The fact that performance in angle discrimination tasks is strongly dependant on the overall stimulus geometry implies that geometric angles are computed by mechanisms that are sensitive to global aspects of the stimulus.

Protecting critical facets in layered manufacturing: implementation and experimental results

In layered manufacturing (LM), a three-dimensional polyhedral object is built by slicing its (virtual) CAD model, and manufacturing the slices successively. During this process, support structures are used to prop up overhangs. An important process-planning step in LM is choosing a suitable build direction, as it affects, among other things, the location of support structures on the part, which in turn impacts process speed and part finish. We describe a robust, exact, and efficient implementation of an algorithm that computes a description of a subset of all build directions for which a prescribed facet is not in contact with supports. We also present test results on models obtained from industry, and on collections of random triangles.

Pathology of collective doxa. Automata models

We use a conventional belief operator to derive a pentad of operators of knowledge, misbelief, delusion, doubt and ignorance. The operators are considered to be states of agent belief, and are called, therefore, doxastic states. The binary composition of doxastic states is constructed and investigated. Some useful algebraic properties of the composition are highlighted. Several derivatives of the composition of doxastic states are analysed and assigned to the so-called naive, conservative, anxious and contradictory agents. The discrete dynamics of reflecting agents and stirred collectives of agents is under study from the automata point of view. The non-stirred, or ordered, collectives of agents exhibit in their evolution the richest range of space-time phenomena from the competing domains of regular patterns to the random trees and triangles. They are analysed in detail, including possible influence of algebraic properties of doxastic compositions on space-time patterns of doxastic derivatives that emerge in the evolution of agent collectives. We determine products of so-called interacting doxastic worlds, where every world is an element of a Boolean of the set of doxastic states, to enforce our insight in domains of attracting and impossible worlds.