Stationary Poisson processes of lines in the plane are studied, whose directional distributions are concentrated on $k\geq 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles’s classical result for the isotropic case is obtained by an approximation argument.

A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.

The shape of the nth in a sequence of random triangle shapes is (xn, yn). It was shown in [8] and [9] that yn → 0, almost surely, as n → ∞. In this paper we show that x„ converges in distribution, as n→∞, to a random variable x, and we find the probability distribution of x. The convergence in law of xn enables us to complete an argument used in one part of [9].

This paper is concerned with the shape-density for a random triangle whose vertices are randomly labelled and i.i.d.-uniform in a compact convex polygon K. In earlier work we have already shown that there is a network of curves (the singular tessellation T(K)) across which suffers discontinuities of form. In two papers which will appear in parallel with this, Hui-lin Le finds explicit formulae for (i) the form of within the basic tile T0 of T(K), and (ii) the jump-functions which link the local forms of on either side of any curve separating two tiles. Here we exploit these calculations to find in the most general case. We describe the geometry of T(K), we examine the real-analytic structure of within a tile, and we establish by analytic continuation an explicit formula giving in an arbitrary tile T as the sum of the basic-tile function and the members of a finite sequence of jump-functions along a ‘stepping-stone' tile-to-tile route from T0 to T. Finally we comment on some of the problems that arise in the use of this formula in studies relating to the applications in archaeology and astronomy.

Consider a random triangle formed by vertices which are independent and distributed uniformly inside a given bounded convex region K. Then a well-known result of W. Blaschke states that, among all regions of fixed area, the area AK of this random triangle is stochastically bounded by Acircle (Ac) and A triangle (At). The distribution of the latter was obtained by Alagar (1977). We use the classical Crofton technique to derive an integral representation for the distribution function Fc of Ac from which Fc is calculated using numerical quadrature. The maximum deviation between Fc and the distribution function of At is less than 0.04. Additionally, we obtain an explicit form for the distribution of Aparallelogram which may be represented as a mixture of functions involving Dirichlet-distributed random vectors.

In this paper we find the distribution of the area X of a randomly chosen triangle within a given triangle T. An extension of the classical Crofton technique is used in setting up a sequence of differential equations to obtain the desired distribution.