Related papers: Triangles Formed via Poisson Nearest Neighbors
Let the Euclidean plane be simultaneously and independently endowed with a Poisson point process and a Poisson line process, each of unit intensity. Consider a triangle T whose vertices all belong to the point process. The triangle is…
In this paper, a theorem about similar triangles is proved. It shows that two small and four large triangles similar to the original triangle can appear if we choose well among several intersections of the perpendicular bisectors of the…
In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent…
Stationary Poisson processes of lines in the plane are studied whose directional distributions are concentrated on $k \ge 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random…
Among the topics we discuss are certain joint densities (for sides and for angles), acuteness probabilities and bivariate Rice moments.
We study a local thinning $T_r$ that retains a point with probability $p(n_r)$, where $n_r$ counts neighbors within radius $r$. For Poisson input with spatially varying intensity, we obtain an exact intensity via a Poisson--mixture formula…
Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary…
In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the…
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known…
We study some properties of a triad of circles associated with a triangle. Each circle is inside the triangle, tangent to two sides of the triangle, and externally tangent to the circle on the third side as diameter. In particular, we find…
Pick n points independently at random in R^2, according to a prescribed probability measure mu, and let D^n_1 <= D^n_2 <= ... be the areas of the binomial n choose 3 triangles thus formed, in non-decreasing order. If mu is absolutely…
The mechanism by which a resonance may be attracted to a sharp threshold is described with several examples. It involves a threshold cusp interfering constructively with either or both (i) a resonance produced via confinement, (ii)…
This paper deals with the typical cell in a Poisson line tessellation in the plane whose directional distribution is concentrated on three equally spread values with possibly different weights. Such a random polygon can only be a triangle,…
We introduce patterns on a triangular grid generated by paperfolding operations. We show that in case these patterns are defined using a periodic sequence of foldings, they can also be generated using substitution rules and compute…
Starting with any nondegenerate triangle we can use a well defined interior point of the triangle to subdivide it into six smaller triangles. We can repeat this process with each new triangle, and continue doing so over and over. We show…
In this article we prove some theorems related to the triplets of triangles, homological two by two. These theorems will be used later to build triplets of triangles two by two tri-homological.
We show that along a density one subsequence of admissible radii, the nearest neighbor spacing between lattice points on circles is Poissonian.
We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable Courant algebroids. The additive group of…
In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…
We present a class of Poisson structures on trivial extension algebras which generalize some known structures induced by Poisson modules. We show that there exists a one-to-one correspondence between such a class of Poisson structures and…