Related papers: The chord-length probability density of the regula…
In this paper we obtain the chord length distribution function for any regular polygon. From this function we conclude the density function and the distribution function of the distance between two uniformly and independently distributed…
This work studies the chord length distribution, in the case where both ends lie on a $N$-dimensional hypersphere ($N \geq 2$). Actually, after connecting this distribution to the recently estimated surface of a hyperspherical cap…
In this paper we obtain the density function and the distribution function of the distance between two uniformly and independently distributed random points in any right-angled triangle. The density function is derived from the chord length…
The volume charge density for a conducting ellipsoid is expressed in simple geometrical terms, and then used to obtain the known surface charge density as well as the uniform charge per length along any principal axis. Corresponding results…
It has been known that the distribution of the random distances between two uniformly distributed points within a convex polygon can be obtained based on its chord length distribution (CLD). In this report, we first verify the existing…
The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the…
Position probability distribution of a set of massive mutually exclusive particles in one dimension has been defined. Examples with a given two mutually exclusive particles system are considered. It is emphasized that quantum particles at…
This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular $\el$-sided polygon. Using this result, we obtain the closed-form probability…
We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a…
In this report, the explicit probability density functions of the random Euclidean distances associated with regular hexagons are given, when the two endpoints of a link are randomly distributed in the same hexagon, and two adjacent…
The Dirac's chord method may be suitable in different areas of physics for the representation of certain six-dimensional integrals for a convex body using the probability density of the chord length distribution. For a homogeneous model…
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…
A flat membrane with given shape is displayed; two points in the membrane are randomly selected; the probability that the separation between the points have a specified value is sought. A simple method to evaluate the probability density is…
The discrete unitary (reversible) analogues of the continuous (irreversible) tent maps are numerically investigated, in particular, the lengths probability distribution of their periodic orbits. It is found that its density can be well…
An earlier paper describes a program to prove the Kepler conjecture on sphere packings. This paper carries out the second step of that program. A sphere packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are divided…
We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…
The study of "random segments" is a classic issue in geometrical probability, whose complexity depends on how it is defined. But in apparently simple models, the random behavior is not immediate. In the present manuscript the following…
If the four triangular facets of a tetrahedron can be partitioned into pairs having the same area, then the triangles in each pair must be congruent to one another. A Heron-style formula is then derived for the volume of a tetrahedron…
We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this…
In [3], algorithms to compute the density of the distance to a random variable uniformly distributed in (a) a ball, (b) a disk, (c) a line segment, or (d) a polygone were introduced. For case (d), the algorithm, based on Green's theorem,…