Markov chains and dynamic geometry of polygons

Abstract

In this paper we construct sequences of polygons from a given nsided cyclic polygon by iterated procedures and study the limiting behaviors of these sequences in terms of nonnegative matrices and Markov chains.

Random geometric subdivisions

“…where σ n = σ (1) , σ (2) , σ (3) , σ (4) , σ (5) , σ (6) is a random permutation of the set of three elements; σ n (a, b, c) takes one of the six possible values…”

“…where σ n = σ (1) , σ (2) , σ (3) , σ (4) , σ (5) , σ with equal probability, and σ n 's are i.i.d. The advantage of this method is that the distribution of the set of the angles of the n-th triangle is completely symmetric with respect to exchanges of its components, even though the components are not independent.…”

GETMe.anis: On geometric polygon transformations leading to anisotropy

“…One direction is the geometric element transformation methods that improved the mesh quality by an iterative application of a geometric transformation to the elements of the mesh [10,18,15,20,12]. See also [17,16,19,14,11,13,4,7,5,6,9,8,3,2,1] for the analysis of various aspects of geometric polygon transformations. In line of this research the current work is extending the topic to anisotropic limit polygons, as the aim of the previous work is to generate meshes with as regular as possible elements.…”

Random Polygon to Ellipse: A Generalization

“…Indeed, there are numerous papers written on the subject of iterative transformations of random polygons, in which limiting behavior is of interest. Many of these papers hint at an underlying invariance and structure of these types of iterations (see [1], [3], [4], and [5]).…”