Subset Of Polygons Research Articles

Writhe induced phase transition in unknotted self-avoiding polygons

Recently it has been argued that weighting the writhe of unknotted self-avoiding polygons can be related to possible experiments that turn double stranded DNA. We first solve exactly a directed model and demonstrate that in such a subset of polygons the problem of weighting their writhe is associated with a phase transition. We then analyse simulations using the Wang-Landau algorithm to observe scaling in the fluctuations of the writhe that is compatible with a second-order phase transition in a undirected self-avoiding polygon model. The transition can be clearly detected when the polygon is stretched with a strong pulling force.

POLS: A versatile tool for sampling polygon GIS layers

We introduce POLS (shorthand for POLygon Sampling), a versatile GIS tool for extracting a random subset of polygons from a vector layer. POLS enables users to optionally (i) set the sampling intensity in terms of percent area of the layer, number of polygons, or both; (ii) specify different strata to be sampled with equal or different intensity; (iii) preclude the occurrence of adjacent polygons in the sample; (iv) ensure that the output sample is spatially balanced; (v) estimate empirically (through simulation) the inclusion probability of each individual polygon; and (vi) compute the Horvitz–Thompson Estimator (HTE) and its confidence interval for target variables measured in the sample polygons. POLS is specially suited for accuracy assessments of thematic maps that use polygons as sampling units, but it can also be applied to any probability-based survey that relies on GIS polygons. The option enforcing non-adjacency potentially increases sampling efficiency by reducing the effect of spatial autocorrelation. The spatial balance option ensures that the polygons in the sample are well distributed across the extent of the layer. When the non-adjacency constraint is used, the tool applies a novel random-selection algorithm that is designed to reduce the impact of this constraint on both the inclusion probability and the spatial distribution of sample polygons. We describe the tool and the algorithm behind it, compare the latter with two other methods that we previously tested, study the impact of the non-adjacency constraint and the spatial balance on the inclusion probability, and demonstrate the estimation of both the HTE and its variance for a sample target variable. The tool is freely available on the internet.

Dynamic acoustic rendering

A method of acoustically rendering a virtual environment is described. The method includes receiving a subset of polygons derived for an acoustic display from a set of polygons generated for a graphical display. Acoustic reflections are determined from a sound source that bounce off of polygons in the subset of polygons to a listener position in the virtual environment. It is determined whether a polygon in the subset of polygons causes an occlusion of the sound source at the listener position, and a play list of sounds is generated based on the reflections and occlusions.

The writhe of a self-avoiding polygon

We discuss the writhe of a self-avoiding polygon on a lattice, as a geometrical measure of its entanglement complexity. We prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon, and use Monte Carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils.

An O( kN.log N) algorithm for decomposing a set of polygons into d-separable components

Given a set of polygons P = { P 1, P 2,…, P N } in the plane, a D-separable component with respect to a given set of directions D = { d 1, d 2,…, d k } is a subset of polygons Q ⊆ P which can be moved freely as a group in any of the directions D without colliding with any polygon not in Q. We present an O( kN.log N) algorithm for computing the D-separable components of a set of N polygons. The algorithm is based on the concept of D-closure defined here and can be viewed as a certain orderly application of the well-known plane-sweep method to specially formed subgroups of polygons. An improper choice of the order of the plane-sweep operations or of the subgroups of polygons can increase the complexity to O( N 2) in the worst case. The notion of separable components is useful for partitioning the free space in VLSI design and for motion planning in robotics.