Transitions in pedestrian fundamental diagrams of straight corridors and T-junctions

Many observations of pedestrian dynamics, including various self-organization phenomena, have been reproduced successfully by different models. But the empirical databases for quantitative calibration are still insufficient, e.g. the fundamental diagram as one of the most important relationships displays non-negligible differences among various studies. To improve this situation, experiments in straight corridors and T-junction are performed. Four different measurement methods are defined to study their effects on the fundamental diagram. It is shown that they have minor influences for {\rho} <3.5 m-2 but only the Voronoi method is able to resolve the fine-structure of the fundamental diagram. This enhanced measurement method permits to observe the occurrence of boundary-induced phase transition. For corridors of different widths we found that the specific flow concept works well for {\rho} <3.5 m-2. Moreover, we illustrate the discrepancies between the fundamental diagrams of a T-junction and a straight corridor.Comment: 17 pages, 10 figures, 3 table

Methods for measuring pedestrian density, flow, speed and direction with minimal scatter

The progress of image processing during recent years allows the measurement of pedestrian characteristics on a "microscopic" scale with low costs. However, density and flow are concepts of fluid mechanics defined for the limit of infinitely many particles. Standard methods of measuring these quantities locally (e.g. counting heads within a rectangle) suffer from large data scatter. The remedy of averaging over large spaces or long times reduces the possible resolution and inhibits the gain obtained by the new technologies. In this contribution we introduce a concept for measuring microscopic characteristics on the basis of pedestrian trajectories. Assigning a personal space to every pedestrian via a Voronoi diagram reduces the density scatter. Similarly, calculating direction and speed from position differences between times with identical phases of movement gives low-scatter sequences for speed and direction. Closing we discuss the methods to obtain reliable values for derived quantities and new possibilities of in depth analysis of experiments. The resolution obtained indicates the limits of stationary state theory.Comment: 16 pages, 10 figs, submitted to Physica

Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings

This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S from the tangent space at v. We characterize the `cut locus' (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R^2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo

Anisotropy in the Hubble constant as modeled by density gradients

The all-sky maps of the observed variation of the Hubble constant can be reproduced from a theoretical point of view by introducing an intergalactic plasma with a variable number density of electrons. The observed averaged value and variance of the Hubble constant are reproduced by adopting a rim model, an auto-gravitating model, and a Voronoi diagrams model as the backbone for an auto-gravitating medium. We also analyze an astronomer's model based on the 3D spatial distribution of galaxies as given by the 2MASS Redshift Survey and an auto-gravitating Lane--Emden ($n=5$) profile of the electrons. The simulation which involves the Voronoi diagrams is done in a cubic box with sides of 100 Mpc. The simulation which involves the 2MASS covers the range of redshift smaller than 0.05.Comment: 26 pages 17 figure

Correcting curvature-density effects in the Hamilton-Jacobi skeleton

The Hainilton-Jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two-dimensional (2-D) shapes. The approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero, while the flux is negative at the locations of skeletal points. Nonetheless, the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and, hence, unbounded near endpoints. This makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter. This problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision. However, here, we develop a method for 2-D skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error. This is done by taking into account variations of density due to boundary curvature. This yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton-Jacobi method