On the number of sides necessary for polygonal approximation of black-and-white figures in a plane

A bound on the number of extreme points or sides necessary to approximate a convex planar figure by an enclosing polygon is described. This number is found to be proportional to the fourth root of the figure's area divided by the square of a maximum Euclidean distance approximation parameter.An extension of this bound, preserving its fourth root quality, is made to general planar figures. This is done by decomposing the general figure into nearly convex sets defined by inflection points, cusps, and multiple windings.A procedure for performing actual encoding of this type is described. Comparisons of parsimony are made with contemporary figure encoding schemes

Subset Warping: Rubber Sheeting with Cuts

Image warping, often referred to as "rubber sheeting" represents the deformation of a domain image space into a range image space. In this paper, a technique is described which extends the definition of a rubber-sheet transformation to allow a polygonal region to be warped into one or more subsets of itself, where the subsets may be multiply connected. To do this, it constructs a set of "slits" in the domain image, which correspond to discontinuities in the range image, using a technique based on generalized Voronoi diagrams. The concept of medial axis is extended to describe inner and outer medial contours of a polygon. Polygonal regions are decomposed into annular subregions, and path homotopies are introduced to describe the annular subregions. These constructions motivate the definition of a ladder, which guides the construction of grid point pairs necessary to effect the warp itself

Modelling potential movement in constrained travel environments using rough space-time prisms

The widespread adoption of location-aware technologies (LATs) has afforded analysts new opportunities for efficiently collecting trajectory data of moving individuals. These technologies enable measuring trajectories as a finite sample set of time-stamped locations. The uncertainty related to both finite sampling and measurement errors makes it often difficult to reconstruct and represent a trajectory followed by an individual in space-time. Time geography offers an interesting framework to deal with the potential path of an individual in between two sample locations. Although this potential path may be easily delineated for travels along networks, this will be less straightforward for more nonnetwork-constrained environments. Current models, however, have mostly concentrated on network environments on the one hand and do not account for the spatiotemporal uncertainties of input data on the other hand. This article simultaneously addresses both issues by developing a novel methodology to capture potential movement between uncertain space-time points in obstacle-constrained travel environments

Control of the Mitotic Cleavage Plane by Local Epithelial Topology

For nearly 150 years, it has been recognized that cell shape strongly influences the orientation of the mitotic cleavage plane (e.g. Hofmeister, 1863). However, we still understand little about the complex interplay between cell shape and cleavage plane orientation in epithelia, where polygonal cell geometries emerge from multiple factors, including cell packing, cell growth, and cell division itself. Here, using mechanical simulations, we show that the polygonal shapes of individual cells can systematically bias the long axis orientations of their adjacent mitotic neighbors. Strikingly, analysis of both animal epithelia and plant epidermis confirm a robust and nearly identical correlation between local cell topology and cleavage plane orientation in vivo. Using simple mathematics, we show that this effect derives from fundamental packing constraints. Our results suggest that local epithelial topology is a key determinant of cleavage plane orientation, and that cleavage plane bias may be a widespread property of polygonal cell sheets in plants and animals.Engineering and Applied Science

Reconstruction of a 3-dimensional transonic rotor flow field from holographic interferogram data

Holographic interferometry and computer-assisted tomography (CAT) are used to determine the transonic velocity field of a model rotor blade in hover. A pulsed ruby laser recorded 40 interferograms with a 2-ft-diam view field near the model rotor-blade tip operating at a tip Mach number of 0.90. After digitizing the interferograms and extracting fringe-order functions, the data are transferred to a CAT code. The CAT code then calculates the perturbation velocity in seeral planes above the blade surface. The values from the holography-CAT method compare favorably with previously obtained numerical computations in most locations near the blade tip. The results demonstrate the technique's potential for three-dimensional transonic rotor flow studies

Three-dimensional, transonic rotor flow field reconstructed from holographic interferogram data

Holographic interferometry and computer-assisted tomography (CAT) are used to determine the transonic flow field of a model rotor blade in hover. A pulsed ruby laser records 40 interferograms with a 61 cm-diam view field near the model rotor-blade tip operating at a tip Mach number of 0.90. After digitizing the interferograms and extracting fringe-order functions, the data are transferred to a CAT code. The CAT code then calculates pressure coefficients in several planes above the blade surface. The values from the holography-CAT method compare favorably with previously obtained numerical computations and laser velocimeter measurements at most locations near the blade tip. The results demonstrate the technique's potential for three-dimensional transonic rotor flow studies