Small Polygons Research Articles

Periglacial landforms at the Phoenix landing site and the northern plains of Mars

We examine potentially periglacial landforms in Mars Orbiter Camera (MOC) and High Resolution Imaging Science Experiment (HiRISE) images at the Phoenix landing site and compare them with numerical models of permafrost processes to better understand the origin, nature, and history of the permafrost and the surface of the northern plains of Mars. Small‐scale (3–6 m) polygonal‐patterned ground is ubiquitous throughout the Phoenix landing site and northern plains. Larger‐scale (20–25 m) polygonal patterns and regularly spaced (20–35 m) rubble piles (localized collections of rocks and boulders) are also common. Rubble piles were previously identified as “basketball terrain” in MOC images. The small polygon networks exhibit well‐developed and relatively undegraded morphology, and they overlay all other landforms. Comparison of the small polygons with a numerical model shows that their size is consistent with a thermal contraction origin on current‐day Mars and are likely active. In addition, the observed polygon size is consistent with a subsurface rheology of ice‐cemented soil on depth scales of about 10 m. The size and morphology of the larger polygonal patterns and rubble piles indicate a past episode of polygon formation and rock sorting in thermal contraction polygons, while the ice table was about twice as deep as it is presently. The pervasive nature of small and large polygons, and the extensive sorting of surface rocks, indicates that widespread overturning of the surface layer to depths of many meters has occurred in the recent geologic past. This periglacial reworking has had a significant influence on the landscape at the Phoenix landing site and over the Martian northern plains.

A comparison of analysis units for associating Lyme disease with forest-edge habitat

The rational definition of spatial analysis units is critical to modeling and understanding large-scale ecological processes. This study assessed the relationship between forest-edge habitat pattern and Lyme disease incidence rate when modeled under three designs for spatial data aggregation. Incidence rates were calculated from passive surveillance data reported for 12 counties in the U.S. State of Maryland during 1996–2000. A design using road-bounded polygons that varied in size from 0.002 km 2 to 368 km 2 ( n = 415) was compared with designs that used grid cells of 10 km 2 ( n = 823) and 36 km 2 ( n = 230). Major roads were chosen to approximate bounded populations of deer and humans engaged in outdoor activity around the home (peridomestic activity). While cell boundaries were arbitrary, cell sizes were chosen to eliminate outliers observed in small polygons, and to standardize the presumed zone of exposure. The single variable that explained the most variation in incidence rate across all study desi...

Solidarity in choosing a location on a cycle

We study the implications of two solidarity conditions on the efficient location of a public good on a cycle, when agents have single-peaked, symmetric preferences. Both conditions require that when circumstances change, the agents not responsible for the change should all be affected in the same direction: either they all gain or they all loose. The first condition, population-monotonicity, applies to arrival or departure of one agent. The second, replacement-domination, applies to changes in the preferences of one agent. Unfortunately, no Pareto-efficient solution satisfies any of these properties. However, if agents’ preferred points are restricted to the vertices of a small regular polygon inscribed in the circle, solutions exist. We characterize them as a class of efficient priority rules.

High latitude patterned grounds on Mars: Classification, distribution and climatic control

Patterned grounds such as polygonal features and slope stripes are the signature of the presence of ground ice and of temperature variations in cold regions on Earth. Identifying similar features on Mars is important to understand its past climate as well as the ground ice distribution. In this study, young patterned grounds are classed and mapped from the systematical analysis of Mars Observer Camera high resolution images. These features are located poleward of 55° latitude which fits the distribution of ground ice found by the Neutron Spectrometer onboard Mars Odyssey. Thermal contraction due to seasonal temperature variations is the predominant process of formation of polygons formed by cracks which sizes vary from 15 to 300 m. The small (<40 m) widespread polygons are very recent and degraded by the desiccation of ground ice from the cracks which enhances the effect of ice sublimation. The large polygons (50 to 300 m) located only around the south CO 2 polar cap indicate the presence of ground ice and thus outline the limit of the CO 2 ice cap. They could be due to the blanketing of water ice deposits by the advances and retreats of the residual CO 2 ice cap during the last thousand years. Large (50–250 m) and homogeneous polygons similar to ice wedge polygons, hillslope stripes and solifluction lobes may indicate that specific environments such as crater floors and hillslopes could have been submitted to freeze–thaw cycles, possibly related to higher summer temperatures in periods of obliquity higher than 35°. These interpretations must be strengthened by higher resolution images such as those of the HiRise mission of the Mars Reconnaissance Orbiter because locations with past seasonal thaw could be of major interest as potential landing sites for the Phoenix mission.

Semisolid microstructure evolution of the predeformed AZ91D alloy during heat treatment

The semisolid microstructure evolution process of the predeformed AZ91D alloy during heat treatment was studied. After being compressed by a ratio of 20.6% in height at 220 °C, all the dendrites oriented in the same way. With increasing heat-treatment temperature, the dissolution of eutectic β-Mg 17Al 12 first took place, resulting in the primary dendritic grains coarsening into interconnected non-dendritic grains. With heating continuously, a large amount of polygon recrystallization grains occur in the microstructure. Subsequently, the residual interdendritic β-Mg 17Al 12 at the edges of the primary grains and the new recrystallization grains melted in succession at higher heat-treatment temperature and the primary grains separated into small polygon grains. After being heat-treated at the semisolid temperature, the amount of liquid increased until the solid–liquid system reached its equilibrium state, and the grains spheroidized.

Present-Day Periglacial Environments in Central Spitsbergen, Svalbard

This report overviews observations on periglacial geomorphology in central Spitsbergen, which have been undertaken by the Japanese geomorphological groups. Focus is given to permafrost-related processes, including solifluction, thermal contraction cracking and pingo growth. Annual freeze-thaw action dominates the ground, resulting in extensive occurrence of solifluction and shallow landslides on soil slopes. Solifluction shows low surface velocity but large volumetric transport, which respectively reflects infrequent diurnal frost creep and deep movement. The latter partly originates from plug-like flow in the basal active layer, where permafrost temperature is low and muddy sediment is thick. Non-sorted polygons with a wide range of diameters develop on lowlands. In colder inland terrains, large polygons (>7m) have ice wedges, whereas smaller polygons have only soil wedges or cracks confined to the active layer. Significant ice-wedge cracking occurs during rapid and intensive cooling in midwinter. In warmer coastal terrains, ice wedges are not common even below large polygons, because higher winter temperature can produce only shallow cracks. Open-system pingos occur in valley bottoms and near shores. Some pingos are still growing under a low artesian pressure fed by constant supply of sub-permafrost water. Following lateral river erosion, a new frost mound emerged at a side of a pingo, reaching 3m high during three years. The observations demonstrate that central Spitsbergen is situated in a High Arctic but relatively warm permafrost environment. The transitional condition between cold and warm permafrost allows diverse periglacial features to coexist within a small area. Even minor climatic change can switch the two thermal regimes, affecting significantly the type and magnitude of periglacial processes.

Automated torch path planning using polygon subdivision for solid freeform fabrication based on welding

A continuous path is one of the most important requirements for solid freeform fabrication (SFF) based on welding. This paper proposes a method for torch path planning applicable to SFF based on welding with an emphasis on minimum human intervention. The suggested approach describes a method based on the subdivision of a two-dimensional (2-D) polygonal section into a set of monotone polygons to generate a continuous path for material deposition. A two-dimensional contour is subdivided into smaller polygons (subpolygons). The path for each individual subpolygon is generated. The final torch path is obtained by connecting individual paths for all the subpolygons and trimming along the points of intersection of the paths for individual subpolygons. The final path is a closed loop; therefore, any point can be selected as the starting point for material deposition. The proposed method can be used to develop the toolpath for CNC milling.

Spatial resolution influence on the identification of land cover classes in the Amazon environment

To evaluate the role played by the spatial resolution in distinguishing land cover classes in the Amazon region, different levels of spatial resolution (60, 100, 120, 200 and 250 meters) were simulated from a Landsat_5 Thematic Mapper (TM) image. Thematic maps were produced by visual interpretation from the original (30 x 30 meters) and simulated set of images. The map legend included primary forest, old and young woody secondary succession, and non-forest. The results indicated that for the discrimination between primary forest and non-forest, spatial resolution did not have great influence for pixel size equal or lower than 200 meters. The contrary was verified for the identification of old and young woody secondary vegetation due to their occurrence in small polygons. To avoid significant changes in the calculated area of these land cover types, a spatial resolution better than 100 meters is required. This result is an indication that the use of the future Brazilian remote sensing satellite (SSR-1) for secondary succession identification may be unreliable, especially for latitudes between S10° and S15° where critical areas of deforestation are located and pixel size is expected to vary within the same scene from 100 meters (S10°) to 200 meters (S15°).

Microstructural evolution of previously deformed ZA27 alloy during partial remelting

The microstructural evolution process of previously deformed ZA27 alloy during partial remelting was studied. After being compressed by a ratio of 20% in length at 250 °C, the microstructure of ZA27 alloy changed from developed dendrites to coarsened short dendrites and near-equiaxed grains, then in turn to united pieces, big lumps, small pieces, polygon particulates, and finally to spherical grains that would grow and agglomerate and become larger in size and irregular in shape during following partial remelting at semi-solid temperature of 460 °C. The structure evolution resulted from the eutectics first melted between boundaries of the primary grains and then penetrated into polygon boundaries that formed due to recrystallization, resulting in the separation of the primary grains into small polygon grains, followed by spheroidizing. This process was mainly discussed through analyzing the phase transformations occurred during partial remelting.

Some Constructions of Small Generalized Polygons

We present several new constructions for small generalized polygons using small projective planes together with a conic or a unital, using other small polygons, and using certain graphs such as the Coxeter graph and the Pappus graph. We also give a new construction of the tilde geometry using the Petersen graph.

Number of Polygons Generated by Map Overlay: The Case of Convex Polygons

This paper analyzes the number of polygons generated by map overlay. Overlay of maps yields a number of small polygons, and often causes problems in computation, handling, and storage of derived polygons. One method to deal with these problems is to estimate the number of derived polygons before executing overlay operation. To this end, a stochastic model is proposed by which the expected number of derived polygons is calculated. The results are summarized as follows: (1) the number of polygons generated by map overlay depends on the number of original polygons and their perimeters; (2) the number of derived polygons increases with the perimeters of original polygons; (3) McAlpine and Cook's (1971) earlier method underestimated the number of derived polygons; and (4) the number of polygons generated by the overlay of maps having the same lattice system is proportional to the square of the number of overlaid maps and the number of polygons on the map, and inversely proportional to the area‐perimeter ratio of the unit cell.

Voronoi tessellation to study the numerical density and the spatial distribution of neurones

The conditions of regularity and isotropy, required by standard morphometric procedures, are generally not fulfilled in the central nervous system (CNS) where cells are distributed in a highly complex manner. The evaluation of the mean numerical density of neuronal or glial cells does not take into account the topographical heterogeneity and thereby misses the information that it contains. A local measurement of the density can be obtained by evaluating the ‘numerical density of one cell’, i.e. the ratio 1/(the volume that the cell occupies). This volume is the region of space that is closer to that cell than to any other. It has the shape of a polyhedron, called Voronoi (or Dirichlet) polyhedron. In 2-D, the Voronoi polyhedron is a polygon, the sides of which are located at mid-distance from the neighbouring cells. The Voronoi polygons are contiguous and their set fills the space without interstice or overlap, i.e. they perform a ‘tessellation’ that may yield a density map when the same colours are used to fill polygons of similar sizes. The use of Voronoi polygons allows computing the confidence interval of a mean numerical density that makes statistical comparisons possible. The tessellation also provides information concerning spatial distribution; the areas of the Voronoi polygons do not vary much when the cells are regularly distributed. On the contrary, small and large polygons are found when cellular clusters are present. The coefficient of variation of the polygon areas is an objective measurement of their variability and helps to define ‘regular’, ‘clustered’ and ‘random’ distributions. When cells are clustered, small polygons are contiguous and may be objectively identified by simple algorithms. Voronoi tessellations are easily performed in 2-D. On an average the area of a polygon times the thickness of the section equals the volume of the corresponding polyhedron. 3-D tessellations that are theoretically possible and for which algorithms have been published remain to be adapted to histological works.

Evidence for recent groundwater seepage and surface runoff on Mars.

Relatively young landforms on Mars, seen in high-resolution images acquired by the Mars Global Surveyor Mars Orbiter Camera since March 1999, suggest the presence of sources of liquid water at shallow depths beneath the martian surface. Found at middle and high martian latitudes (particularly in the southern hemisphere), gullies within the walls of a very small number of impact craters, south polar pits, and two of the larger martian valleys display geomorphic features that can be explained by processes associated with groundwater seepage and surface runoff. The relative youth of the landforms is indicated by the superposition of the gullies on otherwise geologically young surfaces and by the absence of superimposed landforms or cross-cutting features, including impact craters, small polygons, and eolian dunes. The limited size and geographic distribution of the features argue for constrained source reservoirs.

Evaluation of a sonic telemetry system in three habitats of an estuarine environment

A directional sonic telemetry system in small embayment, seagrass, and channel habitats in Core Sound, North Carolina was evaluated. We compared point location estimates calculated after correcting for system bias in three different ways: using test station (i.e. known location transmitting–receiving stations) angle errors, and using a within test site and an among test site mean angle error estimate. Estimates are necessary because, when tracking an animal, system bias cannot be corrected for using test station angle errors. In addition, telemetered animals may move beyond test areas or into different habitats. We found no significant difference ( P>0.05) among point location estimates, suggesting that a within or an among mean angle error was an acceptable estimate. Choosing the appropriate angle error estimate must be done carefully because both presented limitations. A within test site mean angle error was the more conservative approach, avoiding biases caused by significant ( P<0.05) interhabitat angle error variability. An estimate containing interhabitat variability (i.e. among test site angle error) might be more robust for correcting system bias when the instrumented animal moves outside test areas or into a heterogeneous area. Seagrass habitat polygons in southern Core Sound range in size from 0.1 to 3189 ha. Attained levels of accuracy and precision from this study suggest that work could be conducted in areas where polygons are ≥6.9 ha, which represents >97% of the seagrass habitat in Core Sound. Although the majority (80%) of the polygons are small (<10.0 ha), they represent <5% of the total seagrass area. In addition, classifying use of habitat in areas where polygons are ≤6.9 ha is possible because small polygons have a contagious distribution; hence, their areas may be additive. Risks of misclassifying use of habitat can be reduced also by controlling the size of confidence areas ( A e ) by adjusting the distance between observers and the tracked animal. The confidence area as a function of distance can be predicted because location error varied linearly and significantly with geometric mean distance ( D g ). On the basis of this relationship, D g must be <326 m for A e 95 to be <10.0 ha.

Effects of geographic information system vector–raster–vector data conversion on landscape indices

The effects of geographic information system (GIS) data conversion on several polygon-and landscape-level indices were evaluated by using a GIS vegetation coverage from eastern Oregon, U.S.A. A vector–raster–vector conversion process was used to examine changes in GIS data. This process is widely used for data input (digital scanning of vector maps) and somewhat less widely used for data conversion (output of GIS data to specific formats). Most measures were sensitive to the grid cell size used in the conversion process. At the polygon level, using the conversion process with grid cell sizes of 3.05, 6.10, and 10 m produced relatively small changes to the original polygons in terms of ln(polygon area), ln(polygon perimeter), and 1/(fractal dimension). When grid cell size increased to 20 and 30 m, however, polygons were significantly different (p &lt; 0.05) according to these polygon-level indices. At the landscape level, the number of polygons, polygon size coefficient of variation (CV), and edge density increased, while mean polygon size and an interspersion and juxtaposition index (IJI) decreased. The youngest and oldest age-class polygons followed the trends of overall landscape only in terms of number of polygons, mean polygon size, CV, and IJI. One major side effect of the conversion process was that many small polygons were produced in and around narrow areas of the original polygons. An alleviation process (referred to as the dissolving process) was used to dissolve the boundaries between similarly attributed polygons. When we used the dissolving process, the rate of change for landscape-level indices slowed; although the number of polygons and CV still increased with larger grid cell sizes, the increase was less than when the dissolving process was not used. Mean polygon size, edge density, and fractal dimension decreased after use of the dissolving process. Trends for the youngest and oldest age-class polygons were similar to those for the total landscape, except that IJI was greater for these age-classes than for the total landscape.

A study of the impact of automated editing on polygon overlay analysis accuracy

Polygon overlay is among the most widely and frequently used Geographic Information System (GIS) operations. A polygon overlay operation may generate a large number of small polygons even when there exist few digitizing problems. In practical applications, automated editing methods are used to eliminate the small polygons for improving efficiency. However, automated editing may introduce errors because the locations originally occupied by the polygons eliminated are associated with changed attribute values. Knowledge about the errors is important for avoiding errors of unacceptable levels. In this research, the impacts of automated editing on overlay analysis accuracy are studied. The relationship between editing parameters and the errors introduced are investigated. Mathematical models are created to predict the errors. Two automated editing methods are examined and compared.

A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons

In the early 1940s David Kendall conjectured that the shapes of the ‘large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

A heuristic proof of a long-standing conjecture of D. G. Kendall concerning the shapes of certain large random polygons

In the early 1940s David Kendall conjectured that the shapes of the ‘large' (i.e. large area A) convex polygons determined by a standard Poisson line process in the plane tend to circularity (as A increases). Subject only to one heuristic argument, this conjecture and the corresponding two results with A replaced in turn by number of sides N and perimeter S, are proved. Two further similar limiting distributions are considered and, finally, corresponding limiting non-deterministic shape distributions for the small polygons are determined.

Multiple fault diagnosis by sensitizing input pairs

Interpolating subdivision schemes let users adjust coordinates of a 3D object's initial vertices, generate new vertices, and replace the original mesh with more and smaller polygons to make the object's surface smoother. Two new schemes for triangular meshes extend and combine existing schemes to improve the process and its results.

Motion Planning Using Medial Axis

Abstract This paper addresses the problem of planning the motion of a polygonal object through a set of planar obstacles. We propose a two-disk motion planning strategy to navigate the object within the free space between the obstacles from an initial location to a final location. This method makes use of the Medial Axis Transform (MAT) of the free space which can be generated efficiently using the method developed in [11]. We determine two minima] overlapping disks that fully enclose the moving object, and then constrain the centers of the two disks to move continuously aJong a path on the medial axis. In this paper we direct our efforts to the problem of finding the two enclosing disks for a moving object which is considered as a polygon. The problem has been considered as being optimally cutting a polygon into two smaller polygons such that each of smaller polygons can be covered by a minimal disk. We have proved that if the cut is optimal. the resultant minimal disks for two smaller polygons have equal diameters. Based on the geometry of a convex polygon, we formulated the problem as an optimization problem to determine a local optimum for a given edge pair. For a concave polygon, we propose a method to create a hypothetical convex polygon for approximating a given concave polygon, and then perform optimization using the method described above. We have proved that the maximal radii of the disks covering the hypothetical convex polygon generated from a concave polygon is 1.25 times of the radii of the disks that covers the original concave polygon. This shows that the approximation of a concave polygon using a minimal convex polygon does not yield too conservative solution. Simulations are presented for a variety of polygons. The method is being used for disassembly motion planning of a subassembly within its parent subassembly.