Sobre pavimentações do plano euclidiano

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Construções euclidianas e o desfecho de problemas famosos da geometria

Construction techniques with ruler and the compasses, fundamental on Euclidean geometry, have been related to modern algebraic theories such as solving equations and extension of bodies from the works by Paolo Ruffini (1765-1822), Niels Henrik Abel (1802-1829) and Evariste Galois (1811-1832). This relation could provide an answer to some famous problems, from ancient Greece, such as doubling the cube, the trisection Angle, the Quadrature of the Circle and the construction of regular polygons, which remained unsolved for over two thousand years. Also important for our purposes are the notions of algebraic numbers, transcendental and the criteria for constructability, of those numbers. The objective of this study is to reconstruct relevant steps of geometric constructions with ruler (unmarked) and the compasses, from the elementary to the outcome buildings, in the nineteenth century, considering those mentioned problems.

O ensino de geometria através da pavimentação do plano

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Refinamento geométrico de contornos e cumeeiras de telhados de edifícios extraídos de dados laser com uso de imagem aérea

In this paper, a method is proposed to refine the LASER 3D roofs geometrically by using a high-resolution aerial image and Markov Random Field (MRF) models. In order to do so, a MRF description for grouping straight lines is developed, assuming that each projected side contour and ridge is topologically correct and that it is only necessary to improve its accuracy. Although the combination of laser data with data from image is most justified for refining roof contour, the structure of ridges can give greater robustness in the topological description of the roof structure. The MRF model is formulated based on relationships (length, proximity, and orientation) between the straight lines extracted from the image and projected polygon and also on retangularity and corner injunctions. The energy function associated with MRF is minimized by the genetic algorithm optimization method, resulting in the grouping of straight lines for each roof object. Finally, each grouping of straight lines is topologically reconstructed based on the topology of the corresponding LASER scanning polygon projected onto the image-space. The results obtained were satisfactory. This method was able to provide polygons roof refined buildings in which most of its contour sides and ridges were geometrically improved.

Áreas de polígonos via determinantes

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Collision free region determination by modified polygonal Boolean operations

Cutting and packing problems are found in numerous industries such as garment, wood and shipbuilding. The collision free region concept is presented, as it represents all the translations possible for an item to be inserted into a container with already placed items. The often adopted nofit polygon concept and its analogous concept inner fit polygon are used to determine the collision free region. Boolean operations involving nofit polygons and inner fit polygons are used to determine the collision free region. New robust non-regularized Boolean operations algorithm is proposed to determine the collision free region. The algorithm is capable of dealing with degenerated boundaries. This capability is important because degenerated boundaries often represent local optimal placements. A parallelized version of the algorithm is also proposed and tests are performed in order to determine the execution times of both the serial and parallel versions of the algorithm.

Parallel Layout Construction Algorithm for Irregular Shape Packing Problems

Cutting and packing problems arise in a variety of industries, including garment, wood and shipbuilding. Irregular shape packing is a special case which admits irregular items and is much more complex due to the geometry of items. In order to ensure that items do not overlap and no item from the layout protrudes from the container, the collision free region concept was adopted. It represents all possible translations for a new item to be inserted into a container with already placed items. To construct a feasible layout, collision free region for each item is determined through a sequence of Boolean operations over polygons. In order to improve the speed of the algorithm, a parallel version of the layout construction was proposed and it was applied to a simulated annealing algorithm used to solve bin packing problems. Tests were performed in order to determine the speed improvement of the parallel version over the serial algorithm

The Fundamental Gap For Hyperbolic Triangles

In 1983, M. van den Berg made his Fundamental Gap Conjecture about the difference between the first two Dirichlet eigenvalues (the fundamental gap) of any convex domain in the Euclidean plane. Recently, progress has been made in the case where the domains are polygons and, in particular, triangles. We examine the conjecture for triangles in hyperbolic geometry, though we seek an for an upper bound for the fundamental gap rather than a lower bound.

Isospectrality and Heat Content

We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schrodinger operators acting in L-2[0,1] with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.

Effect of Methodological and Ecological Approaches on Heterogeneity of Nest-Site Selection of a Long-Lived Vulture

The application of scientific-based conservation measures requires that sampling methodologies in studies modelling similar ecological aspects produce comparable results making easier their interpretation. We aimed to show how the choice of different methodological and ecological approaches can affect conclusions in nest-site selection studies along different Palearctic meta-populations of an indicator species. First, a multivariate analysis of the variables affecting nest-site selection in a breeding colony of cinereous vulture (Aegypius monachus) in central Spain was performed. Then, a meta-analysis was applied to establish how methodological and habitat-type factors determine differences and similarities in the results obtained by previous studies that have modelled the forest breeding habitat of the species. Our results revealed patterns in nesting-habitat modelling by the cinereous vulture throughout its whole range: steep and south-facing slopes, great cover of large trees and distance to human activities were generally selected. The ratio and situation of the studied plots (nests/random), the use of plots vs. polygons as sampling units and the number of years of data set determined the variability explained by the model. Moreover, a greater size of the breeding colony implied that ecological and geomorphological variables at landscape level were more influential. Additionally, human activities affected in greater proportion to colonies situated in Mediterranean forests. For the first time, a meta-analysis regarding the factors determining nest-site selection heterogeneity for a single species at broad scale was achieved. It is essential to homogenize and coordinate experimental design in modelling the selection of species' ecological requirements in order to avoid that differences in results among studies would be due to methodological heterogeneity. This would optimize best conservation and management practices for habitats and species in a global context.

Patterned-ground plant communities along a bioclimate gradient in the High Arctic, Canada

Non-sorted circles, non-sorted polygons, and earth hummocks are common ground-surface features ill arctic regions. The), are caused by a variety of physical processes that Occur in permafrost regions including contraction cracking and frost heave. Here we describe the vegetation of patterned-ground forms on zonal sites at three location!: along an N-S transect through the High Arctic of Canada. We made 75 releves on patterned-ground features (circles, polygons, earth hummocks) and adjacent tundra (Interpolygon, intercircle, interhummock areas) and identified and classified the vegetation according to the Braun-Blanquet Method. Environmental factors were correlated with the vegetation data using a nonmetric multidimensional scaling ordination (NMDS). We identified eleven commnunities: (1) Puccinellia angustata-Papaver radicalum community in xeromesic non-sorted polygons of subzone A of the Circumpolar Arctic Vegetation Map; (2) Saxifraga-Parmelia omphalodes ssp. glacialis community in hydromesic interpolygon areas of subzone A; (3) Hypogymnia subobscura-Lecanora epibryon community In xeromesic non-sorted polygons of subzone B; (4) Orthotrichum speciosum-Salix arctica community In xeromesic interpolygon areas of subzone B; (5) Cochlearia groenlandica-Luzula nivalis community in hydromesic earth Mocks Of subzone B; (6) Salix arctica-Eriophorum angustifolium ssp. triste community in hygric earth hummocks of subzone 13; (7) Puccinellia angustata-Potentilla vahliana community in xeromesic non-sorted circles and bare patches of subzone Q (8) Dryas integrifolia-Carex rupestris community in xeromesic intercircle areas and vegetated patches of subzone C; (9) Braya glabella ssp. purpurascens-Dryas integrifolia community In hydromesic non-sorted circles of subzone Q (10) Dryas integrifolia-Carex aquatilis community in hydromesic intercircle areas of subzone C; and (11) Eriophorum angustifolium ssp. triste-Carex aquatilis community ill hygric intercircle areas of subzone C. The NMDS ordination displayed the vegetation types with respect to complex environmental gradients. The first axis of the ordination corresponds to a complex soil moisture gradient and the second axis corresponds to a complex geology/elevation/climate gradient. The tundra plots have a greater moss and graminoid cover than the adjacent frost-heave communities. In general, frost-heave features have greater thaw depths, more bare ground, thinner organic horizons, and lower soil moisture than the surrounding tundra. The morphology of the investigated patterned ground forms changes along the climatic gradient, with non-sorted pollygons dominating in the northernmost sites and non-sorted circles dominating, in the southern sites.

Arctic patterned-ground ecosystems: A synthesis of field studies and models along a North American Arctic Transect

Arctic landscapes have visually striking patterns of small polygons, circles, and hummocks. The linkages between the geophysical and biological components of these systems and their responses to climate changes are not well understood. The "Biocomplexity of Patterned Ground Ecosystems" project examined patterned-ground features (PGFs) in all five Arctic bioclimate subzones along an 1800-km trans-Arctic temperature gradient in northern Alaska and northwestern Canada. This paper provides an overview of the transect to illustrate the trends in climate, PGFs, vegetation, n-factors, soils, active-layer depth, and frost heave along the climate gradient. We emphasize the thermal effects of the vegetation and snow on the heat and water fluxes within patterned-ground systems. Four new modeling approaches build on the theme that vegetation controls microscale soil temperature differences between the centers and margins of the PGFs, and these in turn drive the movement of water, affect the formation of aggradation ice, promote differential soil heave, and regulate a host of system propel-ties that affect the ability of plants to colonize the centers of these features. We conclude with an examination of the possible effects of a climate wan-ning on patterned-ground ecosystems.

Analysis of polygonal cracking patterns in chloride-bearing terrains on Mars: Indicators of ancient playa settings

The ancient southern highlands on Mars (~3.5 Gyr old) contain > 600 regions that display spectral evidence in the infrared for the presence of chloride-bearing materials. Many of these locations were previously reported to display polygonal cracking patterns. We studied more than 80 of the chloride-bearing terrains using high-resolution (0.25-0.5 m/pixel) images, as well as near-infrared spectral data, to characterize the surface textures and the associated cracking patterns and mineralogies. Our study indicates that ~75% of the studied locations display polygonal cracks that resemble desiccation cracks, while some resemble salt expansion/thrust polygons. Furthermore, we detect, spectrally, the presence of smectites in association with ~30% of the studied fractured terrains. We note that smectites are a special class of swelling clay minerals that can induce formation of large desiccation cracks. As such, we suggest that the cracking patterns are indicative of the presence of smectite phyllosilicates even in the absence of spectral confirmation. Our results suggest that many chloride-bearing terrains have a lacustrine origin and a geologic setting similar to playas on Earth. Such locations would have contained ephemeral lakes that may have undergone repeated cycles of desiccation and recharging by a near-surface fluctuating water table in order to account for the salt-phyllosilicates associations. These results have notable implications for the ancient hydrology of Mars. We propose that the morphologies and sizes of the polygonal cracks can be used as paleoenvironmental, as well as lithological, indicators that could be helpful in planning future missions.

Potential desiccation cracks on Mars: A synthesis from modeling, analogue-field studies, and global observations

Potential desiccation polygons (PDPs) are polygonal surface patterns that are a common feature in Noachian-to-Hesperian-aged phyllosilicate- and chloride-bearing terrains and have been observed with size scales that range from cm-wide (by current rovers) to 10s of meters-wide. The global distribution of PDPs shows that they share certain traits in terms of morphology and geologic setting that can aid identification and distinction from fracturing patterns caused by other processes. They are mostly associated with sedimentary deposits that display spectral evidence for the presence of Fe/Mg smectites, Al-rich smectites or less commonly kaolinites, carbonates, and sulfates. In addition, PDPs may indicate paleolacustrine environments, which are of high interest for planetary exploration, and their presence implies that the fractured units are rich in smectite minerals that may have been deposited in a standing body of water. A collective synthesis with new data, particularly from the HiRISE camera suggests that desiccation cracks may be more common on the surface of Mars than previously thought. A review of terrestrial research on desiccation processes with emphasis on the theoretical background, field studies, and modeling constraints is presented here as well and shown to be consistent with and relevant to certain polygonal patterns on Mars.

Volcanic terrain and the possible periglacial formation of "excess ice" at the mid-latitudes of Utopia Planitia, Mars

At the mid-latitudes of Utopia Planitia (UP), Mars, a suite of spatially-associated landforms exhibit geomorphological traits that, on Earth, would be consistent with periglacial processes and the possible freeze-thaw cycling of water. The suite comprises small-sized polygonally-patterned ground, polygon-junction and -margin pits, and scalloped, rimless depressions. Typically, the landforms incise a dark-toned terrain that is thought to be ice-rich. Here, we investigate the dark-toned terrain by using high resolution images from the HiRISE as well as near-infrared spectral-data from the OMEGA and CRISM. The terrain displays erosional characteristics consistent with a sedimentary nature and near-infrared spectra characterised by a blue slope similar to that of weathered basaltic-tephra. We also describe volcanic terrain that is dark-toned and periglacially-modified in the Kamchatka mountain-range of eastern Russia. The terrain is characterised by weathered tephra inter-bedded with snow, ice-wedge polygons and near-surface excess ice. The excess ice forms in the pore space of the tephra as the result of snow-melt infiltration and, subsequently, in-situ freezing. Based on this possible analogue, we construct a three-stage mechanism that explains the possible ice-enrichment of a broad expanse of dark-toned terrain at the mid-latitudes of UP: (1) the dark-toned terrain accumulates and forms via the regional deposition of sediments sourced from explosive volcanism; (2) the volcanic sediments are blanketed by atmospherically-precipitated (H2O) snow, ice or an admixture of the two, either concurrent with the volcanic-events or between discrete events; and, (3) under the influence of high obliquity or explosive volcanism, boundary conditions tolerant of thaw evolve and this, in turn, permits the migration, cycling and eventual formation of excess ice in the volcanic sediments. Over time, and through episodic iterations of this scenario, excess ice forms to decametres of depth. (C) 2015 Elsevier B.V. All rights reserved.