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1
Ross, Kenneth G., Michael J. B. Krieger, and D. DeWayne Shoemaker. "Alternative Genetic Foundations for a Key Social Polymorphism in Fire Ants." Genetics 165, no. 4 (December 1, 2003): 1853–67. http://dx.doi.org/10.1093/genetics/165.4.1853.
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Abstract Little is known about the genetic foundations of colony social organization. One rare example in which a single major gene is implicated in the expression of alternative social organizations involves the presumed odorant-binding protein gene Gp-9 in fire ants. Specific amino acid substitutions in this gene invariably are associated with the expression of monogyny (single queen per colony) or polygyny (multiple queens per colony) in fire ant species of the Solenopsis richteri clade. These substitutions are hypothesized to alter the abilities of workers to recognize queens and thereby regulate their numbers in a colony. We examined whether these same substitutions underlie the monogyny/polygyny social polymorphism in the distantly related fire ant S. geminata. We found that Gp-9 coding region sequences are identical in the polygyne and monogyne forms of this species, disproving our hypothesis that one or a few specific amino acid replacements in the protein are necessary to induce transitions in social organization in fire ants. On the other hand, polygyne S. geminata differs genetically from the monogyne form in ways not mirrored in the two forms of S. invicta, a well-studied member of the S. richteri clade, supporting the conclusion that polygyny did not evolve via analogous routes in the two lineages. Specifically, polygyne S. geminata has lower genetic diversity and different gene frequencies than the monogyne form, suggesting that the polygyne form originated via a founder event from a local monogyne population. These comparative data suggest an alternative route to polygyny in S. geminata in which loss of allelic variation at genes encoding recognition cues has led to a breakdown in discrimination abilities and the consequent acceptance of multiple queens in colonies.
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Bettinger, Pete, Gay A. Bradshaw, and George W. Weaver. "Effects of geographic information system vector–raster–vector data conversion on landscape indices." Canadian Journal of Forest Research 26, no. 8 (August 1, 1996): 1416–25. http://dx.doi.org/10.1139/x26-158.
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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 < 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.
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Gau, Y. David, and Lindsay A. Tartre. "The Sidesplitting Story of the Midpoint Polygon." Mathematics Teacher 87, no. 4 (April 1994): 249–56. http://dx.doi.org/10.5951/mt.87.4.0249.
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Understanding what area means and learning ways to calculate the area of various figures are important objectives in geometry. Students as adults will use concepts related to finding areas of polygons in many contexts, such as finding the area of their backyard or knowing how much wallpaper is needed to cover a wall in their dining room. One context for exploring area relationships is comparing the area of a polygon to the area of its associated midpoint polygon, formed by joining the midpoints of consecutive sides of the original polygon. This article describes activities that examine the patterns and relationships between the areas of polygons and those of their associated midpoint polygons for triangles, quadrilaterals, pentagons, and other polygons. We shall also look at the pattern for regular polygons.
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BLACHE, RÉGIS. "NEWTON POLYGONS FOR CHARACTER SUMS AND POINCARÉ SERIES." International Journal of Number Theory 07, no. 06 (September 2011): 1519–42. http://dx.doi.org/10.1142/s1793042111004368.
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In this paper, we precise the asymptotic behavior of Newton polygons of L-functions associated to character sums, coming from certain n variable Laurent polynomials. In order to do this, we use the free sum on convex polytopes. This operation allows the determination of the limit of generic Newton polygons for the sum Δ = Δ1⊕Δ2 when we know the limit of generic Newton polygons for each factor. To our knowledge, these are the first results concerning the asymptotic behavior of Newton polygons for multivariable polynomials when the generic Newton polygon differs from the combinatorial (Hodge) polygon associated to the polyhedron.
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Bernabeu, Melania, Salvador Llinares, and Mar Moreno. "Levels of Sophistication in Elementary Students’ Understanding of Polygon Concept and Polygons Classes." Mathematics 9, no. 16 (August 17, 2021): 1966. http://dx.doi.org/10.3390/math9161966.
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This paper reports sophistication levels in third grade children’s understanding of polygon concept and polygon classes. We consider how children endow mathematical meaning to parts of figures and reason to identify relationships between polygons. We describe four levels of sophistication in children’s thinking as they consider a figure as an example of a polygon class through spatial structuring (the mental operation of building an organization for a set of figures). These levels are: (i) partial structuring of polygon concept; (ii) global structuring of polygon concept; (iii) partial structuring of polygon classes; and (iv) global structuring of polygon classes. These levels detail how cognitive apprehensions, dimensional deconstruction, and the use of mathematical language intervene in the mental process of spatial structuring in the understanding of the classes of polygons.
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Wu, San-Yuan, and Sartaj Sahni. "Fast Algorithms to Partition Simple Rectilinear Polygons." VLSI Design 1, no. 3 (January 1, 1994): 193–215. http://dx.doi.org/10.1155/1994/16075.
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Two algorithms to partition hole-free rectilinear polygons are developed. One has complexity ∼ O(kn) and the other O(n log k) where n is the number of vertices in the polygon and k is the smaller of the number of vertical and horizontal inversions of the polygon, k is a measure of the simplicity of a polygon. Since k is small for most practical polygons, our algorithms are fast in practice. Experimental results comparing our algorithms with that of Imai and Asano [1] are also presented.
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AICHHOLZER, OSWIN, THOMAS HACKL, MATIAS KORMAN, ALEXANDER PILZ, and BIRGIT VOGTENHUBER. "GEODESIC-PRESERVING POLYGON SIMPLIFICATION." International Journal of Computational Geometry & Applications 24, no. 04 (December 2014): 307–23. http://dx.doi.org/10.1142/s0218195914600097.
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Polygons are a paramount data structure in computational geometry. While the complexity of many algorithms on simple polygons or polygons with holes depends on the size of the input polygon, the intrinsic complexity of the problems these algorithms solve is often related to the reflex vertices of the polygon. In this paper, we give an easy-to-describe linear-time method to replace an input polygon [Formula: see text] by a polygon [Formula: see text] such that (1) [Formula: see text] contains [Formula: see text], (2) [Formula: see text] has its reflex vertices at the same positions as [Formula: see text], and (3) the number of vertices of [Formula: see text] is linear in the number of reflex vertices. Since the solutions of numerous problems on polygons (including shortest paths, geodesic hulls, separating point sets, and Voronoi diagrams) are equivalent for both [Formula: see text] and [Formula: see text], our algorithm can be used as a preprocessing step for several algorithms and makes their running time dependent on the number of reflex vertices rather than on the size of [Formula: see text]. We describe several of these applications (including linear-time post-processing steps that might be necessary).
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Wang, Chao, and Zhongzi Wang. "The limit shapes of midpoint polygons in ℝ3." Journal of Knot Theory and Its Ramifications 28, no. 10 (September 2019): 1950062. http://dx.doi.org/10.1142/s0218216519500627.
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For a polygon in the [Formula: see text]-dimensional Euclidean space, we give two kinds of normalizations of its [Formula: see text]th midpoint polygon by a homothetic transformation and an affine transformation, respectively. As [Formula: see text] goes to infinity, the normalizations will approach “regular” polygons inscribed in an ellipse and a generalized Lissajous curve, respectively, where the curves may be degenerate. The most interesting case is when [Formula: see text], where polygons with all its [Formula: see text]th midpoint polygons knotted are discovered and discussed. Such polygonal knots can be seen as a discrete version of the Lissajous knots.
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Verheyen, H. F. "A Single Die-Cut Element for Transformable Structures." International Journal of Space Structures 8, no. 1-2 (April 1993): 127–34. http://dx.doi.org/10.1177/0266351193008001-213.
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Any polyhedral structure composed of identical regular polygons can be turned into an expandable structure by replacing the polygons by pairs of polygons which can rotate about a common central hinge, and connecting a vertex of an upper polygon with the vertex of a lower polygon of an adjacent pair. Most of these structures will collapse in the fully expanded position by losing their rigidity near the final stage, and hence, become deployable. A triangular element which enables one to assemble and dismantle such structures is presented here, together with a series of examples of experimental shapes.
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MELKEMI, MAHMOUD, FREDERIC CORDIER, and NICKOLAS S. SAPIDIS. "A PROVABLE ALGORITHM TO DETECT WEAK SYMMETRY IN A POLYGON." International Journal of Image and Graphics 13, no. 01 (January 2013): 1350002. http://dx.doi.org/10.1142/s0219467813500022.
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This paper deals with the problem of detecting "weak symmetry" in a polygon, which is a special bijective and continuous mapping between the vertices of the given polygon. An application of this work is the automatic reconstruction of 3D polygons symmetric with respect to a plane from free-hand sketches of weakly-symmetric 2D polygons. We formalize the weak-symmetry notion and highlight its many properties which lead to an algorithm detecting it. The closest research work to the proposed approach is the detection of skewed symmetry. Skewed symmetry detection deals only with reconstruction of planar mirror-symmetric 3D polygons while our method is able to identify symmetry in projections of planar as well as nonplanar mirror-symmetric 3D polygons.
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TAN, XUEHOU. "EDGE GUARDS IN STRAIGHT WALKABLE POLYGONS." International Journal of Computational Geometry & Applications 09, no. 01 (February 1999): 63–79. http://dx.doi.org/10.1142/s0218195999000066.
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We study the art gallery problem restricted to edge guards and straight walkable polygons. An edge guard is the guard that patrols individual edges of the polygon. A simple polygon P is called straight walkable if there are two vertices s and t in P and we can move two points montonically on two polygonal chains of P from s to t, one clockwise and the other counterclockwise, such that two points are always mutually visible. For instance, monotone polygons and spiral polygons are straight walkable. We show that ⌊(n+2)/5⌋ edge guards are always sufficient to watch and n-vertex gallery of this type. Furthermore, we also show that if the given polygon is straight walkable and rectilinear, then ⌊(n+3)/6⌋ edge guards are sufficient. Both of our upper bounds match the known lower bounds.
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Carson, Linda, Matthew Millard, Nadine Quehl, and James Danckert. "Polygon-Based Drawing Accuracy Analysis and Positive/Negative Space." Art & Perception 2, no. 1-2 (2014): 213–36. http://dx.doi.org/10.1163/22134913-00002021.
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The study of drawing generally depends on ratings by human critics and self-reported expertise of the drawers. To complement those approaches, we developed an objective continuous performance-based measure of drawing accuracy. This measure represents drawings as sets of landmark points and analyses features of particular research interest by comparing polygons of those features’ landmark points with their counterpart polygons in a veridical image. This approach produces local accuracy measures (for each polygon), a global accuracy measure (the mean across several polygons), and four distinct properties of a polygon for analysis: its size, its position, its orientation and the proportionality of its shape. We briefly describe the method and its potential research applications in drawing education and visual perception, then apply it to a specific research question: Are we more accurate when drawing in the so-called ‘positive space’ (or figure)? In a polygon-based accuracy analysis of 34 representational drawings, expert drawers outperformed less experienced participants on overall accuracy and every dimension of polygon error. Comparing polygons in the positive and negative space revealed an apparent trade-off on the different dimensions of polygon error. People were more accurate at proportionality and position in the positive space than in the negative space, but more accurate at orientation in the negative space. The contribution is the use of an objective, performance-based analysis of geometric deformations to study the accuracy of drawings at different levels of organization, here, in the positive and negative space.
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Harp, Dylan R., Vitaly Zlotnik, Charles J. Abolt, Bob Busey, Sofia T. Avendaño, Brent D. Newman, Adam L. Atchley, Elchin Jafarov, Cathy J. Wilson, and Katrina E. Bennett. "New insights into the drainage of inundated ice-wedge polygons using fundamental hydrologic principles." Cryosphere 15, no. 8 (August 23, 2021): 4005–29. http://dx.doi.org/10.5194/tc-15-4005-2021.
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Abstract. The pathways and timing of drainage from the inundated centers of ice-wedge polygons in a warming climate have important implications for carbon flushing, advective heat transport, and transitions from methane to carbon dioxide dominated emissions. Here, we expand on previous research using a recently developed analytical model of drainage from a low-centered polygon. Specifically, we perform (1) a calibration to field data identifying necessary model refinements and (2) a rigorous model sensitivity analysis that expands on previously published indications of polygon drainage characteristics. This research provides intuition on inundated polygon drainage by presenting the first in-depth analysis of drainage within a polygon based on hydrogeological first principles. We verify a recently developed analytical solution of polygon drainage through a calibration to a season of field measurements. Due to the parsimony of the model, providing the potential that it could fail, we identify the minimum necessary refinements that allow the model to match water levels measured in a low-centered polygon. We find that (1) the measured precipitation must be increased by a factor of around 2.2, and (2) the vertical soil hydraulic conductivity must decrease with increasing thaw depth. Model refinement (1) accounts for runoff from rims into the ice-wedge polygon pond during precipitation events and possible rain gauge undercatch, while refinement (2) accounts for the decreasing permeability of deeper soil layers. The calibration to field measurements supports the validity of the model, indicating that it is able to represent ice-wedge polygon drainage dynamics. We then use the analytical solution in non-dimensional form to provide a baseline for the effects of polygon aspect ratios (radius to thaw depth) and coefficient of hydraulic conductivity anisotropy (horizontal to vertical hydraulic conductivity) on drainage pathways and temporal depletion of ponded water from inundated ice-wedge polygon centers. By varying the polygon aspect ratio, we evaluate the relative effect of polygon size (width), inter-annual increases in active-layer thickness, and seasonal increases in thaw depth on drainage. The results of our sensitivity analysis rigorously confirm a previous analysis indicating that most drainage through the active layer occurs along an annular region of the polygon center near the rims. This has important implications for transport of nutrients (such as dissolved organic carbon) and advection of heat towards ice-wedge tops. We also provide a comprehensive investigation of the effect of polygon aspect ratio and anisotropy on drainage timing and patterns, expanding on previously published research. Our results indicate that polygons with large aspect ratios and high anisotropy will have the most distributed drainage, while polygons with large aspect ratios and low anisotropy will have their drainage most focused near their periphery and will drain most slowly. Polygons with small aspect ratios and high anisotropy will drain most quickly. These results, based on parametric investigation of idealized scenarios, provide a baseline for further research considering the geometric and hydraulic complexities of ice-wedge polygons.
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Mogilski, Wiktor J. "Polygon vertex extremality and decomposition of polygons." Discrete Mathematics 310, no. 17-18 (September 2010): 2231–37. http://dx.doi.org/10.1016/j.disc.2010.04.015.
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Schmerl, James H. "Dividing a polygon into two similar polygons." Discrete Mathematics 311, no. 4 (February 2011): 220–31. http://dx.doi.org/10.1016/j.disc.2010.10.021.
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BAREQUET, GILL, and AYA STEINER. "ON THE MATABILITY OF POLYGONS." International Journal of Computational Geometry & Applications 18, no. 05 (October 2008): 469–506. http://dx.doi.org/10.1142/s0218195908002726.
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Interpolating a piecewise-linear triangulated surface between two polygons lying in parallel planes has attracted a lot of attention in the literature over the past three decades. This problem is the simplest variant of interpolation between parallel slices, which may contain multiple polygons with unrestricted geometries and/or topologies. Its solution has important applications to medical imaging, digitization of objects, geographical information systems, and more. Practically all currently-known methods for surface reconstruction from parallel slices assume a priori the existence of a non-self-intersecting triangulated surface defined over the vertices of the two polygons, which connects them. Gitlin et al. were the first to specify a nonmatable pair of polygons. In this paper we provide proof of the nonmatability of a “simpler” pair of polygons, which is less complex than the example given by Gitlin et al. Furthermore, we provide a family of polygon pairs with unbounded complexity, which we believe to be nonmatable. We also give a few sufficient conditions for polygon matability.
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Levy, Joseph S., James W. Head, and David R. Marchant. "The role of thermal contraction crack polygons in cold-desert fluvial systems." Antarctic Science 20, no. 6 (June 2, 2008): 565–79. http://dx.doi.org/10.1017/s0954102008001375.
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AbstractThermal contraction crack polygons modify the generation, transport, and storage of water in Wright Valley gullies. Water generation is contributed to by trapping of windblown snow in polygon troughs. Water transport is modified by changes to the ice-cement table and active layer topography caused by polygon trough formation. Water storage is modified by sediment grain-size distribution within polygons in gully distal hyporheic zones. Patterned ground morphological variation can serve as an indicator of fluvial modification, ranging from nearly unmodified composite-wedge polygons to polygons forming in association with gully channels. Thermal contraction crack polygons may also constrain the gully formation sequence, suggesting the continuous presence of permafrost beneath the Wright Valley gullies during the entire period of gully emplacement. This analysis provides a framework for understanding the relationships between polygons and gullies observed on Mars. If comparable stratigraphic relationships can be documented, the presence of an analogous impermeable ice-cemented layer beneath the gullies can be inferred, suggesting an atmospheric source for Martian gully-carving fluids.
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Hong, Kyungpyo, and Seungsang Oh. "Bounds on Multiple Self-avoiding Polygons." Canadian Mathematical Bulletin 61, no. 3 (September 1, 2018): 518–30. http://dx.doi.org/10.4153/cmb-2017-072-x.
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AbstractA self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problemto this study, we considermultiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number pm×n of distinct multiple self-avoiding polygons in the m × n rectangular grid on the square lattice. For m = 2, p2×n = 2n−1 − 1. And for integers m, n ≥ 3,
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Kartoziia, Andrei. "Assessment of the Ice Wedge Polygon Current State by Means of UAV Imagery Analysis (Samoylov Island, the Lena Delta)." Remote Sensing 11, no. 13 (July 9, 2019): 1627. http://dx.doi.org/10.3390/rs11131627.
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Modern degradation of Arctic permafrost promotes changes in tundra landscapes and leads to degradation of ice wedge polygons, which are the most widespread landforms of Arctic wetlands. Status assessment of polygon degradation is important for various environmental studies. We have applied the geographic information systems’ (GIS) analysis of data from unmanned aerial vehicles (UAV) to accurately assess the status of ice wedge polygon degradation on Samoylov Island. We used several modern models of polygon degradation for revealing polygon types, which obviously correspond to different stages of degradation. Manual methods of mapping and a high spatial resolution of used UAV data allowed for a high degree of accuracy in the identification of all land units. The study revealed the following: 41.79% of the first terrace surface was composed of non-degraded polygonal tundra; 18.37% was composed of polygons, which had signs of thermokarst activity and corresponded to various stages of degradation in the models; and 39.84% was composed of collapsed polygons, slopes, valleys, and water bodies, excluding ponds of individual polygons. This study characterizes the current status of polygonal tundra degradation of the first terrace surface on Samoylov Island. Our assessment reflects the landscape condition of the first terrace surface of Samoylov Island, which is the typical island of the southern part of the Lena Delta. Moreover, the study illustrates the potential of UAV data GIS analysis for highly accurate investigations of Arctic landscape changes.
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Zhou, X. G., and H. S. Wang. "A QUADTREE SPATIAL INDEX METHOD WITH INCLUSION RELATIONS FOR THE INCREMENTAL UPDATING OF VECTOR LANDCOVER DATABASE." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-4 (September 19, 2018): 757–61. http://dx.doi.org/10.5194/isprs-archives-xlii-4-757-2018.
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<p><strong>Abstract.</strong> In vector landcover database, there are a lot of complex polygons with many holes, even nesting holes. In the incremental updating (i.e., using the change-only information to update the land cover database), a new changed parcel usually has 2-dimensional intersections (e.g., overlap, cover, equal and inside, etc.) with several existing regions, automatic updating operations need to identify the affected objects for the new changes at first. If the existing parcels include complex polygons (i.e., the polygon with holes), it is still needed to determine if there are 2-dimensional intersections between the new changed polygon and each holes of the involved complex polygons. The relation between the complex polygon and its holes has not been presented in the current spatial data indexing methods, only the MBB (Minimum Bounding Box) of the exterior ring of the complex polygon has been stored, the non-involved holes can not be filtered at the first step of spatial access methods. As the refinement geometric operation is costly, therefore the updating process for the complex polygons is very complicated and low efficient using the current spatial data indexing methods. In order to solve this problem, an improved quadtree spatial index method is presented in this paper. In this method, the polygons is divided to two categories according to the relations with the quadrant axes, i.e., disjoint to the axes and intersect with the axes. The intersect polygons are still divided to 5 cases according to the intersection position among the polygons and the different level quadrant axes. The intersection polygons are stored in the different level root nodes in our index tree, and five buckets denoted as <i>XpB, XnB, YpB, YnB, XYB</i> are used to store the polygons intersecting the different level quadrant axes respectively. The polygons disjoint to all quadrant axes are stored in the leaf nodes in this method. The authors developed the spatial index structure with inclusion relations and the algorithms of the corresponding index operations (e.g., insert, delete and query) for the complex polygons. The effectiveness of the improved index is verified by an experiment of land cover data incremental updating. Experimental results show that the proposed index method is significantly more efficient than the traditional quadtree index in terms of spatial query efficiency, and the time efficiency of the incremental updating is increased about 3 times using the proposed index method than that using the traditional quadtree index.</p>
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Akitaya, Hugo A., and Csaba D. Tóth. "Reconstruction of Weakly Simple Polygons from Their Edges." International Journal of Computational Geometry & Applications 28, no. 02 (June 2018): 161–80. http://dx.doi.org/10.1142/s021819591860004x.
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We address the problem of reconstructing a polygon from the multiset of its edges. Given [Formula: see text] line segments in the plane, find a polygon with [Formula: see text] vertices whose edges are these segments, or report that none exists. It is easy to solve the problem in [Formula: see text] time if we seek an arbitrary polygon or a simple polygon. We show that the problem is NP-complete for weakly simple polygons, that is, a polygon whose vertices can be perturbed by at most [Formula: see text], for any [Formula: see text], to obtain a simple polygon. We give [Formula: see text]-time algorithms for reconstructing weakly simple polygons: when all segments are collinear or the segment endpoints are in general position. These results extend to the variant in which the segments are directed. We study related problems for the case that the union of the [Formula: see text] input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems.
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DIAO, YUANAN. "MINIMAL KNOTTED POLYGONS ON THE CUBIC LATTICE." Journal of Knot Theory and Its Ramifications 02, no. 04 (December 1993): 413–25. http://dx.doi.org/10.1142/s0218216593000234.
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The polygons on the cubic-lattice have played an important role in simulating various circular molecules, especially the ones with relatively big volumes. There have been a lot of theoretical studies and computer simulations devoted to this subject. The questions are mostly around the knottedness of such a polygon, such as what kind of knots can appear in a polygon of given length, how often it can occur, etc. A very often asked and long standing question is about the minimal length of a knotted polygon. It is well-known that there are knotted polygons on the lattice with 24 steps yet it is unproved up to this date that 24 is the minimal number of steps needed. In this paper, we prove that in order to obtain a knotted polygon on the cubic lattice, at least 24 steps are needed and we can only have trefoils with 24 steps.
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Berry, Lindsay, Andrew Beveridge, Jane Butterfield, Volkan Isler, Zachary Keller, Alana Shine, and Junyi Wang. "Line-of-Sight Pursuit in Monotone and Scallop Polygons." International Journal of Computational Geometry & Applications 29, no. 04 (December 2019): 307–51. http://dx.doi.org/10.1142/s0218195919500122.
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We study a turn-based game in a simply connected polygonal environment [Formula: see text] between a pursuer [Formula: see text] and an adversarial evader [Formula: see text]. Both players can move in a straight line to any point within unit distance during their turn. The pursuer [Formula: see text] wins by capturing the evader, meaning that their distance satisfies [Formula: see text], while the evader wins by eluding capture forever. Both players have a map of the environment, but they have different sensing capabilities. The evader [Formula: see text] always knows the location of [Formula: see text]. Meanwhile, [Formula: see text] only has line-of-sight visibility: [Formula: see text] observes the evader’s position only when the line segment connecting them lies entirely within the polygon. Therefore [Formula: see text] must search for [Formula: see text] when the evader is hidden from view. We provide a winning strategy for [Formula: see text] in two families of polygons: monotone polygons and scallop polygons. In both families, a straight line [Formula: see text] can be moved continuously over [Formula: see text] so that (1) [Formula: see text] is a line segment and (2) every point on the boundary [Formula: see text] is swept exactly once. These are both subfamilies of strictly sweepable polygons. The sweeping motion for a monotone polygon is a single translation, and the sweeping motion for a scallop polygon is a single rotation. Our algorithms use rook’s strategy during its pursuit phase, rather than the well-known lion’s strategy. The rook’s strategy is crucial for obtaining a capture time that is linear in the area of [Formula: see text]. For both monotone and scallop polygons, our algorithm has a capture time of [Formula: see text], where [Formula: see text] is the number of polygon vertices.
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Zaputryaeva, E. S. "Deformations of Planar Equilateral Polygons with a Constant Index." Modeling and Analysis of Information Systems 20, no. 1 (March 18, 2015): 138–59. http://dx.doi.org/10.18255/1818-1015-2013-1-138-159.
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A carpenter’s rule problem is considered for the case of a self-intersecting planar polygon with additional restriction: the index (turning number) of the polygon should be preserved during deformation. We present a solution for equilateral polygons and state a problem for general ones.
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Wales, Nathan A., Jesus D. Gomez-Velez, Brent D. Newman, Cathy J. Wilson, Baptiste Dafflon, Timothy J. Kneafsey, Florian Soom, and Stan D. Wullschleger. "Understanding the relative importance of vertical and horizontal flow in ice-wedge polygons." Hydrology and Earth System Sciences 24, no. 3 (March 10, 2020): 1109–29. http://dx.doi.org/10.5194/hess-24-1109-2020.
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Abstract. Ice-wedge polygons are common Arctic landforms. The future of these landforms in a warming climate depends on the bidirectional feedback between the rate of ice-wedge degradation and changes in hydrological characteristics. This work aims to better understand the relative roles of vertical and horizontal water fluxes in the subsurface of polygonal landscapes, providing new insights and data to test and calibrate hydrological models. Field-scale investigations were conducted at an intensively instrumented location on the Barrow Environmental Observatory (BEO) near Utqiaġvik, AK, USA. Using a conservative tracer, we examined controls of microtopography and the frost table on subsurface flow and transport within a low-centered and a high-centered polygon. Bromide tracer was applied at both polygons in July 2015 and transport was monitored through two thaw seasons. Sampler arrays placed in polygon centers, rims, and troughs were used to monitor tracer concentrations. In both polygons, the tracer first infiltrated vertically until encountering the frost table and was then transported horizontally. Horizontal flow occurred in more locations and at higher velocities in the low-centered polygon than in the high-centered polygon. Preferential flow, influenced by frost table topography, was significant between polygon centers and troughs. Estimates of horizontal hydraulic conductivity were within the range of previous estimates of vertical conductivity, highlighting the importance of horizontal flow in these systems. This work forms a basis for understanding complexity of flow in polygonal landscapes.
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Lebedev, Pavel, and Vladimir Ushakov. "Methods of optimization of Hausdorff distance between convex rotating figures." Yugoslav Journal of Operations Research 30, no. 4 (2020): 429–42. http://dx.doi.org/10.2298/yjor191125027l.
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We studied the problem of optimizing the Hausdorff distance between two convex polygons. Its minimization is chosen as the criterion of optimality. It is believed that one of the polygons can make arbitrary movements on the plane, including parallel transfer and rotation with the center at any point. The other polygon is considered to be motionless. Iterative algorithms for the phased shift and rotation of the polygon are developed and implemented programmatically, providing a decrease in the Hausdorff distance between it and the fixed polygon. Theorems on the correctness of algorithms for a wide class of cases are proved. Moreover, the geometric properties of the Chebyshev center of a compact set and the differential properties of the Euclidean function of distance to a convex set are essentially used. When implementing the software package, it is possible to run multiple times in order to identify the best found polygon position. A number of examples are simulated.
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Simpson, R. J. "Convex lattice polygons of minimum area." Bulletin of the Australian Mathematical Society 42, no. 3 (December 1990): 353–67. http://dx.doi.org/10.1017/s0004972700028525.
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A convex lattice polygon is a polygon whose vertices are points on the integer lattice and whose interior angles are strictly less than π radians. We define a(2n) to be the least possible area of a convex lattice polygon with 2n vertices. A method for constructing convex lattice polygons with area a(2n) is described, and values of a(2n) for low n are obtained.
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Timukhina, Yelena, Petr Kozlov, Vitaly Kolokolnikov, and Nikolay Tushin. "Modeling of large railway polygons." MATEC Web of Conferences 216 (2018): 02025. http://dx.doi.org/10.1051/matecconf/201821602025.
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When large junctions of railway polygons are studied, detailed modeling is subject to some limitations. Therefore, it is necessary to develop appropriate tools. This paper presents an aggregate (macromodeling) technique for necks and yards of railway stations. Results of calculations based on aggregate models are highly consistent with results of detailed modeling, which confirms the applicability of this modeling method to railway polygons. The proposed principles were implemented on the basis of the IMETRA simulation system. The resulting polygon model makes it possible to determine the total throughput of a polygon and to identify bottlenecks that slow down car traffic.
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ESTRADA, B., and E. MARTÍNEZ. "COORDINATES FOR THE TEICHMÜLLER SPACE OF PLANAR SURFACE N.E.C. GROUPS." International Journal of Mathematics 14, no. 10 (December 2003): 1037–52. http://dx.doi.org/10.1142/s0129167x03002137.
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It is proved that every planar surface NEC group Γ admits a fundamental region, called a normal region, which is a hyperbolic right-angled polygon with several pairs of identified sides according to a pattern. Conversely, each such a polygon can be taken as a fundamental region of a planar surface NEC group. By means of these regions the Teichmüller space of Γ is studied, obtaining a parametrization by means of congruence classes of marked polygons. The parameters are certain lengths in these polygons, and from them explicit matrices of the generators of the associated group are obtained.
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LINGAS, ANDRZEJ, AGNIESZKA WASYLEWICZ, and PAWEŁ ŻYLIŃSKI. "LINEAR-TIME 3-APPROXIMATION ALGORITHM FOR THE r-STAR COVERING PROBLEM." International Journal of Computational Geometry & Applications 22, no. 02 (April 2012): 103–41. http://dx.doi.org/10.1142/s021819591250001x.
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The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has Õ(n17)-time complexity, where Õ(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons.
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Levy, Joseph S., David R. Marchant, and James W. Head. "Distribution and origin of patterned ground on Mullins Valley debris-covered glacier, Antarctica: the roles of ice flow and sublimation." Antarctic Science 18, no. 3 (August 24, 2006): 385–97. http://dx.doi.org/10.1017/s0954102006000435.
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We map polygonally patterned ground formed in sublimation tills that overlie debris-covered glaciers in Mullins Valley and central Beacon Valley, in southern Victoria Land, Antarctica, and distinguish five morphological zones. Where the Mullins Valley debris-covered glacier debouches into Beacon Valley, polygonal patterning transitions from radial (orthogonal) intersections to non-oriented (hexagonal) intersections, providing a time-series of polygon evolution within a single microclimate. We offer the following model for polygon formation and evolution in the Mullins Valley system. Near-vertical cracks that ultimately outline polygons are produced by thermal contraction in the glacier ice. Some of these cracks may initially be oriented radial to maximum surface velocities by pre-existing structural stresses and material weaknesses in the glacier ice. In areas of relatively rapid flow, polygons are oriented down-valley forming an overall fan pattern radial to maximum ice velocity. As glacier flow moves the cracks down-valley, minor variations in flow rate deform polygons, giving rise to deformed radial polygons. Non-oriented (largely hexagonal) polygons commonly form in regions of stagnant and/or near-stagnant ice. We propose that orientation and morphology of contraction-crack polygons in sublimation tills can thus be used as an indicator of rates of subsurface ice flow.
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Kim, J., and K. Yu. "AREAL FEATURE MATCHING BASED ON SIMILARITY USING CRITIC METHOD." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-2/W4 (October 19, 2015): 75–78. http://dx.doi.org/10.5194/isprsarchives-xl-2-w4-75-2015.
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In this paper, we propose an areal feature matching method that can be applied for many-to-many matching, which involves matching a simple entity with an aggregate of several polygons or two aggregates of several polygons with fewer user intervention. To this end, an affine transformation is applied to two datasets by using polygon pairs for which the building name is the same. Then, two datasets are overlaid with intersected polygon pairs that are selected as candidate matching pairs. If many polygons intersect at this time, we calculate the inclusion function between such polygons. When the value is more than 0.4, many of the polygons are aggregated as single polygons by using a convex hull. Finally, the shape similarity is calculated between the candidate pairs according to the linear sum of the weights computed in CRITIC method and the position similarity, shape ratio similarity, and overlap similarity. The candidate pairs for which the value of the shape similarity is more than 0.7 are determined as matching pairs. We applied the method to two geospatial datasets: the digital topographic map and the KAIS map in South Korea. As a result, the visual evaluation showed two polygons that had been well detected by using the proposed method. The statistical evaluation indicates that the proposed method is accurate when using our test dataset with a high F-measure of 0.91.
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Chen, Zhanlong, Xiaochuan Ma, Liang Wu, and Zhong Xie. "An Intuitionistic Fuzzy Similarity Approach for Clustering Analysis of Polygons." ISPRS International Journal of Geo-Information 8, no. 2 (February 23, 2019): 98. http://dx.doi.org/10.3390/ijgi8020098.
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Accurate and reasonable clustering of spatial data results facilitates the exploration of patterns and spatial association rules. Although a broad range of research has focused on the clustering of spatial data, only a few studies have conducted a deeper exploration into the similarity approach mechanism for clustering polygons, thereby limiting the development of spatial clustering. In this study, we propose a novel fuzzy similarity approach for spatial clustering, called Extend Intuitionistic Fuzzy Set-Interpolation Boolean Algebra (EIFS-IBA). When discovering polygon clustering patterns by spatial clustering, this method expresses the similarities between polygons and adjacent graph models. Shape-, orientation-, and size-related properties of a single polygon are first extracted, and are used as indices for measuring similarities between polygons. We then transform the extracted properties into a fuzzy format through normalization and fuzzification. Finally, the similarity graph containing the neighborhood relationship between polygons is acquired, allowing for clustering using the proposed adjacency graph model. In this paper, we clustered polygons in Staten Island, United States. The visual result and two evaluation criteria demonstrated that the EIFS-IBA similarity approach is more expressive compared to the conventional similarity (ConS) approach, generating a clustering result more consistent with human cognition.
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DIAO, YUANAN. "The Knotting of Equilateral Polygons in R3." Journal of Knot Theory and Its Ramifications 04, no. 02 (June 1995): 189–96. http://dx.doi.org/10.1142/s0218216595000090.
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It was proved in [4] that the knotting probability of a Gaussian random polygon goes to 1 as the length of the polygon goes to infinity. In this paper, we prove the same result for the equilateral random polygons in R3. More precisely, if EPn is an equilateral random polygon of n steps, then we have [Formula: see text] provided that n is large enough, where ∊ is some positive constant.
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GITLIN, CAROLE, JOSEPH O’ROURKE, and VINITA SUBRAMANIAN. "ON RECONSTRUCTING POLYHEDRA FROM PARALLEL SLICES." International Journal of Computational Geometry & Applications 06, no. 01 (March 1996): 103–22. http://dx.doi.org/10.1142/s0218195996000071.
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The problem of reconstructing a three-dimensional object from parallel slices has application in computer vision and medicine. Here we explore a specific existence question: given two polygons in parallel planes, is it always possible to find a polyhedron that has those polygons as faces, and whose vertices are precisely the vertices of the two polygons? We answer this question in the negative by providing an example of two polygons that cannot be connected to form a simple polyhedron. One polygon is a triangle, the other a somewhat complicated shape with spiraling pockets.
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Diao, Y., J. C. Nardo, and Y. Sun. "Global Knotting in Equilateral Random Polygons." Journal of Knot Theory and Its Ramifications 10, no. 04 (June 2001): 597–607. http://dx.doi.org/10.1142/s0218216501001025.
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In this paper, we study the knotting probability of equilateral random polygons. It is known that such objects are locally knotted with-probability arbitrarily close to one provided the length is sufficiently large ([4]). For Gaussian random polygons, it has been shown that the probability of global knottedness also tends to one as the length of the polygon tends to infinity [8]. In this paper, we prove that global knotting also occurs in equilateral random polygons with a probability approaching one as the length of the polygons goes to infinity.
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Hormann, Kai, and Alexander Agathos. "The point in polygon problem for arbitrary polygons." Computational Geometry 20, no. 3 (November 2001): 131–44. http://dx.doi.org/10.1016/s0925-7721(01)00012-8.
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Lewandowicz, Elżbieta, and Paweł Flisek. "Base Point Split Algorithm for Generating Polygon Skeleton Lines on the Example of Lakes." ISPRS International Journal of Geo-Information 9, no. 11 (November 15, 2020): 680. http://dx.doi.org/10.3390/ijgi9110680.
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This article presents the Base Point Split (BPSplit) algorithm to generate a complex polygon skeleton based on sets of vector data describing lakes and rivers. A key feature of the BPSplit algorithm is that it is dependent on base points representing the source or mouth of a river or a stream. The input values of base points determine the shape of the resulting skeleton of complex polygons. Various skeletons can be generated with the use of different base points. Base points are applied to divide complex polygon boundaries into segments. Segmentation supports the selection of triangulated irregular network (TIN) edges inside complex polygons. The midpoints of the selected TIN edges constitute a basis for generating a skeleton. The algorithm handles complex polygons with numerous holes, and it accounts for all holes. This article proposes a method for modifying a complex skeleton with numerous holes. In the discussed approach, skeleton edges that do not meet the preset criteria (e.g., that the skeleton is to be located between holes in the center of the polygon) are automatically removed. An algorithm for smoothing zigzag lines was proposed.
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SHIRAKAWA, TOSHIHIRO, and RYUHEI UEHARA. "COMMON DEVELOPMENTS OF THREE INCONGRUENT ORTHOGONAL BOXES." International Journal of Computational Geometry & Applications 23, no. 01 (February 2013): 65–71. http://dx.doi.org/10.1142/s0218195913500040.
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We investigate common developments that can fold into several incongruent orthogonal boxes. It was shown that there are infinitely many orthogonal polygons that fold into two incongruent orthogonal boxes in 2008. In 2011, it was shown that there exists an orthogonal polygon that folds into three boxes of size 1 × 1 × 5, 1 × 2 × 3, and 0 × 1 × 11. However it remained open whether there exists an orthogonal polygon that folds into three boxes of positive volume. We give an affirmative answer to this open problem. We show how to construct an infinite number of orthogonal polygons that fold into three incongruent orthogonal boxes.
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Chu, Wei Jun, Jian Zhao Zhou, Qun Zhang Tu, Si Wei Lai, and Ju Ying Dai. "Study on Modeling Technology Based on Polygon Transform." Advanced Materials Research 655-657 (January 2013): 296–99. http://dx.doi.org/10.4028/www.scientific.net/amr.655-657.296.
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As one of the 3D modeling key technology, polygon transform have some operation and manipulation methods which can be applied neatly and confirmed by the characteristic of practical model. Polygon transform methods such as inserting vertex, inserting polygon, extrusion, transition join, triangulation and iterative triangulation algorithm and so on are discussed by using practical examples of reverse modeling in this paper. The experiment results show that using these methods synthetically have an effect on constructing and transforming complicated 3D model. We can thin polygons, redefine new local models, extrude polygon outwards or inwards and largen or reduce selective polygon borderline, form a new rolling polygon which joins on the original model, and subsection polygon into triangles by defining new borders inside the polygon.
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Roisin, Y., and J. M. Pasteels. "Reproductive mechanisms in termites: Polycalism and polygyny inNasutitermes polygynus andN. costalis." Insectes Sociaux 33, no. 2 (June 1986): 149–67. http://dx.doi.org/10.1007/bf02224595.
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MacKay, J. Ross. "Thermally induced movements in ice-wedge polygons, western arctic coast: a long-term study." Géographie physique et Quaternaire 54, no. 1 (October 2, 2002): 41–68. http://dx.doi.org/10.7202/004846ar.
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AbstractThermally induced seasonal movements of the active layer and subjacent permafrost have been measured in numerous ice-wedge polygons that have varied in age, type, crack frequency, and topographic location. The field observations show that, in winter, thermal contraction, which is inward, is constrained or vanishes at the polygon centres but, in summer, thermal expansion, which is outward, is unconstrained at the ice-wedge troughs. Therefore, there tends to be a small net summer transport of the active layer, to varying depths, into the ice-wedge troughs. The movement has been observed in all polygons studied. The slow net transport of material into the ice-wedge troughs has implications for: permafrost aggradation and the growth of syngenetic wedges in some troughs; the palaeoclimatic reconstruction of some ice- wedge casts; and the interpretation of polygon stratigraphy based upon the assumption that the polygon material has accumulatedin situ.
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Cledumas, ABDULLAH Musa, YUSRI BIN KAMIN, RABIU HARUNA, and SHUAIBU HALIRU. "THE THE APPLICATION OF GENERIC GREEN SKILLS IN TESSELLATION OF REGULAR POLYGONS FOR ECONOMIC AND SOCIAL SUSTAINABILITY." Asia Proceedings of Social Sciences 4, no. 3 (June 16, 2019): 121–24. http://dx.doi.org/10.31580/apss.v4i3.873.
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Abstract
This paper proposes an improved modelling approach for tessellating regular polygons in such a way that it is environmentally sustainable. In this paper, tessellation of polygons that have been innovated through the formed motifs, is an innovation from the traditional tessellations of objects and animals. The main contribution of this work is the simplification and innovating new patterns from the existing regular polygons, in which only three polygons (triangle, square and hexagon) that can free be tessellated are used, compared to using irregular polygons or other objects. This is achieved by reducing the size of each polygon to smallest value and tessellating each of the reduced figure to the right or to left to obtain a two different designs of one unit called motif. These motifs are then combined together to form a pattern. In this innovation it is found that the proposed model is superior than tessellating ordinary regular polygon, because more designs are obtained, more colours may be obtained or introduced to give meaningful tiles or patterns. In particular Tessellations can be found in many areas of life. Art, architecture, hobbies, clothing design, including traditional wears and many other areas hold examples of tessellations found in our everyday surroundings.
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WORMAN, CHRIS, and J. MARK KEIL. "POLYGON DECOMPOSITION AND THE ORTHOGONAL ART GALLERY PROBLEM." International Journal of Computational Geometry & Applications 17, no. 02 (April 2007): 105–38. http://dx.doi.org/10.1142/s0218195907002264.
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A decomposition of a polygon P is a set of polygons whose geometric union is exactly P. We study a polygon decomposition problem that is equivalent to the Orthogonal Art Gallery problem. Two points are r-visible if the orthogonal bounding rectangle for p and q lies within P. A polygon P is an r-star if there exists a point k ∈ P such that for each point q ∈ P, q is r-visible from k. In this problem we seek a minimum cardinality decomposition of a polygon into r-stars. We show how to compute the minimum r-star cover of an orthogonal polygon in polynomial time.
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Deguchi, Tetsuo, and Kyoichi Tsurusaki. "A Statistical Study of Random Knotting Using the Vassiliev Invariants." Journal of Knot Theory and Its Ramifications 03, no. 03 (September 1994): 321–53. http://dx.doi.org/10.1142/s0218216594000241.
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Employing the Vassiliev invariants as tools for determining knot types of polygons in 3 dimensions, we evaluate numerically the knotting probability PK(N) of the Gaussian random polygon being equivalent to a knot type K. For prime knots and composite knots we plot the knotting probability PK(N) against the number N of polygonal nodes. Taking the analogy with the asymptotic scaling behaviors of self-avoiding walks, we propose a formula of fitting curves to the numerical data. The curves fit well the graphs of the knotting probability PK(N) versus N. This agreement suggests to us that the scaling formula for the knotting probability might also work for the random polygons other than the Gaussian random polygon.
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DIAO, YUANAN, NICHOLAS PIPPENGER, and DE WITT SUMNERS. "ON RANDOM KNOTS." Journal of Knot Theory and Its Ramifications 03, no. 03 (September 1994): 419–29. http://dx.doi.org/10.1142/s0218216594000307.
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In this paper, we consider knotting of Gaussian random polygons in 3-space. A Gaussian random polygon is a piecewise linear circle with n edges in which the length of the edges follows a Gaussian distribution. We prove a continuum version of Kesten's Pattern Theorem for these polygons, and use this to prove that the probability that a Gaussian random polygon of n edges in 3-space is knotted tends to one exponentially rapidly as n tends to infinity. We study the properties of Gaussian random knots, and prove that the entanglement complexity of Gaussian random knots gets arbitrarily large as n tends to infinity. We also prove that almost all Gaussian random knots are chiral.
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DeGaetano, Arthur T., Griffin Mooers, and Thomas Favata. "Temporal Changes in the Areal Coverage of Daily Extreme Precipitation in the Northeastern United States Using High-Resolution Gridded Data." Journal of Applied Meteorology and Climatology 59, no. 3 (March 2020): 551–65. http://dx.doi.org/10.1175/jamc-d-19-0210.1.
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AbstractTime-dependent changes in extreme precipitation occurrence across the northeastern United States are evaluated in terms of areal extent. Using gridded precipitation data for the period from 1950 to 2018, polygons are defined that are based on isohyets corresponding to extreme daily precipitation accumulations. Across the region, areal precipitation is characterized on the basis of the annual and seasonal number of extreme precipitation polygons and the area of the polygons. Using the subset of grid points that correspond to station locations in the northeastern United States, gridded precipitation replicates the observed trends in extreme precipitation based on station observations. Although the number of extreme precipitation polygons does not change significantly through time, there is a marked increase in the area covered by the polygons. The median annual polygon area nearly doubles from 1950 to 2013. Consistent results occur for percentiles other than the median and a range of extreme precipitation amount thresholds, with the most pronounced changes observed in spring and summer. Like trends in station data, outside the northeastern United States trends in extreme precipitation polygon area are negative, particularly in the western United States, or they are not statistically significant. Collectively, the results suggest that the increases in heavy precipitation frequency and amount observed at stations in the northeastern United States are a manifestation of an expansion of the spatial area over which extreme precipitation occurs rather than a change in the number of unique extreme precipitation polygons.
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DEVILLERS, OLIVIER. "SIMULTANEOUS CONTAINMENT OF SEVERAL POLYGONS: ANALYSIS OF THE CONTACT CONFIGURATIONS." International Journal of Computational Geometry & Applications 03, no. 04 (December 1993): 429–42. http://dx.doi.org/10.1142/s0218195993000270.
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The main concern of this paper is the detection of double contact configurations for some polygons moving in translation in a polygonal environment. We first establish some general properties about such configurations and give conditions of existence of double contacts for two or three objects. For three convex polygons moving in a polygonal environment or three simple polygons moving in a rectangle there always exists a double contact. Two examples without possibility of double contacts are given, one with three polygons (not convex) moving in a polygonal environment, and one with four convex polygons moving in a rectangle. We deduce an algorithm detecting a double contact position in time O(n2) (resp. O(n3)) for two (resp. three) convex polygons of constant sizes moving in a non-convex polygon of size n.
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Lynch, Thomas B. "Voronoi polygons quantify bias when sampling the nearest plant." Canadian Journal of Forest Research 45, no. 12 (December 2015): 1853–59. http://dx.doi.org/10.1139/cjfr-2015-0238.
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The design bias in the sample mean obtained from sampling the trees nearest to points randomly and uniformly distributed over a forested area can be exactly quantified in terms of the Voronoi polygons (V polygons) surrounding each tree in the forest of interest. For this sampling method, the V polygon for a prospective sample tree is its inclusion zone. The sides of such polygons are perpendicular to a line joining adjacent trees and equidistant from these trees. For any individual tree attribute Y, the design bias in such a sample mean for estimating the population mean of Y will be equal to the covariance between Y and V-polygon area V divided by the mean V-polygon area. The bias as a percent of the population mean of Y is the product of the correlation coefficient between Y and V and the coefficients of variation for Y and V multiplied by 100. This implies that attempts to estimate the means of commonly measured individual tree variables such as DBH, basal area, and crown diameter or the area from sampling trees nearest to randomly located points will likely be positively biased, and the magnitude of that bias will depend on the strength of the linear relationship to the V-polygon area, as well as the variability among the V-polygon areas and the variable of interest. It is not obvious whether increment core data will be positively or negatively biased, because this depends on the characteristics of the forest of interest. The main conclusion of the study is that the bias formula derived for unweighted estimation from sampling the tree nearest to a point indicates that bias in the range of 5%–10% or greater can occur in many forest populations.
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CHEN, DANNY Z., XIAOBO S. HU, and XIAODONG WU. "OPTIMAL POLYGON COVER PROBLEMS AND APPLICATIONS." International Journal of Computational Geometry & Applications 12, no. 04 (August 2002): 309–38. http://dx.doi.org/10.1142/s0218195902000918.
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Polygon cover problems are important in several applied areas, such as material layout, layered manufacturing, radiation therapy and radiosurgery, etc. In this paper, we study three optimal polygon cover problems: monotone polygon cover among obstacles, star-shaped polygon cover among obstacles, and strip cover for trapezoidalized polygons. Based on some interesting geometric observations, we develop efficient algorithms for solving these problems. The complexity bounds of our monotone cover and star-shaped cover algorithms are comparable to those of the previously best known algorithms for simpler cases of the problems without considering obstacles. Our strip cover algorithm improves the quality of the previously best known solutions.