Showing papers on "Polygon published in 2011"

Thermodynamic bubble ansatz

Abstract: Motivated by the computation of scattering amplitudes at strong coupling, we consider minimal area surfaces in AdS 5 which end on a null polygonal contour at the boundary. We map the classical problem of finding the surface into an SU(4) Hitchin system. The polygon with six edges is the first non-trivial example. For this case, we write an integral equation which determines the area as a function of the shape of the polygon. The equations are identical to those of the Thermodynamics Bethe Ansatz. Moreover, the area is given by the free energy of this TBA system. The high temperature limit of the TBA system can be exactly solved. It leads to an explicit expression for a special class of hexagonal contours.

From polygons and symbols to polylogarithmic functions

Abstract: We present a review of the symbol map, a mathematical tool that can be useful in simplifying expressions among multiple polylogarithms, and recall its main properties. A recipe is given for how to obtain the symbol of a multiple polylogarithm in terms of the combinatorial properties of an associated rooted decorated polygon. We also outline a systematic approach to constructing a function corresponding to a given symbol, and illustrate it in the particular case of harmonic polylogarithms up to weight four. Furthermore, part of the ambiguity of this process is highlighted by exhibiting a family of non-trivial elements in the kernel of the symbol map for arbitrary weight.

System and method for geographic based data visualization and extraction

Abstract: A displayed map is overlaid with a map-based graphical user interface (GUI). The GUI provides one or more related translucent views corresponding to the displayed map, where each view is associated with a dataset. The GUI also provides interlocked queries and analysis tools to probe the dataset. Each view comprises a polygon layer corresponding to the displayed map and a point layer corresponding to the displayed map, where the polygon layer and the point layer work in tandem to simultaneously display both aggregated and disaggregated data across the displayed map. A selection of a specific item of the point layer or a sub-region of the polygon layer is received from the user. In response, both the point layer and the polygon layer are modified so as to provide additional selectable information corresponding to the received selection and also additional selectable information that may be of interest to the user.

Linear time algorithms for visibility and shortest path problems inside simple polygons

Abstract: We present linear time algorithms for solving the following problems involving a simple planar polygon P: (i) Computing the collection of all shortest paths inside P from a given source vertex s to all the other vertices of P; (ii) Computing the subpolygon of P consisting of points that are visible from a segment within P; (iii) Preprocessing P so that for any query ray r emerging from some fixed edge e of P, we can find in logarithmic time the first intersection of r with the boundary of P; (iv) Preprocessing P so that for any query point x in P, we can find in logarithmic time the portion of the edge e that is visible from x; (v) Preprocessing P so that for any query point x inside P and direction u, we can find in logarithmic time the first point on the boundary of P hit by the ray at direction u from x; (vi) Calculating a hierarchical decomposition of P into smaller polygons by recursive polygon cutting, as in [Ch]. (vii) Calculating the (clockwise and counterclockwise) “convex ropes” (in the terminology of [PS]) from a fixed vertex s of P lying on its convex hull, to all other vertices of P. All these algorithms are based on a recent linear time algorithm of Tarjan and Van Wyk for triangulating a simple polygon, but use additional techniques to make all subsequent phases of these algorithms also linear.

3-D modeling of tomato canopies using a high-resolution portable scanning lidar for extracting structural information.

Abstract: In the present study, an attempt was made to produce a precise 3D image of a tomato canopy using a portable high-resolution scanning lidar. The tomato canopy was scanned by the lidar from three positions surrounding it. Through the scanning, the point cloud data of the canopy were obtained and they were co-registered. Then, points corresponding to leaves were extracted and converted into polygon images. From the polygon images, leaf areas were accurately estimated with a mean absolute percent error of 4.6%. Vertical profile of leaf area density (LAD) and leaf area index (LAI) could be also estimated by summing up each leaf area derived from the polygon images. Leaf inclination angle could be also estimated from the 3-D polygon image. It was shown that leaf inclination angles had different values at each part of a leaf.

Asymmetric dynamic geo-fencing

Abstract: Architecture that enables location based notifications (eg, geo-fences) using standard polygons the capture of complex regions As applied to geo-fencing, it extends geo-fencing beyond the mere representation of the virtual perimeter (fenced) area More specifically, the architecture takes into consideration geographical and demographical features, such as the layout of the roads and streets, the types of available of transportation (eg, car, bus, walk, biking, etc), the traffic conditions, and the dynamic properties of a point of interest (POI) such as opening hours, total wait time, etc More specifically, the architecture enables the dynamic modification of polygon geo-fence based on POI properties, dynamic modification of polygon geo-fence based on road/street layout, dynamic modification of polygon geo-fence based on means of transportation, dynamic modification of polygon geo-fence based on traffic conditions, dynamic modification of polygon geo-fence based on user's state, and dynamic recalculation of regions and directions tailored to user interests

New developments on inverse polygon mapping to calculate gravitational lensing magnification maps: optimized computations

Abstract: We derive an exact solution (in the form of a series expansion) to compute gravitational lensing magnification maps. It is based on the backward gravitational lens mapping of a partition of the image plane in polygonal cells (inverse polygon mapping, IPM), not including critical points (except perhaps at the cell boundaries). The zeroth-order term of the series expansion leads to the method described by Mediavilla et al. The first-order term is used to study the error induced by the truncation of the series at zeroth order, explaining the high accuracy of the IPM even at this low order of approximation. Interpreting the Inverse Ray Shooting (IRS) method in terms of IPM, we explain the previously reported N –3/4 dependence of the IRS error with the number of collected rays per pixel. Cells intersected by critical curves (critical cells) transform to non-simply connected regions with topological pathologies like auto-overlapping or non-preservation of the boundary under the transformation. To define a non-critical partition, we use a linear approximation of the critical curve to divide each critical cell into two non-critical subcells. The optimal choice of the cell size depends basically on the curvature of the critical curves. For typical applications in which the pixel of the magnification map is a small fraction of the Einstein radius, a one-to-one relationship between the cell and pixel sizes in the absence of lensing guarantees both the consistence of the method and a very high accuracy. This prescription is simple but very conservative. We show that substantially larger cells can be used to obtain magnification maps with huge savings in computation time.

A many-objective test problem for visually examining diversity maintenance behavior in a decision space

Abstract: Recently distance minimization problems in a two-dimensional decision space have been utilized as many-objective test problems to visually examine the behavior of evolutionary multi-objective optimization (EMO) algorithms. Such a test problem is usually defined by a single polygon where the distance from a solution to each vertex is minimized in the decision space. We can easily generate different test problems from different polygons. We can also easily generate test problems with multiple equivalent Pareto optimal regions using multiple polygons of the same shape and the same size. Whereas these test problems have a number of advantages, they have no clear relevance to real-world situations since they are artificially generated unrealistic test problems. In this paper, we generate a distance minimization problem from a real-world map. Our test problem has four objectives, which are to minimize the distances to the nearest elementary school, junior high school, railway station, and convenience store. Using our test problem, we examine the behavior of well-known and frequently-used EMO algorithms in terms of their diversity maintenance ability in the two-dimensional decision space.

Knot Tightening by Constrained Gradient Descent

Abstract: We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of “tight” knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their knot types. While previous authors have minimized ropelength for polygons using simulated annealing, the new idea in our code is to minimize length over the set of polygons of thickness at least one using a version of constrained gradient descent. We rewrite the problem in terms of minimizing the length of the polygon subject to an infinite family of differentiable constraint functions. We prove that the set of variations of a polygon of thickness one that does not decrease thickness to first order is a finitely generated polyhedral cone, and give an explicit set of generators. Using this cone, we give a first-order minimization procedure and a Karush–Kuhn–Tucker criterion for polygonal-ropelen...

Particle shape effects in discrete element modelling of cohesive angular particles

Abstract: This paper investigates the effect of particle angularity on general granular response concentrating on flow and stress-strain behaviour. A 2D polygon DEM model is developed to a 3D polyhedron model. The effect of particle shape on the response of polygons in simple shear and polyhedra under gravity flow is investigated using regular shapes with rounded vertices. The study concentrates on the angularity rather than aspect ratio by comparing circles, near squares and near equilateral triangles in 2D and spheres, tetrahedra and octahedra in 3D. In summary the more angular the particle the greater the resistance to the forcing load and the flowability is reduced. A mix of spheres and octahedra demonstrates an approximate linear combination of effects.

Rendering of specular surfaces in polygon-based computer-generated holograms

Abstract: A technique is presented for realistic rendering in polygon-based computer-generated holograms (CGHs). In this technique, the spatial spectrum of the reflected light is modified to imitate specular reflection. The spectral envelopes of the reflected light are fitted to a spectral shape based on the Phong reflection model used in computer graphics. The technique features fast computation of the field of objects, composed of many specular polygons, and is applicable to creating high-definition CGHs with several billions of pixels. An actual high-definition CGH is created using the proposed technique and is demonstrated for verification of the optical reconstruction of specular surfaces.

Accurate Numerical Methods for Computing 2D and 3D Robot Workspace

Abstract: Exact computation of the shape and size of robot manipulator workspace is very important for its analysis and optimum design First, the drawbacks of the previous methods based on Monte Carlo are pointed out in the paper, and then improved strategies are presented systematically In order to obtain more accurate boundary points of two-dimensional (2D) robot workspace, the Beta distribution is adopted to generate random variables of robot joints And then, the area of workspace is acquired by computing the area of the polygon what is a closed path by connecting the boundary points together For comparing the errors of workspaces which are generated by the previous and the improved method from shape and size, one planar robot manipulator is taken as example A spatial robot manipulator is used to illustrate that the methods can be used not only on planar robot manipulator, but also on the spatial The optimal parameters are proposed in the paper to computer the shape and size of 2D and 3D workspace Finally, we provided the computation time and discussed the generation of 3D workspace which is based on 3D reconstruction from the boundary points

Multivariate Areal Interpolation for Continuous and Count Data

Abstract: Geographic information system (GIS) users often need to disaggregate and reaggregate data collected in polygons, but classical kriging models only allow for data collected in points. We discuss our implementation of areal interpolation, a kriging-based disaggregation technique, in the Geostatistical Analyst extension of ArcGIS 10.1 for Gaussian, binomial, and overdispersed Poisson data. All methods allow for surfaces of prediction standard errors. We also allow for the use of a secondary cokriging variable, which can be any of the three above-mentioned distributions. Our areal interpolation model overcomes several computational problems, such as how to handle polygons of vastly different sizes and how to analyze polygons that are overlapping or disjoint. For Gaussian data averaged over polygons, the output is a surface predicting the value at each individual location. Gaussian polygonal data may arise when continuous point measurements are averaged to polygons in order to protect privacy or reduce overhead, and the original point data is discarded. For polygons containing Poisson counts, the output is a surface predicting the density of counts at each location in the data domain. Our model allows for overdispersed counts and for different observation times between polygons. The output for binomial data is a surface predicting the underlying risk at each location of seeing an individual with a certain trait. Each polygon of the input data must contain a count and a population value. The latter indicates the number of individuals sampled, and the former indicates the number of sampled individuals with a certain trait. Once a prediction surface has been created, predictions can be aggregated back to a new set of polygons. This allows for the collection of data over one set of polygons and the prediction for a different set of polygons. We discuss diagnostic options for determining how well the data fits a model, and we demonstrate areal interpolation with three case studies.

Improved Approximation for Guarding Simple Galleries from the Perimeter

Abstract: We provide an O(log log opt)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building e-nets of size $O(\frac{1}{\varepsilon}\log\log\frac{1}{\varepsilon})$ for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Bronnimann and Goodrich to build an approximation algorithm from this e-net finder. Along with a simple polygon P, our algorithm takes as input a finite set of potential guard locations that must include the polygon’s vertices. If a finite set of potential guard locations is not specified, e.g., when guards may be placed anywhere on the perimeter, we use a known discretization technique at the cost of making the algorithm’s running time potentially linear in the ratio between the longest and shortest distances between vertices. Our algorithm is the first to improve upon O(log opt)-approximation algorithms that use generic net finders for set systems of finite VC-dimension.

Polygon formation and surface flow on a rotating fluid surface

Abstract: We present a study of polygons forming on the free surface of a water flow confined to a stationary cylinder and driven by a rotating bottom plate as described by Jansson et al. (Phys. Rev. Lett., vol. 96, 2006, 174502). In particular, we study the case of a triangular structure, either completely ‘wet’ or with a ‘dry’ centre. For the dry structures, we present measurements of the surface shapes and the process of formation. We show experimental evidence that the formation can take place as a two-stage process: first the system approaches an almost stable rotationally symmetric state and from there the symmetry breaking proceeds like a low-dimensional linear instability. We show that the circular state and the unstable manifold connecting it with the polygon solution are universal in the sense that very different initial conditions lead to the same circular state and unstable manifold. For a wet triangle, we measure the surface flows by particle image velocimetry (PIV) and show that there are three vortices present, but that the strength of these vortices is far too weak to account for the rotation velocity of the polygon. We show that partial blocking of the surface flow destroys the polygons and re-establishes the rotational symmetry. For the rotationally symmetric state our theoretical analysis of the surface flow shows that it consists of two distinct regions: an inner, rigidly rotating centre and an outer annulus, where the surface flow is that of a point vortex with a weak secondary flow. This prediction is consistent with the experimentally determined surface flow

Approximate Boolean Operations on Large Polyhedral Solids with Partial Mesh Reconstruction

Abstract: We present a new approach to compute the approximate Boolean operations of two freeform polygonal mesh solids efficiently with the help of Layered Depth Images (LDIs). After applying the LDI sampling-based membership classification, the most challenging part, a trimmed adaptive contouring algorithm, is developed to reconstruct the mesh surface from the LDI samples near the intersected regions and stitch it to the boundary of the retained surfaces. Our method of approximate Boolean operations holds the advantage of numerical robustness as the approach uses volumetric representation. However, unlike other methods based on volumetric representation, we do not damage the facets in nonintersected regions, thus preserving geometric details much better and speeding up the computation as well. We show that the proposed method can successfully compute the Boolean operations of free-form solids with a massive number of polygons in a few seconds.

Computing Cartograms with Optimal Complexity

Abstract: In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight of the corresponding vertex. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine the representation into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal rectangular layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs.

Multi-patterning method

Abstract: A method includes receiving data representing a layout of a DPT-layer of an integrated circuit generated by a place and route tool. The layout includes a plurality of polygons to be formed in the DPT-layer by a multi-patterning process. First and second ones of the plurality of polygons to be formed using first and second photomasks, respectively are identified. Any intervening polygons along a first path connecting the first polygon to the second polygon, and separator regions between adjacent polygons along the first path are identified. The separator regions have sizes less than a minimum threshold distance between polygons formed on the first photomask. The separator regions are counted. A multi-patterning conflict is identified, if the count of separator regions is even, prior to assigning all remaining ones of the plurality of polygons to the first or second masks.

A framework for the automatic generation of surface topologies for abdominal aortic aneurysm models.

Abstract: Patient-specific abdominal aortic aneurysms (AAAs) are characterized by local curvature changes, which we assess using a feature-based approach on topologies representative of the AAA outer wall surface. The application of image segmentation methods yields 3D reconstructed surface polygons that contain low-quality elements, unrealistic sharp corners, and surface irregularities. To optimize the quality of the surface topology, an iterative algorithm was developed to perform interpolation of the AAA geometry, topology refinement, and smoothing. Triangular surface topologies are generated based on a Delaunay triangulation algorithm, which is adapted for AAA segmented masks. The boundary of the AAA wall is represented using a signed distance function prior to triangulation. The irregularities on the surface are minimized by an interpolation scheme and the initial coarse triangulation is refined by forcing nodes into equilibrium positions. A surface smoothing algorithm based on a low-pass filter is applied to remove sharp corners. The optimal number of iterations needed for polygon refinement and smoothing is determined by imposing a minimum average element quality index with no significant AAA sac volume change. This framework automatically generates high-quality triangular surface topologies that can be used to characterize local curvature changes of the AAA wall.

Magnetic and electronic toy construction systems and elements

Abstract: Magnetic and electronic toy construction systems and elements are provided that include an assembly of at least two panels that have at least three magnets located around the perimeter of the panel such that the dipole axes of magnets in a single panel are coplanar and intersect to define a polygon. When attaching two adjacent panels, at least two ferromagnetic spheres are used such that the dipole axes of one magnet from each of the adjacent panels are collinear, and such that the dipole axes are collinear with the centers of the ferromagnetic spheres. In this manner, several panels configured in this way may be nested together to form great varieties of constructions.

Memory-Constrained Algorithms for Simple Polygons

Abstract: A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of $O(\log n)$ bits, where $n$ is the size of the input. We assume that a simple $n$-gon is given by the ordered sequence of its vertices. We show that we can find a triangulation of a plane straight-line graph in $O(n^2)$ time. We also consider preprocessing a simple polygon for shortest path queries when the space constraint is relaxed to allow $s$ words of working space. After a preprocessing of $O(n^2)$ time, we are able to solve shortest path queries between any two points inside the polygon in $O(n^2/s)$ time.

Temperature interpolation based on local information: the example of France

Abstract: Methods of interpolation, whether based on regressions or on kriging, are global methods in which all the available data for a given study area are used. But the quality of results is affected when the study area is spatially very heterogeneous. To overcome this difficulty, a method of local interpolation is proposed and tested here with temperature in France. Starting from a set of weather stations spread across the country and digitized as 250 m-sided cells, the method consists in modelling local spatial variations in temperature by considering each point of the grid and the n weather stations that are its nearest neighbours. The procedure entails a series of steps: recognition of the n stations closest to the cell to be evaluated and subdivision of the study area into polygons defined by a neighbourhood rule, elaboration of a local model by multiple regression for each polygon, and application of the parameter estimate from the regression to obtain a predicted value of temperature at each point of the polygon under consideration. These results are compared with results from three global interpolation methods: (1) regression, (2) ordinary kriging, and (3) regression with kriging of residuals. We then develop the original results from local interpolation such as mapping of the coefficients of determination and of the parameter estimate related to altitude and to distance to the sea. These developments highlight the processes that dictate the spatial variation of climate

Work machine safety device

Abstract: Disclosed is a work machine safety device that instantly, easily and precisely makes an operator aware of current stability when carrying out work such as front assembly operations and rotating operations. In a work machine safety device, a control device is provided with: a ZMP calculation means (60f) that uses position information, acceleration information, and external force information for each moving part of a body and a traveling body, such as a front assembly, to calculate ZMP coordinates; and a stability computation means (60d) that calculates a support polygon formed by a plurality of ground contact points for the ground under the work machine, and generates a tilting warning when the ZMP is contained in the warning area formed inside of the periphery of the support polygon. Also provided is a display device (61d) that displays the top view of the work machine and the ZMP position of the work machine with respect to the support polygon. The ZMP calculation means and the stability computation means compute and display the support polygon, which includes the ZMP and the warning area, and generate a tilting warning when the computed ZMP position is contained in the warning area formed inside of the periphery of the support polygon.