Showing papers on "Polygon published in 2008"

Monitoring Environmental Boundaries With a Robotic Sensor Network

Abstract: In this brief, we propose and analyze an algorithm to monitor an environmental boundary with mobile agents. The objective is to optimally approximate the boundary with a polygon. The mobile sensors rely only on sensed local information to position some interpolation points and define an approximating polygon. We design an algorithm that distributes the vertices of the approximating polygon uniformly along the boundary. The notion of uniform placement relies on a metric inspired by approximation theory for convex bodies. The algorithm is provably convergent for static boundaries and efficient for slowly-moving boundaries because of certain input-to-state stability properties.

Methods for rapidly processing angular masks of next-generation galaxy surveys

Abstract: As galaxy surveys become larger and more complex, keeping track of the completeness, magnitude limit and other survey parameters as a function of direction on the sky becomes an increasingly challenging computational task. For example, typical angular masks of the Sloan Digital Sky Survey contain about N = 300 000 distinct spherical polygons. Managing masks with such large numbers of polygons becomes intractably slow, particularly for tasks that run in time O(N 2 ) with a naive algorithm, such as finding which polygons overlap each other. Here we present a 'divide-and-conquer' solution to this challenge: we first split the angular mask into pre-defined regions called 'pixels', such that each polygon is in only one pixel, and then perform further computations, such as checking for overlap, on the polygons within each pixel separately. This reduces O(N 2 ) tasks to 0(N), and also reduces the important task of determining in which polygon(s) a point on the sky lies from 0(N) to O(1), resulting in significant computational speedup. Additionally, we present a method to efficiently convert any angular mask to and from the popular HEALPIX format. This method can be generically applied to convert to and from any desired spherical pixelization. We have implemented these techniques in a new version of the MANGLE software package, which is freely available at http://space.mit.edu/home/tegmark/mangle/, along with complete documentation and example applications. These new methods should prove quite useful to the astronomical community, and since MANGLE is a generic tool for managing angular masks on a sphere, it has the potential to benefit terrestrial mapmaking applications as well.

Computer generated holograms from three dimensional meshes using an analytic light transport model

Abstract: We present a method to analytically compute the light distribution of triangles directly in frequency space. This allows for fast evaluation, shading, and propagation of light from 3D mesh objects using angular spectrum methods. The algorithm complexity is only dependent on the hologram resolution and the polygon count of the 3D model. In contrast to other polygon based computer generated holography methods we do not need to perform a Fourier transform per surface. The theory behind the approach is derived, and a suitable algorithm to compute a digital hologram from a general triangle mesh is presented. We review some first results rendered on a spatial-light-modulator-based display by our proof-of-concept software.

A geometric model for cluster categories of type Dn

Abstract: We give a geometric realization of cluster categories of type D n using a polygon with n vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.

The Pentagram map: a discrete integrable system

Abstract: The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons (a twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation). We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.

Size-constrained Region Merging (SCRM): An Automated Delineation Tool for Assisted Photointerpretation

Abstract: The manual delineation of vegetation patches or forest stands is a costly and crucial stage in any land-cover mapping project or forest inventory based upon photointerpretation. Recent computer techniques have eased the task of the interpreter; however, a good deal of craftsmanship is still required in the delineation. In an effort to contribute to the automation of this process, we introduce Size-Constrained Region Merging (SCRM), a recently implemented software tool that provides the interpreter with an initial template of the tobe-mapped area that may reduce the manual digitization portion of the interpretation. In essence, SCRM transforms an ortho-rectified aerial or satellite image (single or multichannel) into a polygon vector layer that resembles the work of a human interpreter, whom with no a priori knowledge of the scene, was given the task of partitioning the image into a number of homogeneous polygons all exceeding a minimum size. We provide background information on SCRM foundations and workflow, and illustrate its application on three different types of satellite images.

Satellite Constellation Design for Complex Coverage

Abstract: Most traditional satellite constellation design methods are associated with a simple zonal or global, continuous or discontinuous coverage connected with a visibility of points on the Earth’s surface. A new geometric approach for more complex coverage of a geographic region is proposed. Full and partial coverage of regions is considered. It implies that, at any time, the region is completely or partially within the instantaneous access area of a satellite of the constellation. The key idea of themethod is a two-dimensional space application formaps of the satellite constellation and coverage requirements. The space dimensions are right ascension of ascending node and argument of latitude. Visibility requirements of each region can be presented as a polygon and satellite constellation as a uniformmoving grid. At any time, at least one grid vertex must belong to the polygon. The optimal configuration of the satellite constellation corresponds to the maximum sparse grid. The method is suitable for continuous and discontinuous coverage. In the last case, a vertex belonging to the polygon should be examined with a revisit time. Examples of continuous coverage for a space communication network and of the United States are considered. Examples of discontinuous coverage are also presented.

Supramolecule-to-supramolecule transformations of coordination-driven self-assembled polygons.

Abstract: Two types of supramolecular transformations, wherein a self-assembled Pt(II)-pyridyl metal−organic polygon is controllably converted into an alternative polygon, have been achieved through the reaction between cobalt carbonyl and the acetylene moiety of a dipyridyl donor ligand. A [6 + 6] hexagon is transformed into two [3 + 3] hexagons, and a triangle−square mixture is converted into [2 + 2] rhomboids. 1H and 31P NMR spectra are used to track the transformation process and evaluate the yield of new self-assembled polygons. Such transformed species are identified by electrospray ionization (ESI) mass spectrometry. This new kind of supramolecule-to-supramolecule transformations provides a viable means for constructing, and then converting, new self-assembled polygons.

Floor decompositions of tropical curves : the planar case

Abstract: In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams. The focusing on dimension 2 is motivated by the special combinatoric of floor diagrams compared to arbitrary dimension. We treat a general toric surface case in this dimension: the curve is given by an arbitrary lattice polygon and include computation of Welschinger invariants with pairs of conjugate points. See also cite{FM} for combinatorial treatment of floor diagrams in the projective case.

VecGCA: a vector-based geographic cellular automata model allowing geometric transformations of objects

Abstract: Cellular automata (CA) can reproduce global patterns and behavior from local interactions of cells and they are used increasingly to simulate complex natural and human systems. Among their attributes are their computational simplicity and their explicit representation of space and time. However, the classic definition of CA limits their application to problems that involve a discrete space, and similar rules and neighborhoods for all cells. In addition, the standard raster-based CA model is sensitive to spatial scale. This paper presents a new vector-based geographic cellular automata model, called the VecGCA model, which defines space as a collection of irregular geographic objects. Each object has a geometric representation (a polygon) that evolves through time according to a transition function that depends on the influence of neighboring polygons. In this model, the neighborhood is defined as the region of influence on each geographic object, and the neighbors are all geographic objects located within...

Caging Polygons with Two and Three Fingers

Abstract: We present algorithms for computing all placements of two and three fingers that cage a given polygonal object with n edges in the plane. A polygon is caged when it is impossible to take the polygon to infinity without penetrating one of the fingers. Using a classification into squeezing and stretching cagings, we provide an algorithm that reports all caging placements of two disc fingers in O(n 2logn) time. Our result extends and improves a recent solution for point fingers. In addition, we construct a data structure requiring O(n 2) storage that can answer in O(logn) whether two fingers in a query placement cage the polygon. We also study caging with three point fingers. Given the placements of two so-called base fingers, we report all placements of the third finger so that the three fingers jointly cage the polygon. Using the fact that the boundary of the set of placements for the third finger consists of equilibrium grasps, we present an algorithm that reports all placements of the third finger in O(n 6log2 n) deterministic time and O(n 6logn(loglogn)3) expected time. Our results extend previous solutions that only apply to convex polygons.

Polygon Characterization With the Multiplicatively Weighted Voronoi Diagram

Abstract: Many landscape features are represented as polygons in GIS This paper characterizes polygon shapes with the multiplicatively weighted Voronoi (MW-Voronoi) diagram and improves its understanding The MW-Voronoi diagram's composition is implemented with topological overlay, growth simulation, and vertex calculation methods The decomposition is done by reversing a polygon to MW-Voronoi point pairs by segment It is a new approach to record, characterize, and compare polygons with form and process The implementation also serves as a geographic education and visualization tool Applications of the methods are presented with precipitation, fire polygon, and population change data

Simple Robots with Minimal Sensing: From Local Visibility to Global Geometry

Abstract: We consider problems of geometric exploration and self-deployment for simple robots that can only sense the combinatorial (non-metric) features of their surroundings. Even with such a limited sensing, we show that robots can achieve complex geometric reasoning and perform many non-trivial tasks. Specifically, we show that one robot equipped with a single pebble can decide whether the workspace environment is a simply connected polygon and, if not, it can also count the number of holes in the environment. Highlighting the subtleties of our sensing model, we show that a robot can decide whether the environment is a convex polygon, yet it cannot resolve whether a given vertex is convex. Finally, we show that by using such local and minimal sensing a robot can compute a proper triangulation of a polygon and that the triangulation algorithm can be implemented collaboratively by a group of m such robots, each with Θ(n/m) word memory. As a corollary of the triangulation algorithm, we derive a distributed analog of the well-known Art Gallery Theorem. A group of [n/3] (bounded memory) robots in our minimal sensing model can self-deploy to achieve visibility coverage of an n-vertex art gallery (polygon). This resolves an open question raised recently.

Comparison of Triple Patterning Decomposition Algorithms Using Aperiodic Tiling Patterns

Abstract: Double patterning has gained prominence as the most likely methodology to help keep Moore's law going towards 22nm 1/2 pitch lithography. However, most designs cannot be blindly shrunk to run using only two patterning layers and a variety of constraints must be imposed on designs to allow for correct decomposition. These constraints are more onerous for the contact layer than for line/space patterns because they more easily form odd cycles on the 2D plane, which cannot be broken using polygon cutting. As this can adversely limit packing density, especially in bit cells, a triple patterning decomposition capability could be attractive for the contact layer. Pattern decomposition for contacts can be likened to coloring a map where minimum spaces between contacts are replaced with borders. It is well known that 4 colors can color any map, but it is an NP-complete problem to compute the minimum number of colors needed to color any given map. This should place an upper limit on the scalability of any algorithm able to color large networks. A variety of test patterns that are known 3-colorable are needed to compare suitable algorithms. It has been proved that a set of aperiodic tiling known as "Penrose Tiles" is 3-colorable. This paper compares the scalability of different coloring algorithms using a variety of contact patterns based on Penrose Tiles.

Toric Surface Codes and Minkowski Length of Polygons

Abstract: In this paper we prove new lower bounds for the minimum distance of a toric surface code $\mathcal{C}_P$ defined by a convex lattice polygon $P\subset\mathbb{R}^2$. The bounds involve a geometric invariant $L(P)$, called the full Minkowski length of $P$. We also show how to compute $L(P)$ in polynomial time in the number of lattice points in $P$.

A Cluster Expansion Formula ($A_n$ case)

Abstract: We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\Sigma$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $\Sigma$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\Sigma$.

Geodesic Fréchet Distance Inside a Simple Polygon.

Abstract: We unveil an alluring alternative to parametric search that applies to both the non-geodesic and geodesic Fr{'\e}chet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We present the first algorithm for the geodesic Fr{'\e}chet distance between two polygonal curves $A$ and $B$ inside a simple bounding polygon $P$. The geodesic Fr{'\e}chet decision problem is solved almost as fast as its non-geodesic sibling and requires $O(N^{2log k)$ time and $O(k+N)$ space after $O(k)$ preprocessing, where $N$ is the larger of the complexities of $A$ and $B$ and $k$ is the complexity of $P$. The geodesic Fr{'\e}chet optimization problem is solved by a randomized approach in $O(k+N^{2log kNlog N)$ expected time and $O(k+N^{2)$ space. This runtime is only a logarithmic factor larger than the standard non-geodesic Fr{'\e}chet algorithm (Alt and Godau 1995). Results are also presented for the geodesic Fr{'\e}chet distance in a polygonal domain with obstacles and the geodesic Hausdorff distance for sets of points or sets of line segments inside a simple polygon $P$.

A Sharpness-Dependent Filter for Recovering Sharp Features in Repaired 3D Mesh Models

Abstract: This paper presents a sharpness-based method for hole-filling that can repair a 3D model such that its shape conforms to that of the original model. The method involves two processes: interpolation-based hole-filling, which produces an initial repaired model, and postprocessing, which adjusts the shape of the initial repaired model to conform to that of the original model. In the interpolation-based hole-filling process, a surface interpolation algorithm based on the radial basis function creates a smooth implicit surface that fills the hole. Then, a regularized marching tetrahedral algorithm is used to triangulate the implicit surface. Finally, a stitching and regulating strategy is applied to the surface patch and its neighboring boundary polygon meshes to produce an initial repaired mesh model, which is a regular mesh model suitable for postprocessing. During postprocessing, a sharpness-dependent filtering algorithm is applied to the initial repaired model. This is an iterative procedure whereby each iteration step adjusts the face normal associated with each meshed polygon to recover the sharp features hidden in the repaired model. The experiment results demonstrate that the method is effective in repairing incomplete 3D mesh models.

The role of thermal contraction crack polygons in cold-desert fluvial systems

Abstract: Thermal 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.

A heuristic for nesting problems of irregular shapes

Abstract: Layout has a close relationship with product cost in the vein of how to most efficiently cut product patterns from raw materials. This is the so-called ''nesting problem'', which occurs frequently in sheet metal and furniture industries, wherein material utilization needs to be maximized. In this paper, a quick location and movement (QLM) algorithm is proposed to solve the situation of irregular shapes nested on multiple irregular sheets. This approach includes two major parts: it first approximates irregular shapes to a polygon with the use of a cluster of straight lines, and second, it arranges the approximated shapes one-by-one with the proposed step-by-step rule. Finally, this study investigates and compares examples presented by other authors. The results show that the QLM algorithm takes less time to calculate a layout and the material utilization efficiency is higher compared to other methods.

An Algorithm for Computing the Minimum Area Bounding Rectangle of an Arbitrary Polygon

Abstract: The minimum area bounding rectangle(MABR) of an arbitrary polygon is an important tool in the geographic information systems and computer graphics,but how to precisely calculate it is of great difficulty.Firstly,the MABR of an arbitrary polygon shares a common edge with the convex hull of the polygon is proved.Secondly,an algorithm for computing MABR based on this idea is presented and the efficiency of the algorithm is discussed.Finally,some experiments are given to show the feasibility and reliability of the algorithm.

High Performance LED Package

Abstract: A light emitting diode lamp is disclosed that includes a resin package that defines a recess in the shape of a solid polygon or another three-dimensional solid. The recess includes a floor, two side walls along the respective longer sides of the floor, and two end walls along the respective shorter sides of the floor. The two side walls define an angle therebetween greater than 3°, and the two end walls define an angle therebetween greater than 40°. A light emitting diode chip is positioned on the rectangular floor of the package.

Invariant polygons in systems with grazing-sliding.

Abstract: The paper investigates generic three-dimensional nonsmooth systems with a periodic orbit near grazing-sliding. We assume that the periodic orbit is unstable with complex multipliers so that two dominant frequencies are present in the system. Because grazing-sliding induces a dimension loss and the instability drives every trajectory into sliding, the system has an attractor that consists of forward sliding orbits. We analyze this attractor in a suitably chosen Poincare section using a three-parameter generalized map that can be viewed as a normal form. We show that in this normal form the attractor must be contained in a finite number of lines that intersect in the vertices of a polygon. However the attractor is typically larger than the associated polygon. We classify the number of lines involved in forming the attractor as a function of the parameters. Furthermore, for fixed values of parameters we investigate the one-dimensional dynamics on the attractor.

Detection of skewed symmetry.

Abstract: This study examined the ability of human observers to discriminate between symmetric and asymmetric planar figures from perspective and orthographic images. The first experiment showed that the discrimination is reliable in the case of polygons, but not dotted patterns. The second experiment showed that the discrimination is facilitated when the projected symmetry axis or projected symmetry lines are known to the subject. A control experiment showed that the discrimination is more reliable with orthographic, than with perspective images. Based on these results, we formulated a computational model of symmetry detection. The model measures the asymmetry of the presented polygon based on its single orthographic or perspective image. Performance of the model is similar to that of the subjects.