Set Of Triangles Research Articles

АНАЛИЗ КОНСТРУКЦИЙ И РАБОЧИХ ПРОЦЕССОВ ТЕХНИЧЕСКИХ СРЕДСТВ ДЛЯ ПРОФИЛАКТИКИ И ТУШЕНИЯ ЛЕСНЫХ ПОЖАРОВ ГРУНТОМ

The analysis of designs of domestic and foreign ground-throwing machines and conducted research on extinguishing a forest grass-roots fire by throwing soil. It is established that for carrying out high-quality preventive and forest fire works, it is advisable to use technical means that combine active and passive working bodies in their design. This will allow combining the process of pre-treatment of soil and further release of loosened soil in a given direction, as well as significantly increase the efficiency of the technological process of creating mineralized strips. Based on the mathematical model, the working surfaces of the soil-laying machine are represented as a set of elementary triangles with vertex coordinates and the nature of soil movement is studied. The main part of the flying soil (about 70 %) settles in the band from 1.5 to 3.0 m from the border of the machine. However, part of the ground gets a high initial velocity and such an angle of release that it reaches a distance of 20 to 40 m. To study the movement of the elements of the ground-thrower, equations of motion of the mechanical system in the form of Lagrange equations of the second kind are compiled. It was found that when the values of the depth of the casing-Ripper are from 0.25 to 0.3 m, the best indicators of the range of soil discharge and productivity are observed (20 m and 0.07 m3 / s, respectively), and the power consumption is in the permissible range (17 kW). It is concluded that the ground-throwing equipment should be considered only as part of a combined machine that will allow you to remove the forest floor from the path of the unit, preventing its release into the fire zone, and protecting the working bodies of the unit from the root system of trees. The highly efficient design of the unit with technical solutions is proposed, which allows performing work on stopping and localizing fire.

CALCULATED EVALUATION OF THE DEGREE OF SUBSOLIDUS STRUCTURE COMPLEXITY OF THE PHYSICO-CHEMICAL THREE-COMPONENT SYSTEMS

The study of state diagrams of multicomponent physicochemical systems is the most science-intensive task of materials science. Without knowledge about the structure of the state diagrams of such systems, it is impossible for technologists to predict the phase composition of materials during their production and use of products from them; the systematic analysis of the results of experimental studies to optimize the properties of the materials being developed becomes much more complicated. In refractory technology, the defining stage of production is solid-phase sintering, which makes the information on the subsolidus structure of state diagrams of physicochemical systems represented by a set of components in accordance with the planned phase composition of materials particularly important. Three-component systems, the simple components of which are refractory oxides, constitute the physicochemical basis of most mass-produced refractories and their subsolidus structure is quite clearly displayed in the concentration triangle of the system by a set of triangles, the vertices of which are the points of the composition of the compounds. The study is devoted to the establishment of an analytical relationship between the number of double and triple compounds and the number of all possible segments of connecting lines between the points of the composition of the connections, as well as the points of intersection of the compounds between themselves. During the research, the general principles of systems analysis, logical methods and terminology of physicochemical analysis of multicomponent systems, as well as information on elementary mathematics from the sections on numerical series, the basics of combinatorics and algebra were used. The corresponding analytical expressions make it possible to calculate the quantitative classification characteristics in the taxonomy of multicomponent systems by the degree of complexity of the structure of their subsolidus regions, in particular, when comparing the complexity of studies of three-component oxide systems and their typification. The formulas obtained were tested for calculations using examples of specific oxide systems. The research results allow one to obtain important quantitative characteristics for assessing the degree of complexity of the subsolidus structure of three-component systems.

Fast and robust mesh arrangements using floating-point arithmetic

We introduce a novel algorithm to transform any generic set of triangles in 3D space into a well-formed simplicial complex. Intersecting elements in the input are correctly identified, subdivided, and connected to arrange a valid configuration, leading to a topologically sound partition of the space into piece-wise linear cells. Our approach does not require the exact coordinates of intersection points to calculate the resulting complex. We represent any intersection point as an unevaluated combination of input vertices. We then extend the recently introduced concept of indirect predicates [Attene 2020] to define all the necessary geometric tests that, by construction, are both exact and efficient since they fully exploit the floating-point hardware. This design makes our method robust and guaranteed correct, while being virtually as fast as non-robust floating-point based implementations. Compared with existing robust methods, our algorithm offers a number of advantages: it is much faster, has a better memory layout, scales well on extremely challenging models, and allows fully exploiting modern multi-core hardware with a parallel implementation. We thoroughly tested our method on thousands of meshes, concluding that it consistently outperforms prior art. We also demonstrate its usefulness in various applications, such as computing efficient mesh booleans, Minkowski sums, and volume meshes.

On the triangles in certain types of line arrangements

In this paper, we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been described for both the types and some properties of the nomenclature have been proved. Later, using the nomenclature, we describe the triangles present in both types of line arrangements in main Theorems A and B. We also prove that the set of triangles uniquely determines, in a certain precise sense, the line arrangements with global cyclicity and not the infinity type line arrangements, where counter examples have been provided. In Theorem 9.1, given a nomenclature, we characterize when a particular line symbol in the nomenclature is a line at infinity for the arrangement determined by the nomenclature.

Ensemble size judgments account for size constancy.

The natural environment is full of redundant information that the visual system compresses into an ensemble representation by averaging features of groups of items. Ensemble perception has been shown to operate with remarkable flexibility, efficiently integrating information across a variety of visual domains. In the current set of experiments, we tested whether average size representations reflect the physical size of objects displayed on a screen or perceptual transformations due to size constancy. We induced a perceptual change by presenting sets of triangles with linear perspective cues - lines converging at the horizon. Assuming a constant size, these cues cause individual objects "in the distance" to appear larger than objects without distance cues, due to size constancy heuristics. Observers viewed sets of triangles with and without linear perspective cues and judged whether a subsequently presented test triangle was larger or smaller than the average size of the preceding set. Results revealed ensemble size representations took size constancy into account, reflecting the perceived size of the triangles rather than their absolute size. Interestingly, the amount of bias exhibited was well characterized by the summed bias associated with each of the three triangles presented individually. Other pictorial cues to depth, such as occlusion and height-in-field, did not elicit the same bias when those were the only depth cues available. Overall, our results complement and extend other work showing that average size reflects the perceptual size of individual items in a set.

Smooth reconstruction method for surfaces defined by a continuous set of triangles

The paper considers the problem of triangular grids distortion in three-dimensional space, approximating the boundaries and interfaces of interacting solid, liquid and gaseous media. Unacceptable deformations of the mesh cells size and shape can occur during the numerical solution of continuous media dynamics problems. A method and algorithm for rebuilding grids using smooth piecewise polynomial interpolation by algebraic polynomials of the third degree is proposed.

Large triangle packings and Tuza’s conjecture in sparse random graphs

AbstractThe triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

Affine-invariant querying of spatial data using a triangle-based logic

If the same land area is photographed from two different angles, then a computer can recognize the two images to show the same land area by proving that the images are affine-invariant. In practice, images are abstracted as a type of spatial data that is a finite set of triangles. Hence we propose a simple affine-generic query language with variables over finite sets of triangles. The proposed query language is intuitive and has the same expressive power as the affine-generic fragment of first-order logic over the reals on triangle databases.

Tile‐based polygon overlay of vector data in a virtual globe

AbstractThe demand for analysis of large‐scale data has increased with increased access to spatial vector data. Polygon overlay of vector data in a virtual globe requires proprietary data structures and proprietary analysis algorithms. A vector data structure is designed for rapid polygon overlay in a virtual globe by recording the metadata of the triangles (TriMeta) that constitute polygons. A polygon overlay algorithm is proposed based on this data structure. The overlay of two complex GIS polygons is transformed into the intersection computation of their 3D triangular presentations. The intersection computation of two sets of triangles is reduced considerably by utilizing TriMeta to filter out the disjoint triangle pairs and rapidly identify the contained triangle. The new method improves the overlay efficiency in a virtual globe because the amount of computation required to calculate the intersections of two large polygons and drape the intersections onto a terrain surface is reduced.

Anticomplementary Triangles for Efficient Coverage in Sensor Network-Based IoT

The coverage-lifetime tradeoff is a perennial research problem in sensor-based internet of things framework. This article proposes a centralized scheme to achieve energy efficient maximum area COverage in Sensor network Based IoT (COSBIoT) for detection of events in dense and randomly deployed network. The sensor nodes are scheduled to function in sleep–awake cycle such that network is virtually a set of contiguous anticomplementary triangles (ACT) of corresponding Miquel triangles (MT). The node activation is accomplished by randomly choosing the first Miquel point as reference point to form anticomplementary triangles. Extensive experiments and simulations have been performed to test the efficacy of the scheme and results have been shown in different possible scenarios. Comparison with other schemes shows that the proposed scheme outperforms the existing schemes like, MPSO, BBO, VF-BBO, and MIA in terms of area coverage of region of interest. Similarly, in terms of network lifetime and energy efficiency, an average performance increase is achieved compared to MCLCT, OCCH, CWGC, Greedy CSC, and GIECC.

Novel Copy Move Forgery Detection Based on Repeated Feature Extraction and Delaunay Triangulation

Nowadays new and creative methods of forging images are developed with the invention of sophisticated softwares like Adobe photoshop. Tools available in such softwares will make the forged image look real which cannot be even identified by a naked eye. In this paper, key point based approach of taking out features using Scale Invariant Feature Transform (SIFT) is used. The feature points thus extracted are then modeled to get a set of triangles using Delaunay Triangulation method. These triangles are matched using mean vertex descriptor and the removal of false positives is done using the method of Random Sample Consensus (RANSAC). Implementation show that the proposed approach outdoes the equivalent methods

Natural element method for axisymmetric elastoplasticproblems

Based on the meshless natural element method and incremental theory of plasticity, a new algorithm for elastoplastic analysis of a symmetric structure is put forward. The meshless method is a recently developed numerical method, which only requires the nodal information and can effectively overcome the difficulties brought about by the meshes. As widely known, conventional meshfree methods suffer from the numerical difficulties in dealing with the essential boundary conditions. Differing from most of meshfree methods, the natural element method (NEM) is a Galerkin-based meshless method that is built upon the notion of the natural neighbor interpolation. It is noteworthy that, there are two natural neighbor interpolants: Natural neighbor-based Sibson interpolation and Laplace interpolation (non-Sibsonian interpolation). The interpolation relies on the concepts of Delaunay triangulations and Dirichlet tessellations of a set of nodes to build the shape function. Compared with the moving least squares approximation widely used in many meshless methods, the natural neighbor interpolation does not involve complex matrix inversion, needs no artificial parameter but can improve the computational efficiency. Because the constructed shape functions from the natural neighbor interpolation possess the Kronecker delta function property, the essential boundary conditions can be imposed accurately without any special techniques. Meanwhile, the NEM does not require extra efforts to generate the background because it utilizes a set of Delaunay triangles, which are automatically identified in process of the basis function definition, as its background cell. Therefore, the NEM is a most promising numerical method which can be widely applied to nonlinear analysis of materials, three-dimensional calculation and large deformation problems. Generally, plasticity would make many projects more reasonable, and would fully utilize intensity potentially of structures. The NEM has been successfully used to solve elastoplastic problems of plane structures. In the present work, the NEM is applied to elastoplastic analysis of axisymmetric structures. Firstly, a brief description of the rudiments of Voronoi diagrams and Delaunay triangles in the context of natural neighbor interpolation is delineated. The basic principle of Sibson natural neighbor interpolation approach is also presented. A three-dimensional axisymmetric structure is created by the rotation of the cross section around the axis of symmetry and a set of properly scattered nodes within the cross section are employed. In the analysis of elastoplastic problems, the increments of stress and strain for axisymmetric problems are utilized to characterize the elastoplastic constitutive relationship. Owing to the nonlinear nature of plastic deformation, the computation at each load step employs an iterative solution method, either the Newton-Raphson method or the modified Newton-Raphson method. In order to reduce the expensive operations of stiffness matrix evaluation and factorization, the modified Newton-Raphson method with initial elastic stiffness matrix is adopted in the paper. The tangent predictor-radial return algorithm is applied to integrate the constitutive equations. The elastoplastic analyses of two classical axisymmetric structures have been successfully carried out and the fact that the present results show a good agreement with the analytical solutions demonstrate the effectiveness and accuracy of the proposed method for axisymmetric elastoplastic analyses.

МЕТОД РАСПАДА РАЗРЫВОВ В ТРЕХМЕРНОЙ ДИНАМИКЕ УПРУГОПЛАСТИЧЕСКИХ СРЕД

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies with large displacements and deformations based on the method of disintegration of discontinuities according to the Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The Riemann's solver for an elastic medium in the case of an arbitrary stress state are obtained and presented. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact template for moving Eulerian – Lagrangian grids. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can vary. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian – Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing to determine the parameters on the boundary and contact surfaces. The values of the underdetermined parameters near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions and with the experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

PERFECT TRIANGLES ON THE CURVE

A Heron triangle is a triangle that has three rational sides $(a,b,c)$ and a rational area, whereas a perfect triangle is a Heron triangle that has three rational medians $(k,l,m)$. Finding a perfect triangle was stated as an open problem by Richard Guy [Unsolved Problems in Number Theory (Springer, New York, 1981)]. Heron triangles with two rational medians are parametrized by the eight curves $C_{1},\ldots ,C_{8}$ mentioned in Buchholz and Rathbun [‘An infinite set of heron triangles with two rational medians’, Amer. Math. Monthly 104(2) (1997), 106–115; ‘Heron triangles and elliptic curves’, Bull. Aust. Math.Soc. 58 (1998), 411–421] and Bácskái et al. [Symmetries of triangles with two rational medians, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.6533, 2003]. In this paper, we reveal results on the curve $C_{4}$ which has the property of satisfying conditions such that six of seven parameters given by three sides, two medians and area are rational. Our aim is to perform an extensive search to prove the nonexistence of a perfect triangle arising from this curve.

Reversible data hiding using B‐tree triangular decomposition based prediction

A novel reversible data hiding (RDH) technique has been proposed in this study using a B-tree triangular decomposition-based prediction of image pixels. The superiority of a prediction error expansion-based RDH scheme depends on a good prediction strategy for image pixels. In the proposed scheme, the B-tree triangular decomposition is used to recursively decompose the image into a set of right-angled triangles. The vertices of such triangles serve as the reference pixels to predict (interpolate) the intensity values of other non-reference pixels within the triangle. Data bits are reversibly embedded into these non-reference pixels by expanding the prediction error. Moreover, according to the proposed scheme, the number of bits being embedded in these pixels is varied (either one or two bits) based on an estimated local complexity of the triangle. The local complexity of a triangle is computed from the intensity of vertices of the triangle. The superior performance of the proposed method is verified through extensive experiments.

On Tuza's Conjecture for Triangulations and Graphs with Small Treewidth

Tuza (1981) conjectured that the cardinality τ(G) of a minimum set of edges that intersects every triangle of a graph G is at most twice the cardinality ν(G) of a maximum set of edge-disjoint triangles of G. I this paper we present three results regarding Tuza's Conjecture. We verify it for graphs with treewidth at most 6; and we show that τ(G)≤32ν(G) for every planar triangulation G different from K4; and that τ(G)≤95ν(G)+15 if G is a maximal graph with treewidth 3.

PatCC1: an efficient parallel triangulation algorithm for spherical and planar grids with commonality and parallel consistency

Abstract. Graphs are commonly gridded by triangulation, i.e., the generation of a set of triangles for the points of the graph. This technique can also be used in a coupler to improve the commonality of data interpolation between different horizontal model grids. This paper proposes a new parallel triangulation algorithm, PatCC1 (PArallel Triangulation algorithm with Commonality and parallel Consistency, version 1), for spherical and planar grids. Experimental evaluation results demonstrate the efficient parallelization of PatCC1 using a hybrid of MPI (message passing interface) and OpenMP (Open Multi-Processing). They also show PatCC1 to have greater commonality than existing parallel triangulation algorithms (i.e., it is capable of handling more types of model grids) and that it guarantees parallel consistency (i.e., it achieves exactly the same triangulation result under different parallel settings).

Improper sum-list colouring of 2-trees

Let G be a graph and P be a class of graphs. We consider list colouring of vertices of G in which the sizes of lists assigned to different vertices can be different. We colour G from the lists in such a way that vertices of each colour class induce a graph in P. The smallest possible sum of all the list sizes, such that, according to the rules, G is colourable for any particular assignment of the lists of these sizes is denoted by χscP(G).Let D1, I1 be classes of acyclic and triangle-free graphs, respectively. Given a graph G, we consider a graph HG whose vertex set is the set of triangles of G and whose two vertices t1,t2 are adjacent if the triangles t1, t2 share an edge in G.In this work we provide an upper bound of n+α(HG) on χscD1(G), where G is an arbitrary n-vertex 2-tree and α(HG) is the independence number of HG. We give a polynomial-time algorithm to find the assignment of list sizes that realizes this bound. Moreover, we prove that this bound is achieved for an arbitrary n-vertex 2-tree G that is 3-sun-free (equivalently, Hajoś-graph-free, see Fig. 7) and for such an n-vertex 2-tree G whose graph HG is a tree with α(HG)=⌈n2⌉−1. Finally, we show that χscI1(G)=χscD1(G) for each 2-tree G.

KTMiner: Distributed k-truss detection in big graphs

Discovering cohesive subgraphs is an important issue in the analysis of massive graphs. A new type of cohesive subgraphs, k-truss, has gained a lot of attentions in recent years. Although different methods have been proposed to extract k-truss subgraphs, they are unable to deal with big graphs, and they suffer from the lack of efficiency. To deal with the problem of finding k-truss subgraphs in big graphs, we propose a novel efficient distributed algorithm, called KTMiner, based on the MapReduce paradigm and the key–value structure. KTMiner is deployed on the Spark platform, a big data framework. Given a specific k value, KTMiner finds edges belonging to the k-truss subgraphs. It includes three consequent phases. First, a novel distributed k-core routine is employed to prune unnecessary vertices from the input graph. Then, the triangle information per edge in the form of a new data structure, called Triangle Set (TSet), is produced. Finally, using a distributed iterative procedure, the desired k-truss subgraphs are detected. KTMiner perfectly caches the reusable data in the distributed memory. In addition, it forms an efficient load balancing mechanism by designing appropriate data structures. This results in a fine-grained parallelism. The superiority of our solution over the state-of-the-art methods is shown by the experiments on real-world graphs.

Geometrically Non-linear Analysis of Elastic Structures by Petrov-Galerkin Natural Element Method

This paper is concerned with the geometrically non-linear analysis of elastic structures using the natural element method, particularly a Petrov-Galerkin Natural Element Method (PG-NEM). The geometrically non-linear problem is linearized and formulated by the Taylor series expansion and the Total Lagrangian (TL) method. The Voronoi polygon-based Laplace Interpolation (LI) functions are used to approximate the displacement increments, while the Delaunay triangle-supported Constant-Strain (CS) basis functions are adopted for the expansion of the virtual displacement. The iterative computation is performed by the Newton-Raphson method, and a set of Delaunay triangles serves as a background cell for the traditional Gauss integration rule-based numerical integration. The proposed PG-NEM is illustrated through the numerical experiments, and its numerical accuracy is compared with MSC/Marc, CS-FEM and BG-NEM.