Polygonal Research Articles

FOLDING MOTION IN THAT SHRINKS A GOOD POLYGONAL ANNULUS TO ARBITRARY SMALL NEIGHBORHOOD OF THE INNER BOUNDARY

In 2004, Demaine et al. showed that if is a simply connected polygonal region in the plane, then for any piecewise- folded state of , there exists a folding motion from to [1]. We have shown that if is not simply connected, then the conclusion of the theorem does not necessarily hold, in fact, there exists an annulus in such that there exists a piecewise- folded state of which does not admit folding motions from by using "knot and link theory". From the viewpoint of knot and link theory, it is reasonable to expect that the folded state admits a folding motion from if and only if forms a trivial link in (Conjecture). Note that the 'only if part of the Conjecture is obvious. In this paper, we give a proof of the following statement that will be used in an expected proof of the 'if part of the Conjecture for special kind of annulus . If is an annulus such that each of the boundary components bounds a polygonal convex region, then for any , there exists a flat folded state such that is contained in the -neighborhood of the inner boundary of in , denoted by with a strongly flat folding motion from to which satisfies the following: (1) For any , the number of 1 -simplices of the 1 -complex is finite, where denotes the canonical crease pattern of . (2) For any is a good polygonal annulus. (3) For any with , we have .

Associations between sense of place and the geometric shape complexity of corresponding self-reported spatial footprints from Lisbon, Portugal

ABSTRACT This article investigates how people translate the complex concept of sense of place into reductionist geospatial footprints. Utilising a map-supported survey conducted in Lisbon, statistical shape complexity measures, and logistic regression, we show that the number of mapped polygonal footprint vertices correlates with the sense-of-place dimensions of place identity and place attachment. Furthermore, we show that places to which people connect particularly close lead to more complex forms of shape complexity beyond the number of mapped points. Our findings contribute to the discourse on place-based information and to an improved understanding of the nexus between place and its geometric representation.

Bicriteria Shapes: Hierarchical Grouping and Aggregation of Polygons with an Efficient Graph-Cut Approach

An important task of pattern recognition and map generalization is to partition a set of disjoint polygons into groups and to aggregate the polygons within each group to a representative output polygon. We introduce a new method for this task called bicriteria shapes. Following a classical approach, we define the output polygons by merging the input polygons with a set of triangles that we select from a conforming Delaunay triangulation of the input polygons’ exterior. The innovation is that we control the selection of triangles with a bicriteria optimization model that is efficiently solved via graph cuts. In particular, we minimize a weighted sum that combines the total area of the output polygons and their total perimeter. In a basic problem, we ask for a single solution that is optimal for a preset parameter value. In a second problem, we ask for a set containing an optimal solution for every possible value of the parameter. We discuss how this set can be approximated with few solutions and show that it is hierarchically nested. Hence, the output is a hierarchical clustering that corresponds to multiple levels of detail. An evaluation with building footprints as input and a comparison with α -shapes that are based on the same underlying triangulation conclude the article. An advantage of bicriteria shapes compared to α -shapes is that the sequence of solutions for decreasing values of the parameter is monotone with respect to the total perimeter of the output polygons, resulting in a monotonically decreasing visual complexity.

Precipitation Behavior of Secondary γ′ Phase of Nickel-based Superalloy during Gradient Cooling Process

Abstract Nickel-based superalloy is an indispensable strategic material in aerospace, national defense, and other fields. Heat treatment is an important process for preparing nickel-based superalloys because of its great influence on the microstructure and mechanical properties. In this paper, the end-quench test and numerical simulation were applied to study the relationship between the cooling rate and precipitation behavior of the secondary γ′ phase of the nickel-based superalloy efficiently. The in situ end-quench test system was established to reduce temperature error caused by heat loss during specimen movement, providing an accurate initial temperature distribution for quenching. Besides, the process including the induction heating and the gradient cooling was simulated to obtain the cooling rate at any position of the specimen. The results show that in the cooling rate range of 51~1574 K /min, when the cooling rate decreases, the morphology of the secondary γ′ phase changes from spherical to nearly square or irregular polygon, and the size increases from 27.8nm to 161.4nm. Furthermore, a double logarithmic correlation was found between the average cooling rate and the size of the secondary γ′ phase, which had great meaning in designing and optimizing the heat treatment process of the nickel-based superalloy.

Exposure of Proso millet starch to superheated steam: Effect on physicochemical, techno-functional, rheological behavior, digestibility, and related mechanism.

In the present study, proso millet starch (PMS) was treated with SHS (120-160°C for 1-5min) to investigate the molecular interactions and techno-functional, rheological, and digestible properties. Exposure to SHS induced the degradation of helical structure, and longer chains, reducing amylose, blue value, optical density, and relative crystallinity. Meanwhile, SHS treatment might have introduced hydro‑carbonyl groups, eventually increasing water absorption capacity, swelling power, and transparency. As per SEM images, SHS-treated PMS had rough and irregular polygon surfaces with small pinholes. Compared to control, SHS treatment slightly improved the elastic nature of PMS samples. Furthermore, SHS treatment at 140 and 160°C for 3 and 5min increased the slowly digestible and resistant starch content. Multivariate analysis suggests that SHS treatment could be performed at 140°C for 3min to modify the PMS.

다각형의 둘레와 넓이를 활용한 공변 지도

While there are many studies on elementary school students' functional thinking, there is a lack of research that focuses on the core concept of covariation. The purpose of this study is to present a case study of teaching covariation to fifth-grade students using perimeter and area of polygons. Three lessons were conducted in the first semester of fifth grade, following the study of rules and correspondences in Unit 3 and the perimeter and area of polygons in Unit 6. The first lesson was about exploring and expressing changes in perimeters as the length of a side of a square changes, and the other lessons were about exploring and expressing changes in area of a polygon. In these lessons, we provided tasks that naturally built on perimeter and area to see if elementary students could understand covariations and variables. As a result, students were able to understand the situations in which two quantities change simultaneously and to represent the change of two quantities in various ways. Students also used the given function table to reason about the relationship between two quantities, making it easier to perceive and represent relationships between expressions. Based on these findings, this study discussed implications for the teaching of covariation to elementary school students.

Visual Geometric Thinking — an investigation in a short course/worksh

The research, of a qualitative nature, sought to answer the following question: how can participants in a didactic workshop develop geometric visual thinking based on activities with folds on an A4 sheet involving fundamental elements of euclidean geometry? To this end, the general objective was to analyze the geometric visual geometric thinking of participants in a didactic workshop when carrying out activities with folds on a sheet of paper and identifying geometric elements contained in these folds. The results showed a limitation in the visualization of straight lines, half-lines, straight segments, reflections, polygons and polygonal regions, which characterizes a lack of developing visual geometric thinking in the teacher of training of mathematics teachers.

Study of the residual stress distribution in SiCp/Al composites under thermal cycling conditions

In this study, a three-dimensional random representative volume element model was constructed using the finite element method combined with Python script to investigate the residual stress distribution in SiCp/Al composites with varying volume fractions, particle shapes, and particle size under thermal cycling conditions. The model has two different particle morphologies: spheres and irregular polygons. The results show that the irregular polygonal SiC particles exhibit higher residual stresses than spherical particles before and after thermal cycling due to their complex load transfer pathways and higher stress concentration at particle corners, which makes them more consistent with the characterization of SiCp/Al composites under practical application conditions. As the volume fraction of SiC particles increases, the particle spacing decreases, hindering the release of thermal stresses through plastic deformation. Notably, the relief rate of residual stress reached 53% in the 12 vol% SiCp/Al composites, with the stress decreasing from 698 MPa to 328 MPa after 50 thermal cycles. In contrast, the reduction rate for 18 vol% composites is only 13% after thermal cycling. Furthermore, compared to larger particles (7.3 µm, 8.9 µm), the smaller particles (4.3 µm, 5.9 µm) exhibit a more uniform stress distribution and lower thermal stresses before and after thermal cycling. Consequently, our work provides a significant theoretical basis for the design and application of SiCp/Al composites, especially for the application of materials under extreme temperature changes.

Gaussian Mixture Model Based Voronoi Partitioning Scheme in Multi-Beam Satellites

With advances in technology and growing user needs, satellite communication is changing over time. Efficiently managing resources across diverse satellite coverage areas can be significantly enhanced through the application of Artificial Intelligence (AI) and Machine Learning (ML). The intended project seeks to employ fundamental machine learning algorithms, including Weighted k-means and Gaussian Mixture Model (GMM), to cluster user demands within a designated geographical region. This approach aims to optimize the utilization of satellite bandwidth in a multi-beam satellite system. A comparison of the results obtained using two techniques indicates improved performance of the GMM in terms of allocating the user demands. To ensure effective coverage, the clusters are transformed into shapes that match the coverage areas of the satellite. A technique called Voronoi Tessellation is used to create irregular polygons, and then these polygons are approximated as ellipses to cover the clusters more effectively.

An approach to apply the Jaya optimization algorithm to the nesting of irregular patterns

Abstract The problem of nesting frequently arises in the industrial environment, and it has a strong ecological and economic impact in the manufacturing processes. It basically consists of placing a set of pieces (polygons) on a material sheet, making sure that the pieces do not overlap and that they do not exceed the boundaries of the sheet. With regard to irregular 2D polygons, the problem is NP-complete. Therefore, different heuristics have been developed so as to cope with the problem. In this paper, the application of the Jaya metaheuristic algorithm to the nesting problem is proposed. This algorithm has been already applied to several engineering problems and has generally demonstrated better results than most metaheuristic algorithms. In this paper, the Jaya algorithm has been adapted to the specific features of the nesting problem so as to optimize the placement of pieces on a sheet, with the objective of minimizing material waste and computational time. The results of our experimentation demonstrate the algorithm's effectiveness in reducing the convex hull area across various datasets, showing potential in solving complex, irregular shape nesting problems. This research provides a new application of the Jaya algorithm and opens ways for future work in optimization techniques and parameter-free heuristic algorithms for nesting.

A Novel Model Based on Ensemble Learning for Phishing Attack

With the increase in the speed of the internet environment and the development of the infrastructures used, people have started to perform most of their work online. As much as this makes life easier, it also increases the possibility of being attacked by malicious people. Attackers can activate a phishing attack that aims to steal information from victims by creating copied, fake websites. While this attack is very old and somewhat simple, it can still be effective due to low IT literacy. People can enter their information on these fake websites out of spontaneity or ignorance or good intentions and be exposed to Phishing attacks. The compromise of a user's account information also puts at risk the security of the organization or institution to which it is connected. In this study, we propose a new machine learning-based ensemble model with feature selection methods to detect phishing attacks. Also, an ablation study is presented to measure the effect of different feature selection methods. The proposed model which we named as NaiveStackingSymmetric (NSS) is analyzed using the widely used accuracy (ACC), the area under curve (AUC), and F-score metrics as well as the polygon area metric (PAM), and it is shown that it outperforms other studies in the literature using the same dataset.

Detective generalized multiscale hybridizable discontinuous Galerkin(GMsHDG) method for porous media

The Detective Generalized Multiscale Hybridizable Discontinuous Galerkin (Detective GMsHDG) method aims to further reduce the computational cost of the GMsHDG method. The GMsHDG method itself reduces the computational cost of the HDG method by employing an upscaled structure on a two-grid mesh. Given a PDE within a specified domain, we subdivide the domain into polygonal subdomains and transforms a HDG problem into globular and local problems. Globular problem concerns whether the solutions on smaller domains glue well to form a globular solution. The process involves generation of multiscale spaces, which is a vector space of functions defined on edges of the polygonal regions. A naive approximation by polynomials fails, especially in porous media, necessitating the generation of problem-specific spaces. The Detective GMsHDG method improves this process by replacing the generation of the multiscale space with the detective algorithm. The Detective GMsHDG method has two stages. First is called an offline stage. During the offline stage, we construct a detective function which, given a permeability distribution, it gives a multiscale space. Later stage is called the offline stage where, given the multiscale space, we use GMsHDG method to solve a given PDE numerically. We show numerical results to argue the liability of the solution using the detective GMsHDG method.

An interface tracking method with triangle edge cuts

This paper introduces a volume-conserving interface tracking algorithm on unstructured triangle meshes. We propose to discretize the interface via triangle edge cuts which represent the intersections between the interface and the triangle mesh edges using a compact 6 numbers per triangle. This enables an efficient implicit representation of the sub-triangle polygonal material regions without explicitly storing connectivity information. Moreover, we propose an efficient advection algorithm for this interface representation that is based on geometric queries and does not require an optimization process. This advection algorithm is extended via an area correction step that enforces volume-conservation of the materials. We demonstrate the efficacy of our method on a variety of advection problems on a triangle mesh and compare its performance to existing interface tracking methods including VOF and MOF.

Curvature continuous corner cutting

Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor (1987) shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu (1996) discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios.

Statistical Law between Areas and Perimeters Created by a Moving Trajectory

Based on our interest in properties of human movement, we investigated Japanese GPS data, and arrived at the following three observations: (1) there is a strong correlation between the area of polygons created by human movement trajectories and their perimeters; (2) short-distance movement trajectories less than 5 km tend to enclose a large area like a circle; and (3) long-distance movement trajectories over 5 km tend to be straight. We also clarified the following four observations on individual attributes and external factors related to long-distance movements: (1) women tend to travel more linearly than men; (2) linearity is stronger on weekends and national holidays in areas with a large theme park; (3) linearity is weaker on weekends and holidays in areas with many historical tourist attractions; and (4) these variations are created by people visiting such areas. These properties should be incorporated when modeling the movement trajectories of humans.

Analytical Solution for the Problem of Point Location in Arbitrary Planar Domains

This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve.

Optimum Size and Heat Transfer and Velocity Correlations of the Outer and Inner Surfaces of Vertical Hollow Polygonal Cylinders in Natural Convection

AbstractLaminar natural convection heat transfer from vertical hollow polygonal cylinders with a wide range of cross-sectional areas is investigated. The buoyancy-driven three-dimensional (3D) flow around hollow polygonal cylinders immersed in quiescent ambient air with equal outer and inner surface temperatures is analyzed. The governing equations are numerically solved in nondimensional variables using the finite volume method. The numerical solution is validated using available experimental and numerical data. Results of the mean Nusselt number for the outer (Nu¯ho) and inner (Nu¯hi) surfaces are obtained by varying a number of key parameters. These parameters are the Rayleigh number based on the cylinder height (Rah) in the range 103≤ Rah≤ 107, the nondimensional cross-sectional area (AC) in the range 0.006 ≤ AC≤ 0.5, and the number of sides of the polygon (N) in the range 6 ≤ N ≤∞. In all cases, a Prandtl number (Pr) of 0.7 has been assumed. The study shows that at a certain Rayleigh number and a certain number of sides, the heat transfer rate from the inner surface decreases (by as much as 79.8%) as the polygon area decreases (by as much as 83.32%), whereas the heat transfer rate on the outer surface increases (by as much as 133.3%) as the polygon area decreases (by as much as 83.32%). It has also been found that the behavior of the buoyancy-driven flow in the vicinity of the outer surface is fundamentally different than that near the inner surface. Additional details about this fundamental difference are presented in the Results and Discussion section of the paper. New correlations to calculate the average velocity at the exit surface of the cylinder inner core and the mean Nusselt number for both the outer and inner surfaces have also been developed. Also, correlations have been developed for selecting the optimal cross-sectional area for purposes of identifying the regions where the thermal and velocity boundary layers overlap within the inner core of the cylinder.

Unsupervised machine learning with different sampling strategies and topographic factors for distinguishing between landslide source and runout areas to improve landslide inventory production

This study derived 12 topographical and hydrological factors related to landslides from a 10-m digital elevation model. Three unsupervised machine learning algorithms were employed to distinguish between the features of landslide sources and runout areas for Typhoon Morakot. Two sampling strategies were designed in this study. The first strategy involved creating a data set by pairing landslide sources and runout areas on the basis of the size of polygonal areas recorded in the inventory and then exploring the effectiveness of feature separation with k-means++ as the primary method and EM (expectation maximization) and DBSCAN (Density-Based Spatial Clustering of Applications with Noise) as supplementary methods. Because of practical challenges associated with the inability to determine whether remote sensing results correspond to landslide sources or runout areas, the first sampling strategy was used to test the feasibility of the adopted algorithms. In the second sampling strategy, the effectiveness of feature separation was tested by dividing polygon samples into several intervals on the basis of their sizes, thereby verifying the practicability of the adopted algorithms in real-world operations. This study verified that combining unsupervised machine learning algorithms with topographic factors facilitates the effective separation of landslide sources and runout areas, thereby enhancing the quality of landslide inventories and reducing the cost of landslide inventory creation. Moreover, it indicated that meter-scale topographic data yield classification results similar to those obtained from manually created landslide inventories based on centimeter-scale stereo aerial photos. The proposed method effectively balances the cost and quality of landslide inventories.

Enhanced Hybrid Algorithms for Segmentation and Reconstruction of Granular Grains From X‐Ray Micro Computed‐Tomography Images

ABSTRACTAccurate three‐dimensional (3D) reconstruction of granular grains from x‐ray micro‐computed tomography (µCT) images is a long‐standing challenge, particularly for dense soil samples. This study develops a machine learning (ML) enhanced approach to automatically reconstruct granular grains from µCT images. The novel academic contributions of this paper include (a) a hierarchical strategy based on parameter‐independent polygonal approximation, area, and concavity analysis, for the first time, to identify and eliminate both intergranular and intragranular voids; (b) incorporation of a recursive segmentation scheme and ML‐based grain classifier to avoid over‐segmentation; (c) novel modifications on the determination of splitting paths to enhance segmentation accuracy; and (d) an effective approach of assigning initial level set functions for reconstructing granular grains automatically. The hybrid ML algorithm is applied to µCT images of dense Mojave Mars Simulant. The results indicate that the proposed method can accurately segment grain clumps with unclear boundaries. The new automatic reconstruction algorithm eliminates ineffective operations and achieves a three‐fold increase in computational speed than previous methods documented in the literature. Ninety‐one percent of grains with distinct boundaries can be reconstructed and the reconstruction ratio reaches 81% even for grains without distinct boundaries. The overall reconstruction ratio of grains increases by 20% compared with previous methods, achieving a step‐change improvement for one‐to‐one mapping of real soil samples.

Full Coverage Path Planning for Torpedo-Type AUVs’ Marine Survey Confined in Convex Polygon Area

In this paper, we present a full coverage path planning (CPP) algorithm for the marine surveys conducted in the convex polygon shaped search area. The survey is supposed to carry out by torpedo-type AUVs (autonomous underwater vehicles). Due to their nonholonomic mechanical characteristics, these vehicles have nonzero minimum turning radius. For any given polygon shaped search area, it can always be partitioned into one or more convex polygons. With this in mind, this paper proposes a novel search algorithm called CbSPSA (Calculation based Shortest Path Search Algorithm) for full coverage of any given convex polygon shaped search area. By aligning the search inter-tracks alongside the edge with the minimum height, we can guarantee the minimum number of the vehicle’s turns. In addition, the proposed method can guarantee the planned path is strictly located inside the polygon area without overlapped or crossed path lines, and also has the total path length as short as possible. Considering the vehicle’s nonzero minimum turning radius, we also propose a sort of smoothing algorithm which can smooth the waypoint path searched by CbSPSA so that the vehicle can exactly follow it. The smoothed path is also guaranteed to be strictly located inside the polygon. Numerical simulation analyses are also carried out to verify the effectiveness of the proposed schemes.