Non-convex Polygons Research Articles

A two-dimensional vertex model for curvy cell-cell interfaces at the subcellular scale.

Cross-sections of cell shapes in a tissue monolayer typically resemble a tiling of convex polygons. Yet, examples exist where the polygons are not convex with curved cell-cell interfaces, as seen in the adaxial epidermis. To date, two-dimensional vertex models predicting the structure and mechanics of cell monolayers have been mostly limited to convex polygons. To overcome this limitation, we introduce a framework to study curvy cell-cell interfaces at the subcellular scale within vertex models by using a parametrized curve between vertices that is expanded in a Fourier series and whose coefficients represent additional degrees of freedom. This extension to non-convex polygons allows for cells with the same shape index, or dimensionless perimeter, to be, for example, either elongated or globular with lobes. In the presence of applied, anisotropic stresses, we find that local, subcellular curvature or buckling can be energetically more favourable than larger scale deformations involving groups of cells. Inspired by recent experiments, we also find that local, subcellular curvature at cell-cell interfaces emerges in a group of cells in response to the swelling of additional cells surrounding the group. Our framework, therefore, can account for a wider array of multicellular responses to constraints in the tissue environment.

Mixed-integer programming models for irregular strip packing based on vertical slices and feasibility cuts

The irregular strip-packing problem, also known as nesting or marker making, is defined as the automatic computation of a non-overlapping placement of a set of non-convex polygons onto a rectangular strip of fixed width and unbounded length, such that the strip length is minimized. Nesting methods based on heuristics are a mature technology, and currently, the only practical solution to this problem. However, recent performance gains of the Mixed-Integer Programming (MIP) solvers, together with the known limitations of the heuristics methods, have encouraged the exploration of exact optimization models for nesting during the last decade. Despite the research effort, there is room to improve the efficiency of the current family of exact MIP models for nesting. In order to bridge this gap, this work introduces a new family of continuous MIP models based on a novel formulation of the NoFit-Polygon Covering Model (NFP-CM), called NFP-CM based on Vertical Slices (NFP-CM-VS). Our new family of MIP models is based on a new convex decomposition of the feasible space of relative placements between pieces into vertical slices, together with a new family of valid inequalities, symmetry breakings, and variable eliminations derived from the former convex decomposition. Our experiments show that our new NFP-CM-VS models outperform the current state-of-the-art MIP models. Ten instances are solved up to optimality within one hour for the first time, including one with 27 pieces. Finally, we provide a detailed reproducibility protocol and dataset as supplementary material to allow the exact replication of our models, experiments, and results.

A separation and compaction algorithm for the two-open dimension nesting problem using penetration-fit raster and obstruction map

Nesting Problems, which are important subjects in the cutting and packing field, involve convex and nonconvex polygons and are common in several industries. These irregular open dimensional problems have been studied for decades, particularly the variant with one open dimension. However, in real-world applications, situations that are better suited to a two open dimensional model may arise and, in this sense, the literature is very limited. We here propose new separation and compaction algorithms for two-open dimension nesting problem. The paper develops an adaptation of the no-fit polygon to consider the penetration depth of pieces. The approach is based on an iterative compaction scheme, in which the key step is an obstruction map-based separation algorithm. The algorithms proposed found optimal solutions for artificial instances with up to 28 items within a small runtime. The results of benchmark instances indicate that the new algorithm is competitive when compared with other literature algorithms. It improved 14 of 15 benchmark instances when considering literature approaches on two open dimensions. In addition, the new algorithm achieved better occupation for some open dimension instances than the state of the art.

Robot path planning based on artificial potential field with deterministic annealing

In the context of motion planning in robotics, the problem of path planning based on artificial potential fields has been examined using different algorithms to avoid trapping in local minima. With this objective, this paper proposes a novel method based on a deterministic annealing strategy to improve the potential field function by introducing a temperature parameter to increase the robot’s obstacle avoidance efficiency. The annealing and tempering strategies prevent the robot from being trapped at the local minima and allow it to continue towards its destination. The initial path is optimised using an annealing algorithm to enhance the overall performance. The time, length and success rate of the planned path measures the quality of the solution. Simulation results and comparative experiments demonstrate that the proposed algorithm can solve path planning in different environments. The proposed algorithm is suitable for complex environments with convex or non-convex polygon obstacles.

Quantifying burned area of wildfires in the western United States from polar-orbiting and geostationary satellite active-fire detections

Background Accurately estimating burned area from satellites is key to improving biomass burning emission models, studying fire evolution and assessing environmental impacts. Previous studies have found that current methods for estimating burned area of fires from satellite active-fire data do not always provide an accurate estimate. Aims and methods In this work, we develop a novel algorithm to estimate hourly accumulated burned area based on the area from boundaries of non-convex polygons containing the accumulated Visible Infrared Imaging Radiometer Suite (VIIRS) active-fire detections. Hourly time series are created by combining VIIRS estimates with Fire Radiative Power (FRP) estimates from GOES-17 (Geostationary Operational Environmental Satellite) data. Conclusions, key results and implication We evaluate the performance of the algorithm for both accumulated and change in burned area between airborne observations, and specifically examine sensitivity to the choice of the parameter controlling how much the boundary can shrink towards the interior of the area polygon. Results of the hourly accumulation of burned area for multiple fires from 2019 to 2020 generally correlate strongly with airborne infrared (IR) observations collected by the United States Forest Service National Infrared Operations (NIROPS), exhibiting correlation coefficient values usually greater than 0.95 and errors <20%.

An extended model formulation for the two-dimensional irregular strip packing problem considering general industry-relevant aspects

Two-dimensional cutting and packing problems including irregular items (nesting problems) are common, e.g., in the clothing, paper, glass and leather industries. In the irregular strip packing problem considered in this work a finite number of rotations of convex as well as non-convex polygons with and without holes are permitted. To deal with the geometry of irregular items, direct trigonometry is applied. The focus is on aspects and characteristics that are typical for many affected industries and have been neglected so far. In the mentioned sectors, it is conceivable that items can be created from smaller parts by assembling them using various techniques. There might be several possible combinations of parts to be joined together to result in the desired item, i.e., there might be several cutting patterns to choose from. Also, whether the large material, i.e., large object is single-colored or has a particular structure or design is of great importance. In the latter case, special attention must be paid to the rotation of certain items or parts in order to achieve the desired (uniform or non-uniform) appearance of the final product. The utilized data structure is introduced, to address the mentioned aspects in the presented mixed-integer linear model which is an extension of a formulation published by previous authors. Furthermore, the method of calculating “critical vertices” is introduced, which requires only a reduced number of comparisons between edges and vertices of two items to ensure overlap-free positioning. Industry-relevant examples are highlighted in the computational study to illuminate the versatility of the model.

Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh

The Galerkin finite element solutionuhu_hof the Poisson equation−Δu=f-\Delta u=funder the Neumann boundary condition in a possibly nonconvex polygonΩ\varOmega, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability:‖uh‖L∞(Ω)≤Cℓh‖u‖L∞(Ω),\begin{align*} \|u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u\|_{L^{\infty }(\varOmega )} , \end{align*}whereℓh=ln⁡(2+1/h)\ell _h = \ln (2+1/h)for piecewise linear elements andℓh=1\ell _h=1for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds:‖u−uh‖L∞(Ω)≤Cℓh‖u−Ihu‖L∞(Ω),\begin{align*} \|u-u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u-I_hu\|_{L^{\infty }(\varOmega )} , \end{align*}whereIhI_hdenotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm:‖u−uh‖L∞(Ω)≤{Cℓhhk+2−2p‖f‖Wk,p(Ω)amp;if r≥k+1,Cℓhhk+1‖f‖Hk(Ω)amp;if r=k,\begin{align*} \|u-u_h\|_{L^\infty (\varOmega )} \le \begin {cases} C\ell _h h^{k+2-\frac {2}{p}} \|f\|_{W^{k,p}(\varOmega )} &\text {if $r\ge k+1$}, \\ C\ell _h h^{k+1} \|f\|_{H^{k}(\varOmega )} &\text {if $r=k$}, \end{cases} \end{align*}wherer≥1r\ge 1is the degree of finite elements,kkis any nonnegative integer no larger thanrr, andp∈[2,∞)p\in [2,\infty )can be arbitrarily large.

The Harmonic Gbc Function Map is a Bijection If the Target Domain is Convex

Harmonic generalized barycentric coordinates (GBC) functions have been used for cartoon animation since an early work in 2006\cite{JMDGS06}. A computational procedure was further developed in \cite{SH15} for deformation between any two polygons. The bijectivity of the map based on harmonic GBC functions is still murky in the literature. In this paper, we present an elementary proof of the bijection of the harmonic GBC map transforming from one arbitrary polygonal domain $V$ to a convex polygonal domain $W$. This result is further extended to a more general harmonic map from one simply connected domain $V$ to a convex domain $W$ if the harmonic map preserves the orientation of the boundary of the domain $V$. In addition, we shall point out that the harmonic GBC map is also a diffeomorphism over the interior of $V$ to the interior of $W$. Finally, we remark on how to construct a harmonic GBC map from $V$ to $W$ when the number of vertices of $V$ is different from the number of vertices of $W$ and how to construct harmonic GBC functions over a polygonal domain with a hole or holes. We also point out that it is possible to use the harmonic GBC map to deform a nonconvex polygon $V$ to another nonconvex polygon $W$ by a good arrangement of the boundary map between $\partial V$ and $\partial W$. Several numerical deformations based on images are presented to show the effectiveness of the map based on bivariate spline approximation of the harmonic GBC functions.

Repulsion-Based p-Dispersion with Distance Constraints in Non-Convex Polygons.

Motivated by the question of optimal facility placement, the classical p-dispersion problem seeks to place a fixed number of equally sized non-overlapping circles of maximal possible radius into a subset of the plane. While exact solutions to this problem may be found for placement into particular sets, the problem is provably NP-complete for general sets, and existing work is largely restricted to geometrically simple sets. This paper makes two contributions to the theory of p-dispersion. First, we propose a computationally feasible suboptimal approach to the p-dispersion problem for all non-convex polygons. The proposed method, motivated by the mechanics of the p-body problem, considers circle centers as continuously moving objects in the plane and assigns repulsive forces between different circles, as well as circles and polygon boundaries, with magnitudes inversely proportional to the corresponding distances. Additionally, following the motivating application of optimal facility placement, we consider existence of additional hard upper or lower distance bounds on pairs of circle centers, and adapt the proposed method to provide a p-dispersion solution that provably respects such constraints. We validate our proposed method by comparing it with previous exact and approximate methods for p-dispersion. The method quickly produces near-optimal results for a number of containers.

Regular Two-Dimensional Packing of Congruent Objects: Cognitive Analysis of Honeycomb Constructions

A new approach to investigate the two-dimensional, regular packing of arbitrary geometric objects (GOs), using cognitive visualization, is presented. GOs correspond to congruent non-convex polygons with their associated coordinate system. The origins of these coordinate systems are accepted by object poles. The approach considered is based on cognitive processes that are forms of heuristic judgments. According to the first heuristic judgment, regular packing of congruent GOs on the plane have a honeycomb structure, that is, each GO contacts six neighboring GO, the poles of which are vertices of the pole hexagon in the honeycomb construction of packing. Based on the visualization of the honeycomb constructions a second heuristic judgment is obtained, according to which inside the hexagon of the poles, there are fragments of three GOs. The consequence is a third heuristic judgment on the plane covering density with regular packings of congruent GOs. With the help of cognitive visualization, it is established that inside the hexagon of poles there are fragments of exactly three objects. The fourth heuristic judgment is related to the proposal of a triple lattice packing for regular packing of congruent GOs.

Flight planning in multi-unmanned aerial vehicle systems: Nonconvex polygon area decomposition and trajectory assignment

Nowadays, it is quite common to have one unmanned aerial vehicle (UAV) working on a task but having a team of UAVs is still rare. One of the problems that prevent us from using teams of UAVs more frequently is flight planning. In this work, we present the first open-source solution ( https://pypi.org/project/pode/ ) for splitting any complex area into multiple parts. The area of interest can be convex or nonconvex and can include any number of no-flight zones. Four solutions, based on the algorithm of Hert and Lumelsky, are tested with the aim of improving the compactness of the partitions. We also show how the shape of the partitions influences flight performance in a real case scenario.

Cognitive approach to analysis of regular packing of congruent objects on plane

New approach to investigate the two-dimensional regular packing of arbitrary geometric objects (GOs) using cognitive visualization is presented. GO correspond to congruent non-convex polygons with their associated coordinate system. The origins of these coordinate systems are accepted by object poles. The approach considered is based on cognitive processes that are forms of heuristic judgments. According to the first heuristic judgment, regular packing of congruent GOs on the plane have a honeycomb structure, that is, each GO contacts six neighboring GO, the poles of which are vertices of the pole hexagon in honeycomb construction of packing. Based on the visualization of the honeycomb constructions a second heuristic judgment is obtained, according to which inside the hexagon of poles there are fragments of three GOs. The consequence is a third heuristic judgment on the plane covering density with regular packings of congruent GOs. With the help of cognitive visualization it is established that inside the hexagon of poles there are fragments of exactly three objects. The fourth heuristic judgment is related to the proposal of a triple lattice packing for regular packing of congruent GOs.

Polylidar3D-Fast Polygon Extraction from 3D Data.

Flat surfaces captured by 3D point clouds are often used for localization, mapping, and modeling. Dense point cloud processing has high computation and memory costs making low-dimensional representations of flat surfaces such as polygons desirable. We present Polylidar3D, a non-convex polygon extraction algorithm which takes as input unorganized 3D point clouds (e.g., LiDAR data), organized point clouds (e.g., range images), or user-provided meshes. Non-convex polygons represent flat surfaces in an environment with interior cutouts representing obstacles or holes. The Polylidar3D front-end transforms input data into a half-edge triangular mesh. This representation provides a common level of abstraction for subsequent back-end processing. The Polylidar3D back-end is composed of four core algorithms: mesh smoothing, dominant plane normal estimation, planar segment extraction, and finally polygon extraction. Polylidar3D is shown to be quite fast, making use of CPU multi-threading and GPU acceleration when available. We demonstrate Polylidar3D’s versatility and speed with real-world datasets including aerial LiDAR point clouds for rooftop mapping, autonomous driving LiDAR point clouds for road surface detection, and RGBD cameras for indoor floor/wall detection. We also evaluate Polylidar3D on a challenging planar segmentation benchmark dataset. Results consistently show excellent speed and accuracy.

Between the Cracks: Filling Space with Polygonal Shapes

A proof is given, for a large class of polygons, of a conjecture of Shier concerning his algorithm for the disjoint but otherwise random placement of successively smaller copies of an arbitrary shape within a bounded region whose area is the infinite sum of all the shape areas. If the shapes decrease successively in size according to a power law with an exponent falling in a specified range of values, dependent on the particular shape, Shier conjectured there will always be sufficient room within the bounded region to place the next shape in the sequence, disjoint from all previous shapes. The conjecture implies the shapes would fill the bounded region, except for a possible set of measure zero. Our proof confirms the conjecture for all convex polygons, as well as those nonconvex polygons that satisfy a certain condition that we call double containment. Double containment seems to us a natural condition to place on the nonconvexity of polygons and may prove of interest beyond its use in this article. Empirically, Shier’s conjecture appears true for a much larger range of exponents and variety of shapes. Some possible reasons for this are discussed.

Polylidar - Polygons From Triangular Meshes

This paper presents Polylidar, an efficient algorithm to extract non-convex polygons from 2D point sets, including interior holes. Plane segmented point clouds can be input into Polylidar to extract their polygonal counterpart, thereby reducing map size and improving visualization. The algorithm begins by triangulating the point set and filtering triangles by user configurable parameters such as triangle edge length. Next, connected triangles are extracted into triangular mesh regions representing the shape of the point set. Finally each region is converted to a polygon through a novel boundary following method which accounts for holes. Real-world and synthetic benchmarks are presented to comparatively evaluate Polylidar speed and accuracy. Results show comparable accuracy and more than four times speedup compared to other concave polygon extraction methods.

A Rendezvous Strategy With $\mathbb{R}^{2}$ Reachability for Kinematic Agents

In this article, we present a consensus strategy based on cyclic pursuit that ensures rendezvous at any desired point in 2-D space ( $\mathbb{R}^{2}$ reachability) starting with any nonsingular initial configuration of the agents. Choice of a negative gain expands the reachable set beyond the convex hull of the initial configuration of agents. However, the entire $\mathbb{R}^{2}$ is not reachable in general. We first find the necessary and sufficient condition for the initial configuration of the agents, which ensures $\mathbb{R}^{2}$ reachability. We show that any desired point can be reached with finite controller gains under basic cyclic pursuit if the initial configuration of agents forms a nonconvex polygon. Next, we employ a control strategy using deviated cyclic pursuit to reach the special configuration mentioned above starting from an arbitrary initial configuration. With a switching between these two strategies we can achieve $\mathbb{R}^{2}$ reachability with finite gains. We also present an implementation algorithm to realize this strategy including computation of the finite gains.

An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons

An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons

A numerical solution for the inhomogeneous Dirichlet boundary value problem on a non-convex polygon

In this paper, we introduce an effective finite element scheme for the Poisson problem with inhomogeneous Dirichlet boundary condition on a non-convex polygon. Due to the corner singularities, the solution is composed of a singular part not belonging to the Sobolev space H2 and a smoother regular part. We first provide a generalized extraction formula for the coefficient of singular part, which is depending on the inhomogeneous boundary condition and the regular part. We then propose a stable finite element method for the regular part by the use of the derived formula. We show the H1 and L2 error estimates of the finite element solution for the regular part and the absolute error estimate of the approximation for the coefficient of singular part. Finally, we give some numerical examples to confirm the efficiency and reliability of the proposed method.

High-frequency behaviour of corner singularities in Helmholtz problems

We analyze the singular behaviour of the Helmholtz equation set in a non-convex polygon. Classically, the solution of the problem is split into a regular part and one singular function for each re-entrant corner. The originality of our work is that the “amplitude” of the singular parts is bounded explicitly in terms of frequency. We show that for high frequency problems, the “dominant” part of the solution is the regular part. As an application, we derive sharp error estimates for finite element discretizations. These error estimates show that the “pollution effect” is not changed by the presence of singularities. Furthermore, a consequence of our theory is that locally refined meshes are not needed for high-frequency problems, unless a very accurate solution is required. These results are illustrated with numerical examples that are in accordance with the developed theory.

Probability distributions related to tilings of non-convex polygons

In this paper, we study random lozenge tilings of non-convex polygonal regions. The interaction of the non-convexities (cuts) leads to new kernels and thus new statistics for the tiling fluctuations near these regions. This paper gives new probability distributions and joint probability distributions for the fluctuation of tiles along lines in between the cuts.