Polygonal Partitions Research Articles

A general algorithm for convex fair partitions of convex polygons

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. The existence of such a partition for any number of regions remains an open question. In this paper, we address this issue by developing an algorithm to find such a convex fair partition without restrictions on the number of regions. Our approach utilizes the normal flow algorithm (a generalization of Newton’s method) to find a zero for the excess areas and perimeters of the convex hulls of the regions. The initial partition is generated by applying Lloyd’s algorithm to a randomly selected set of points within the polygon, after appropriate scaling. We performed extensive experimentation, and our algorithm can find a convex fair partition for 100% of the tested problem. Our findings support the conjecture that a convex fair partition always exists.

Uniformly monotone partitioning of polygons

We study the problem of partitioning a simple polygon P with n vertices (including R reflex vertices) and no holes into a minimum number of uniformly monotone subpolygons using diagonals that lie inside P, each connecting two distinct vertices of P. A set of polygons is uniformly monotone if there exists a line such that all of the polygons are monotone with respect to the line. We present an O(nRlog⁡n+R5)-time algorithm for the problem. When Steiner points can be placed on the boundary of P and each Steiner point is considered as a vertex of P, we present an O(n+R5)-time algorithm for the problem. We present an O(n+R4)-time algorithm when Steiner points can be placed anywhere in P. Our algorithms improve upon the previously best ones. We also present simple and efficient 2-approximation algorithms.

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method’s key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete H1 and L2 norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

Robust methods for multiscale coarse approximations of diffusion models in perforated domains

For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the H1-projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains.

3D Surface-Closed Mesh Clipping Based on Polygonal Partitioning for Surgical Planning.

How to create an efficient and accurate interactive tool for triangular mesh clipping is one of the key problems to be solved in computer-assisted surgical planning. Although the existing algorithms can realize three-dimensional model clipping, problems still remain unsolved regarding the flexibility of clipping paths and the capping of clipped cross-sections. In this study, we propose a mesh clipping algorithm for surgical planning based on polygonal convex partitioning. First, two-dimensional polygonal regions are extended to three-dimensional clipping paths generated from selected reference points. Second, the convex regions are partitioned with a recursive algorithm to obtain the clipped and residual models with closed surfaces. Finally, surgical planning software with the function of mesh clipping has been developed, which is capable to create complex clipping paths by normal vector adjustment and thickness control. The robustness and efficiency of our algorithm have been demonstrated by surgical planning of craniomaxillofacial osteotomy, pelvis tumor resection and cranial vault remodeling.

An algorithmic approach to convex fair partitions of convex polygons

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. In this article we describe an algorithm that finds such fair partition.•The Fair Partitions method finds a fair partition for any given convex polygon and any given number of regions.•Our method relies on two well-known methods: Lloyd's algorithm and the Normal Flow Algorithm.•The method proposed in this article can be used in various contexts and many real-world applications.

Метод триангуляции для приближенного решения вариационных задач нелинейной теории упругости

A variational problem for the minimum of the stored energy functional is considered in the framework of the nonlinear theory of elasticity, taking into account admissible deformations. An algorithm for solving this problem is proposed, based on the use of a polygonal partition of the computational domain by the Delaunay triangulation method. Conditions for the convergence of the method to a local minimum in the class of piecewise affine mappings are found.

A numerical implementation for the high-order 2D virtual element method in MATLAB

We present a numerical implementation for the Virtual Element Method that incorporates high order spaces. We include all the required computations in order to assemble the stiffness and mass matrices, and right hand side. Convergence of the method is verified for different polygonal partitions. An specific mesh-free application that allows to approximate harmonic functions is discussed, which is based on high-order projections onto polynomial spaces of degree k; this approach only requires to solve a k(k − 1)/2 linear system, reducing significantly the number of operations compared to usual finite or virtual element methods, and can be modified for different virtual spaces and elliptic partial differential equations.

Simplification of 2D Polygonal Partitions via Point‐line Projective Duality, and Application to Urban Reconstruction

AbstractWe address the problem of simplifying two‐dimensional polygonal partitions that exhibit strong regularities. Such partitions are relevant for reconstructing urban scenes in a concise way. Preserving long linear structures spanning several partition cells motivates a point‐line projective duality approach in which points represent line intersections, and lines possibly carry multiple points. We propose a simplification algorithm that seeks a balance between the fidelity to the input partition, the enforcement of canonical relationships between lines (orthogonality or parallelism) and a low complexity output. Our methodology alternates continuous optimization by Riemannian gradient descent with combinatorial reduction, resulting in a progressive simplification scheme. Our experiments show that preserving canonical relationships helps gracefully degrade partitions of urban scenes, and yields more concise and regularity‐preserving meshes than common mesh‐based simplification approaches.

Polygon Structure-Guided Hyperspectral Image Classification With Single Sample for Strong Geometric Characteristics Scenes

Combining spectral and spatial information can significantly improve the classification performance of hyperspectral image (HSI). Currently, a lot of spectral–spatial HSI classification methods have been proposed. However, the task of HSI classification has remained challenging since the number of training samples is limited in real scenarios. In this article, we propose a novel HSI classification framework with single sample, in which the spectral self-similarity and spatial polygon structure information are fully combined to improve the classification performance. On the one hand, spectral self-similarity is used to expand training samples, which makes it possible to obtain sufficient samples with minimal cost. On the other hand, polygonal partition is introduced to acquire the geometrical structure of land covers in man-made environments. Specifically, the edge information of geometric objects is captured by polygonal partition, which can be utilized to constrain the spatial range of sample expansion and optimize the classification results. Experimental results on three real HSIs illustrate that the proposed method performs very well under small training sample size even when the number of samples is single per class.

Construction of C1 polygonal splines over quadrilateral partitions

In this paper we construct smooth bivariate spline functions over a polygonal partition, e.g. a convex quadrilateral partition by using vertex spline techniques. Vertex splines, introduced in Chui and Lai (1985) are smooth piecewise polynomial functions supported over a collection of triangles sharing a vertex. In this paper, we extend the concept of vertex splines to the partition of polygons and describe how to construct C1 vertex polygonal splines over a collection of quadrilaterals. We begin with our construction of C1 vertex splines over a collection of parallelograms, although they may not be axis-orientated. Then the construction is generalized to the setting of general convex quadrilaterals. We will use various monomials of Wachspress GBC functions of degrees 5 and 7 to explain how to construct C1 vertex splines together with additional special splines called edge and face splines. With these splines at hand, we construct quasi-interpolatory formulas, whose approximation properties will be shown. Numerical interpolation and approximation results will be presented. Finally, three applications of these splines are explained: the first one is to form smooth locally supported GBC functions, the second one is to construct smooth suitcase corners, and the third one to construct C1 surfaces over quadrilateral partitions with extraordinary points (EP). Several examples will be demonstrated to show the convenience of using these splines.

Computation of the distance between a polygon and a point in spatial analysis

ABSTRACT Distance is one of the most important concepts in geography and spatial analysis. Since distance calculation is straightforward for points, measuring distances for non-point objects often involves abstracting them into their representative points. For example, a polygon is often abstracted into its centroid, and the distance from/to the polygon is then measured using the centroid. Despite the wide use of representative points to measure distances of non-point objects, a recent study has shown that such a practice might be problematic and lead to biased coefficient estimates in regression analysis. The study proposed a new polygon-to-point distance metric, along with two computation algorithms. However, the efficiency of these distance calculation algorithms is low. This research provides three new methods, including the random point-based method, polygon partitioning method, and axis-aligned minimum areal bounding box-based (MABB-based) method, to compute the new distance metric. Tests are provided to compare the accuracy and computational efficiency of the new algorithms. The test results show that each of the three new methods has its advantages: the random point-based method is easy to implement, the polygon partitioning method is most accurate, and the MABB-based method is computationally efficient.

Regular Polygonal Partitions of a Tverberg Type

A seminal theorem of Tverberg states that any set of $$T(r,d)=(r-1)(d+1)+1$$ points in $${\mathbb {R}}^{d}$$ can be partitioned into r subsets whose convex hulls have non-empty r-fold intersection. Almost any collection of fewer points in $${\mathbb {R}}^{d}$$ cannot be so divided, and in these cases we ask if the set can nonetheless be P(r, d)-partitioned, i.e., split into r subsets so that there exist r points, one from each resulting convex hull, which form the vertex set of a prescribed convex d-polytope P(r, d). Our main theorem shows that this is the case for any generic $$T(r,2)-2$$ points in the plane and any $$r\ge 3$$ when $$P(r,2)=P_{r}$$ is a regular r-gon, and moreover that $$T(r,2)-2$$ is tight. For higher dimensional polytopes and $$r=r_{1}\cdots r_{k}$$ , $$r_{i} \ge 3$$ , this generalizes to $$T(r,2k)-2k$$ generic points in $${\mathbb {R}}^{2k}$$ and orthogonal products $$P(r,2k)=P_{r_{1}}\times \cdots \times P_{r_{k}}$$ of regular polygons, and likewise to $$T(2r,2k+1)-(2k+1)$$ points in $${\mathbb {R}}^{2k+1}$$ and the product polytopes $$P(2r,2k+1)=P_{r_{1}}\times \cdots \times P_{r_{k}}\times P_{2}$$ . As with Tverberg’s original theorem, our results admit topological generalizations when r is a prime power, and, using the “constraint method” of Blagojević, Frick, and Ziegler, allow for dimensionally restricted versions of a van Kampen–Flores type and colored analogues in the fashion of Soberón.

Floorplan generation from 3D point clouds: A space partitioning approach

We propose a novel approach to automatically reconstruct the floorplan of indoor environments from raw sensor data. In contrast to existing methods that generate floorplans under the form of a planar graph by detecting corner points and connecting them, our framework employs a strategy that decomposes the space into a polygonal partition and selects edges that belong to wall structures by energy minimization. By relying on a efficient space-partitioning data structure instead of a traditional and delicate corner detection task, our framework offers a high robustness to imperfect data. We demonstrate the potential of our algorithm on both RGBD and LIDAR points scanned from simple to complex scenes. Experimental results indicate that our method is competitive with respect to existing methods in terms of geometric accuracy and output simplicity.

Construction strategy and performance analysis of large-scale spherical solar concentrator for the space solar power station

The space solar power station is a gigantic power satellite to provide the earth with continuous energy. The front-end system of space solar power station, solar concentrator, has significant influences on the optical performance. Regarding the proposed orb-shaped membrane energy gathering array scheme, this paper deals with the construction strategy of its large-scale spherical concentrator to reduce the complexity of manufacturing and improve the optical properties. The optical principle of the spherical concentrator is described and the spatial motion characteristic is presented. A novel construction strategy for the large-scale concentrator is proposed via partitioning octahedron on the sphere and polygonal partition. This paper evaluates the optical performance of the concentrator constructed by the novel strategy using ray tracing method. Performance parameters such as optical concentration ratio and collection efficiency are obtained. Comparison with the other strategies are also implemented. In addition, influence factors such as the sun's non-parallelism, the tracking precision, the orbital periodic motion, and the potential blockage effects of microwave transmitting antenna and supporting structure are mathematically discussed and evaluated. The numerical results indicate that the proposed strategy achieves a high collection rate, a small efficiency fluctuation, appropriate energy distribution, and relatively small splicing modules, which is conducive to improve optical performance and decrease difficulty in manufacturing.

Simplified weak Galerkin and new finite difference schemes for the Stokes equation

This article presents a simplified formulation for the weak Galerkin finite element method for the Stokes equation without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak Galerkin algorithm. The simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions. A particular application of the simplified weak Galerkin on rectangular partitions yields a new class of 5- and 7-point finite difference schemes for the Stokes equation. An explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements. Error estimates of optimal order are established for the simplified weak Galerkin finite element method in the H^1 and L^2 norms. Furthermore, a superconvergence of order O(h^{1.5}) is established on rectangular partitions for the velocity approximation in the H^1 norm at cell centers, and a similar superconvergence is derived for the pressure approximation in the L^2 norm at cell centers. Some numerical results are reported to confirm the convergence and superconvergence theory.

A transmission problem on a polygonal partition: regularity and shape differentiability

ABSTRACTWe consider a transmission problem on a polygonal partition for the two-dimensional conductivity equation. For suitable classes of partitions we establish the exact behaviour of the gradient of solutions in a neighbourhood of the vertexes of the partition. This allows to prove shape differentiability of solutions and to establish an explicit formula for the shape derivative.

A weak Galerkin finite element method for a coupled Stokes–Darcy problem on general meshes

In this paper, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity–pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. By using the weak Galerkin approach, we consider the two-dimensional problem with the usual polynomials of degree k ≥ 1 for the velocity and hydraulic head, while polynomials of degree k − 1 for the pressure, the velocity and hydraulic head is enhanced by polynomials of degree k − 1 on the edge of a finite element partition. This new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained. Moreover, numerical experiences are presented to illustrate the good performance, confirm the optimal order of convergence and verify the efficiency of the proposed weak Galerkin method in this paper. We also deal with finite element partitions consisting of general meshes, such as triangular mesh, quadrilateral mesh, hexagonal-dominant mesh and Voronoi mesh for the numerical weak Galerkin approximation.

A weak Galerkin finite element method for the Navier–Stokes equations

In this paper, we propose and analyze a weak Galerkin finite element method for the Navier–Stokes equations. The new formulation hinges upon the introduction of weak gradient, weak divergence and weak trilinear operators. Moreover, by choosing the matching finite element triples, this new method not only obtains stability and optimal error estimates but also has a lot of attractive computational features: general finite element partitions of arbitrary polygons or polyhedra with certain shape regularity and parameter free. Finally, several numerical experiments assess the convergence properties of the new method and show its computational advantages.

Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT

In this paper, we develop a shape optimization-based algorithm for the electrical impedance tomography (EIT) problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements. The key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition. Numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions.