We devise a polynomial-time algorithm for partitioning a simple polygon $P$ into a minimum number of star-shaped polygons. The question of whether such an algorithm exists has been open for more than four decades [Avis and Toussaint, Pattern Recognit., 1981] and it has been repeated frequently, for example in O'Rourke's famous book [Art Gallery Theorems and Algorithms, 1987]. In addition to its strong theoretical motivation, the problem is also motivated by practical domains such as CNC pocket milling, motion planning, and shape parameterization. The only previously known algorithm for a non-trivial special case is for $P$ being both monotone and rectilinear [Liu and Ntafos, Algorithmica, 1991]. For general polygons, an algorithm was only known for the restricted version in which Steiner points are disallowed [Keil, SIAM J. Comput., 1985], meaning that each corner of a piece in the partition must also be a corner of $P$. Interestingly, the solution size for the restricted version may be linear for instances where the unrestricted solution has constant size. The covering variant in which the pieces are star-shaped but allowed to overlap--known as the Art Gallery Problem--was recently shown to be $\exists\mathbb R$-complete and is thus likely not in NP [Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J. ACM 2022]; this is in stark contrast to our result. Arguably the most related work to ours is the polynomial-time algorithm to partition a simple polygon into a minimum number of convex pieces by Chazelle and Dobkin [STOC, 1979 & Comp. Geom., 1985]. 68 pages. This is the TheoretiCS journal version

In the context of forecasting rapidly developing hazardous natural phenomena leading to emergencies, an algorithm for studying the avalanche hazard of a territory proposed based on a nearly periodic data analysis, which resulted in the formation of intervals of uniform behavior of a nearly periodic function. These intervals divided by uniformly spaced boundaries that form a rectangular grid in the space of linearized data. An algorithm based on a nearly periodic analysis of linearized data obtained during the polygonal transformation of the original satellite image of the snow mass proposed for their effective analysis. Based on the results of applying the algorithm, a composition of uniform longitudinal and transverse intervals of uniform behavior of linearized data is determined, which forms a system of critical spatial barriers of the original image that determine the degree of avalanche hazard of nearby territories. Based on the presented mechanism of linearization and further processing of linearized data, a software product implemented for the application of almost periodic analysis using polygonal partitioning for nonlinear structures in the image in order to take measures to counteract the dangerous natural phenomenon of snow avalanches.

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

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

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

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.

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.

In this paper, we view parking functions viewed as labeled Dyck paths in order to study a notion of pattern avoidance first considered by Remmel and Qiu.In particular, we enumerate the parking functions avoiding any set of two or more patterns of length 3, and we obtain a number of well-known combinatorial sequences as a result.Along the way, we find bijections between specific sets of pattern-avoiding parking functions and a number of combinatorial objects such as partitions of polygons and trees with certain restrictions.

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.

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.

Abstract We 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.

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

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.

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

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

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

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.