Planar Polygons Research Articles

On Minkowski Addition of Convex Polygons in the Plane

Summary We investigate the Minkowski addition of convex polygons in the plane. It is well-known that the Minkowski addition of two planar convex polygons with m and n vertices is a convex polygon with at most m + n vertices. We give an explicit formula for calculating the exact number of vertices of the resulting polygon.

A note on the shift theorem for the Laplacian in polygonal domains

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces.

Computing the cut locus, Voronoi diagram, and signed distance function of polygons

This paper presents a new method for the computation of the generalized Voronoi diagram of planar polygons. First, we show that the vertices of the cut locus can be computed efficiently. This is achieved by enumerating the tripoints of the polygon, a superset of the cut locus vertices. This is the set of all points that are of equal distance to three distinct topological entities. Then our algorithm identifies and connects the appropriate tripoints to form the cut locus vertex connectivity graph, where edges define linear or parabolic boundary segments between the Voronoi regions, resulting in the generalized Voronoi diagram. Our proposed method is validated on complex polygon soups. We apply the algorithm to represent the exact signed distance function of the polygon by augmenting the Voronoi regions with linear and radial functions, calculating the cut locus both inside and outside.

Roof plane parsing towards LoD-2.2 building reconstruction based on joint learning using remote sensing images

Building models are important for urban studies. Remote sensing multi-spectral (MS) images are widely used for its rich semantic information. The lack of geometry features is fulfilled by introducing photogrammetry derived digital surface models (DSMs), resulting in pairs of DSMs and MS images. Utilizing such pairs and a convolutional neural network, level of detail (LoD) 2.2 building models, which contain roof planes and major roof elements (e.g. dormers), are reconstructed in this work. Leveraging both raster and vector predictions, 3-D building models with straight edges and sharp corners are obtained. The proposed two-stage method first extracts vectorized roof lines from pairs of DSMs and RGB images, followed by generation of detailed 2-D and 3-D polygonal building models. We conducted our experiments based on two datasets: a custom dataset in Landsberg am Lech in Germany, and an open dataset named Roof3D. For the custom dataset, our proposed model achieved mean average precision (mAP) for building roof vertices of 64.3% and for building roof lines of 54.5% at highest. Mean precision and recall for reconstructed 2-D building roof plane polygons are 52.2% and 54.7% respectively. For the Roof3D dataset, mAP is reported to be 25.3% and 12.4% for the extracted building roof lines and roof plane polygons respectively.

Reduced polygons in the hyperbolic plane

For a hyperplane H supporting a convex body C in the hyperbolic space Hd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^d$$\\end{document}, we define the width of C determined by H as the distance between H and a most distant ultraparallel hyperplane supporting C. The minimum width of C over all supporting H is called the thickness Δ(C)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (C)$$\\end{document} of C. A convex body R⊂Hd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R \\subset \\mathbb {H}^{d}$$\\end{document} is said to be reduced if Δ(Z)<Δ(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta (Z) < \\Delta (R)$$\\end{document} for every convex body Z properly contained in R. We describe a class of reduced polygons in H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^{2}$$\\end{document} and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses.

Synthesis and cross-section design of a new family of single-loop 7R deployable polygon mechanisms

In this paper, two design methods for different stages in the development of a single-degree-of-freedom (DOF) single-loop 7R deployable polygon mechanism (DGM) without overconstraints are presented, encompassing schematic synthesis and cross-section design methods for 7R DGMs. Firstly, two classes of revolute pair axes in DGMs are reviewed, detailing their value ranges and parameter definitions. A schematic synthesis method for single-DOF single-loop 7R DGMs is then proposed, followed by the development of a cross-section design method to enable desired cross-sections in these mechanisms. By applying these methods, three types of 7R DGMs were created and prototyped, with their folding sequences demonstrated and input singularity analysis conducted. Through the utilization of the presented design method, all single-DOF single-loop 7R DGMs can be designed. This work introduces a new family of DGMs and addresses the gap in the domain of general spatial single-DOF single-loop DGMs that are not overconstrained. It demonstrates the feasibility of designs whose expanded shapes form general planar polygons. This effort expands the variety and quantity of DGMs, paving the way for more innovative deployable network mechanisms.

On polytope intersection by half-spaces and hyperplanes for unsplit geometric volume of fluid methods on arbitrary grids

Several tools with improved accuracy and computational efficiency, along with added capabilities, are presented for performing analytic and geometric operations involving intersections between half-spaces or hyperplanes and 2- or 3-polytopes (polygons or polyhedra, either convex or non-convex) that typically arise in advanced volume of fluid (VOF) methods on arbitrary grids. In particular, tools for the intersection between a polyhedron and a half-space, and for the computation of the interface position to cut off a given area/volume fraction from an arbitrary polygonal/polyhedral grid cell in PLIC (piecewise linear interface calculation) reconstruction have been improved in terms of accuracy and computational efficiency. In addition, specific tools are also proposed for (1) the intersection in 3D between a planar polygon and a half-space, and the computation of the resulting intersected area, which can be useful for unsplit methods based on wetted-area time-integration advection (WATIA) schemes, and for (2) the efficient computation of the geometric center of the set of clipped vertices and volume resulting from the intersection between an arbitrary polyhedron and a half-space, which can be useful for, respectively, several PLIC reconstruction methods and unsplit methods based on fluid-volume intersection advection (FVIA) schemes. Finally, a convenient strategy to perform the recursive intersections required during the advection step in advanced geometric VOF methods based on FVIA schemes is also proposed. To assess the performance and accuracy of the proposed tools, different tests were carried out for several convex and non-convex polytopes. Speedups of ×5 and improvements in accuracy of up to 5 orders of magnitude over previous tools have been achieved for certain geometries and operations. The proposed tools have been included in two advection methods (an FVIA method and a new WATIA method proposed in this work) and in a PLIC reconstruction method to demonstrate their improved performance and capabilities on structured and unstructured grids.

Building planar polygon spaces from the projective braid arrangement

Abstract The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the moduli space of distinct points on the real projective line as an open dense subset. Kapranov showed that the real points of the Deligne–Mumford–Knudson compactification can be obtained from the projective Coxeter complex of type 𝐴 (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper, we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type 𝐴 by performing an iterative cellular surgery along a subcollection of the minimal building set. Interestingly, this subcollection is determined by the combinatorial data associated with the length vector called the genetic code.

Ocular Lichen Planus: A clinicopathologic review.

Lichen planus is a chronic inflammatory dermatosis that can affect the skin, mucous membranes and nails. Cutaneous lichen planus lesions are best described by the "six Ps" - purple pruritic polygonal planar papules and plaques. Mucous membrane lesions are commonly associated with cutaneous lichen planus. Ocular involvement with lichen planus is rare and conjunctival involvement usually predominates, it can however be visually devastating. Ocular lichen planus often progresses to extensive conjunctival scarring which can be impossible to distinguish clinically from other cicatrising conjunctivitis, requiring histopathological confirmation. Here we review the ocular pathology of this condition.

MCTS with Refinement for Proposals Selection Games in Scene Understanding.

We propose a novel method applicable in many scene understanding problems that adapts the Monte Carlo Tree Search (MCTS) algorithm, originally designed to learn to play games of high-state complexity. From a generated pool of proposals, our method jointly selects and optimizes proposals that minimize the objective term. In our first application for floor plan reconstruction from point clouds, our method selects and refines the room proposals, modelled as 2D polygons, by optimizing on an objective function combining the fitness as predicted by a deep network and regularizing terms on the room shapes. We also introduce a novel differentiable method for rendering the polygonal shapes of these proposals. Our evaluations on the recent and challenging Structured3D and Floor-SP datasets show significant improvements over the state-of-the-art both in speed and quality of reconstructions, without imposing hard constraints nor assumptions on the floor plan configurations. In our second application, we extend our approach to reconstruct general 3D room layouts from a color image and obtain accurate room layouts. We also show that our differentiable renderer can easily be extended for rendering 3D planar polygons and polygon embeddings. Our method shows high performance on the Matterport3D-Layout dataset, without introducing hard constraints on room layout configurations.

ROOF3D: A REAL AND SYNTHETIC DATA COLLECTION FOR INDIVIDUAL BUILDING ROOF PLANE AND BUILDING SECTIONS DETECTION

Abstract. Deep learning is a powerful tool to extract both individual building and roof plane polygons. But deep learning requires a large amount of labeled data. Hence, publicly available level of detail (LoD)-2 datasets are a natural choice to train fully convolutional neural networks (FCNs) models for both building section and roof plane instance segmentation. Since publicly available datasets are often automatically derived, e.g. based on laser scanning, they lack on annotation accuracy. To complement such a dataset, we introduce manually annotated and synthetically generated data. Manually annotated data is domain-specific and has a high annotation quality but is expensive to obtain. Synthetically generated data has high-quality annotations by definition, but lacks domain-specificity. Moreover, we not only detect individual building section instances, but also roof plane instances. We predict separations not only between individual buildings, but also by a class that describes the line which separates roof planes. The predicted building and roof plane instances are polygonized by a simple tree search algorithm. To achieve more regular polygons, we utilize the Douglas-Peucker polygon simplification algorithm. We describe our dataset in detail to allow comparability between successive methods. To facilitate future works in building and roof plane prediction, our Roof3D dataset is accessible at &lt;code&gt;https://github.com/dlrPHS/GPUB&lt;/code&gt;.

Numerical modelling of DMLS Ti6Al4V(ELI) polygon structures

Numerical modelling is particularly advantageous for analysing structures with complex behaviour. It is used to predict the mechanical properties of structures. Analytical modelling, on the contrary, has limited capacity for predicting the behaviour, particularly of structures, because it is based on mathematical equations that do not always exactly represent the geometry of the model. In such cases, numerical modelling is used for predicting structural bending, axial deformation, and buckling behaviour. This study documents numerical modelling of different types of polygon structures. To reduce computation costs, planar and extruded Ti6Al4V(ELI) hexagonal shell structures were used to predict stresses in the out-of-plane and in-plane directions. This was followed by numerical modelling of different types of planar polygon structures to predict their load-bearing capacity and stiffness. Thereafter, the hexagonal polygon was subjected to out-of-plane and in-plane uniaxial compression loads. This was done to compare the bending and buckling behaviour of finite element (FE) models to analytical models. The numerical and analytical results were then compared to determine how the ratio (t/L) of the wall thickness (t) and length of the polygon members (L) influenced the effective stiffness of the hexagonal polygon. The triangular polygon was seen to have the greatest load-bearing capacity and stiffness of all polygons that were modelled. The hexagonal model was observed to generate deformations due to compression, similar to those reported in literature. The critical buckling loads for the analytical honeycomb (HC) models were found to be below the yield stress for (1-, 1.125-, and 1.25-mm wall thicknesses) and above the yield stress for all FE HC models, respectively. The effective stiffness of the HC models were observed to increase with the increasing (t/L) ratio, for both the numerical and analytical models.

A Novel Analytical Model for Mutual Inductance Calculations Between Two Nonidentical N-Sided Polygonal Planar Coils Arbitrarily Positioned in 3-D Space for Wireless Power Transfer

Mutual inductance (MI) calculation is of great importance, because there are many systems that operate on the principle of inductive coupling. This paper presents a novel analytical model that calculates the MI between two non-identical n-sided polygonal planar coils located in arbitrary position in 3D space. The model depends upon misalignment of both the primary and the secondary coils of a wireless power transfer (WPT) system operating based on inductive coupling. The polygonal coils used in this model may differ from each other in terms of size, shape and turns number. The misalignment cases considered in the proposed model include lateral, angular, lateral-angular and vertical distance variation between the primary and secondary coils. Based on Stokes Theorem, instead of using the magnetic flux density vector, the magnetic vector potential (MVP) is used to reach the required magnetic flux in the calculation of the MI. A single analytical solution function that can calculate the MI between two planar polygonal coils of any shape in a simple manner and short time duration is obtained by performing difficult mathematical operations through MVP. Results obtained from the developed new model are compared with finite element analysis (FEA), experimental and other studies in the literature. The good agreement of the results of this study with the other results has proved the accuracy and validity of the proposed new analytical model.

A Study of Certain Sharp Poincaré Constants as Set Functions of Their Domain

For bounded, convex sets Omega subset mathbb {R}^d, the sharp Poincaré constant C(Omega ), which appears in ||f-bar{f}_{_{Omega }}||_{L^{infty }(Omega )} le C(Omega )||nabla f||_{L^{infty }(Omega )}, is given by C(Omega )=max _{_{partial Omega }}zeta for a specific convex function zeta [Bevan et al. in Proc Am Math Soc 151:1071–1085, 2023 (Theorem 1.1)]. We study C(cdot ) as a function on convex sets, in particular on polyhedra, and find that while a geometric characterization of C(Omega ) for triangles is possible, for other polyhedra the problem of ordering zeta (V_i), where V_i are the vertices of Omega , can be formidable. In these cases, we develop estimates of C(Omega ) from above and below in terms of more tractable quantities. We find, for example, that a good proxy for C(Q) when Q is a planar polygon with vertices V_i and centroid gamma (Q) is the quantity D(Q)=max _{i}|V_i-gamma (Q)|, with an error of up to sim 8%. A numerical study suggests that a similar statement holds for k-gons, this time with a maximal error across all k-gons of sim 13%. We explore the question of whether there is, for each Omega , at least one point M capable of ordering the zeta (V_i) according to the ordering of the |V_i-M|. For triangles, M always exists; for quadrilaterals, M seems always to exist; for 5-gons and beyond, they seem not to.

A new method for researching and constructing spherical bicentric polygons based on geometric mapping

A bicentric polygon is a special planar polygon that has both a circumcircle and an incircle. Many interesting properties of a planar bicentric polygon have been discovered. However, searching the geometric properties of a spherical bicentric polygon with elementary geometric method is very difficult. In this paper, we propose a special mapping for a pair of circles between a plane and a sphere. This mapping provides us a new method to research the spherical bicentric polygons. With this mapping, we can translate the Fuss relation, which describes the relation of the circumcircle and incircle of a bicentric polygon, from a planar bicentric polygon to a spherical polygon. Based on the mapping, we can derive the geometric properties of spherical bicentric polygons from the planar bicentric polygons, and construct a series of spherical bicentric n-polygons from a planar bicentric n-polygon.

Iterated and mixed discriminants

Classical work by Salmon and Bromwich classified singular intersections of two quadric surfaces. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in the plane and used by Schäfli for the study of hyperdeterminants. More recently, the problem has been revisited with similar tools in the context of geometric modeling and a generalization to the case of two higher dimensional quadric hypersurfaces was given by Ottaviani. We propose and study a generalization of this question for systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant . We define a related polynomial called the multivariate iterated discriminant , generalizing the classical Schäfli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre–Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.

Symmetric configuration spaces of linkages

A configuration of a linkage $$\Gamma $$ is a possible positioning of $$\Gamma $$ in $${{\mathbb {R}}}^{d}$$ , and the collection of all such forms the configuration space $${\mathcal {C}}(\Gamma )$$ of $$\Gamma $$ . We here introduce the notion of the symmetric configuration space of a linkage, in which we identify configurations which are geometrically indistinguishable. We show that the symmetric configuration space of a planar polygon has a regular cell structure, provide some principles for calculating this structure, and give a complete description of the symmetric configuration space of all quadrilaterals and of the equilateral pentagon.

Polygon recutting as a cluster integrable system

Recutting is an operation on planar polygons defined by cutting a polygon along a diagonal to remove a triangle, and then reattaching the triangle along the same diagonal but with opposite orientation. Recuttings along different diagonals generate an action of the affine symmetric group on the space of polygons. We show that this action is given by cluster transformations and is completely integrable. The integrability proof is based on interpretation of recutting as refactorization of quaternionic polynomials.

Long‐diagonal pentagram maps

AbstractThe pentagram map on polygons in the projective plane was introduced by R. Schwartz in 1992 and is by now one of the most popular and classical discrete integrable systems. In the present paper we introduce and prove integrability of long‐diagonal pentagram maps on polygons in , by now the most universal pentagram‐type map encompassing all known integrable cases. We also establish an equivalence of long‐diagonal and bi‐diagonal maps and present a simple self‐contained construction of the Lax form for both. Finally, we prove that the continuous limit of all these maps is equivalent to the ‐KdV equation, generalizing the Boussinesq equation for .

The algebraic dynamics of the pentagram map

AbstractThe pentagram map, introduced by Schwartz [The pentagram map. Exp. Math.1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J.162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.