Polygonal Space Research Articles

Singular fibers of the bending flows on the moduli space of 3D polygons

In this paper, we prove that in the system of bending flows on the moduli space of polygons with fixed side lengths introduced by Kapovich and Millson, the singular fibers are isotropic homogeneous submanifolds. The proof covers the case where the system is defined by any maximal family of disjoint diagonals. We also take in account the case where the fixed side lengths are not generic. In this case, the phase space is an orbispace, and our result holds in the sense that singular fibers are isotropic orbispaces.

Geometry of uncertainty relations for linear combinations of position and momentum

For a quantum particle with a single degree of freedom, we derive preparational sum and product uncertainty relations satisfied by N linear combinations of position and momentum observables. The state-independent bounds depend on their degree of incompatibility defined by the area of a parallelogram in an N-dimensional coefficient space. Maximal incompatibility occurs if the observables give rise to regular polygons in phase space. We also conjecture a Hirschman-type uncertainty relation for N observables linear in position and momentum, generalizing the original relation which lower-bounds the sum of the position and momentum Shannon entropies of the particle.

A 24-Sided Voltage Space Vector Based IM Drive with Low-Order Harmonic Elimination for the Full Speed Range

This paper proposes an inverter topology to generate a 24-sided polygonal voltage space vector structure for variable speed drives using a single dc source. The topology consists of a two-level inverter cascaded with two low-voltage floating capacitor-fed H-bridge cells. The 24-sided structure eliminates harmonics up to and including seventeenth and nineteenth orders from phase voltage output resulting in a smooth torque profile over the entire speed range. This structure also enhances the linear modulation range to 99.42% of the base speed. A reduced switching sequence based pulse width modulation technique achieves minimum pole voltage transitions for the two-level inverter, thereby controlling switching losses. The lower order harmonics are eliminated by switching action of low-voltage H-bridge cells. Experimental results, simulation studies on inverter losses, and harmonic performance validate the effectiveness of proposed drive scheme.

Variable guidance for pedestrian evacuation considering congestion, hazard, and compliance behavior

A methodology for optimizing variable pedestrian evacuation guidance in buildings with convex polygonal interior spaces is proposed. The optimization of variable guidance is a bi-level problem. The calculation of variable guidance based on the prediction of congestion and hazards is the upper-level problem. The prediction of congestion provided the variable guidance is the lower-level problem. A local search procedure is developed to solve the problem. The proposed methodology has three major contributions. First, a logistic regression model for guidance compliance behavior is calibrated using a virtual reality experiment and the critical factors for the behavior are identified. Second, the guidance compliance and following behaviors are considered in the lower-level problem. Third, benchmarks are calculated to evaluate the performance of optimized variable guidance, including the lower bound of the maximum evacuation time and the maximum evacuation time under a fixed guidance. Finally, the proposed methodology is validated with numerical examples. Results show that the method has the potential to reduce evacuation time in emergencies.

Construction, PPy loading and photocatalytic performance of a polyoxometalate-templated metallacycle compound

A polyoxometalate-templated metallacycle compound, [Ag6trz6][H3PMo12O40]2·6H2O (trz = 1,2,3-triazole), has been isolated and structurally characterized. The compound represents the first example of a trigonal antiprism based metal-organic framework (MOFs) with complicated Zr6Br12 type topology, in which the trigonal antiprisms as building blocks are constructed by the Ag6trz6 space polygons and the bifunctional POMs (both as template and building block). In addition, to improve the photocatalytic activity of polyoxometalate-templated metallacycle compound, polypyrrole was loaded on its surface through a chemical oxidation polymerization process in situ. Finally, the photocatalytic activity of [Ag6trz6][H3PMo12O40]2 and PPy@[Ag6trz6][H3PMo12O40]2 under visible light was explored, and the results demonstrate that the PPy loading on the surface of POMs-based hybrid compound is feasible to improve their photocatalytic performance.

On the moduli space of polygons with area center

A point p is said to be an area center of a polygon if all of the triangles composed of p and its edges have one and the same area. We construct a moduli space ACn of such n-gons and study its geometry and arithmetic. For every n≥5, the moduli space is proved to be a rational complete intersection subvariety in An. With the help of some subvarieties of low degree in ACn, we also find a unified method of construction of good-looking polygons with area center.

Multiplication operators on the Bergman spaces of polygons

In this paper, we will show that multiplication operators on the Bergman space of a polygon and some von Neumann algebras induced by these operators have deep connections with the geometry of the polygon.

Solving nonlinear equilibrium equations of deformable systems by method of embedded polygons

ABSTRACTSolving of nonlinear algebraic equations is an obligatory stage of studying the equilibrium paths of nonlinear deformable systems. The iterative method for solving a system of nonlinear algebraic equations stated in an explicit or implicit form is developed in the present work. The method consists of constructing a sequence of polygons in Euclidean space that converge into a single point that displays the solution of the system. Polygon vertices are determined on the assumption that individual equations of the system are independent from each other and each of them is a function of only one variable. Initial positions of vertices for each subsequent polygon are specified at the midpoints of certain straight segments determined at the previous iteration. The present algorithm is applied for analytical investigation of the behavior of biaxially compressed nonlinear-elastic beam-column with an open thin-walled cross-section. Numerical examples are made for the I-beam-column on the assumption that its material follows a bilinear stress-strain diagram. A computer program based on the shooting method is developed for solving the problem. The method is reduced to numerical integration of a system of differential equations and to the solution of a system of nonlinear algebraic equations between the boundary values of displacements at the ends of the beam-column. A stress distribution at the beam-column cross-sections is determined by subdividing the cross-section area into many small cells. The equilibrium path for the twisting angle and the lateral displacements tend to the stationary point when the load is increased. Configuration of the path curves reveals that the ultimate load is reached shortly once the maximal normal stresses at the beam-column fall outside the limit of the elastic region. The beam-column has a unique equilibrium state for each value of the load, that is, there are no equilibrium states once the maximum load is reached.

Iterating Evolutes and Involutes

This paper concerns iterations of two classical geometric constructions, the evolutes and involutes of plane curves, and their discretizations: evolutes and involutes of plane polygons. In the continuous case, our main result is that the iterated involutes of closed locally convex curves with rotation number one (possibly, with cusps) converge to their curvature centers (Steiner points), and their limit shapes are hypocycloids, generically, astroids. As a consequence, among such curves only the hypocycloids are homothetic to their evolutes. The bulk of the paper concerns two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices (\(\mathcal {P}\)-evolutes), or by the incenters of the triples of consecutive sides (\(\mathcal {A}\)-evolutes). For equiangular polygons, the theory is parallel to the continuous case: we define discrete hypocycloids (equiangular polygons whose sides are tangent to hypocycloids) and a discrete Steiner point. The space of polygons is a vector bundle over the space of the side directions; our main result here is that both kinds of evolutes define vector bundle morphisms. In the case of \(\mathcal {P}\)-evolutes, the induced map of the base is 4-periodic, and the dynamics reduces to the linear maps on the fibers. We prove that the spectra of these linear maps are symmetric with respect to the origin. The asymptotic dynamics of linear maps is determined by their eigenvalues with the maximum modulus, and we show that all types of behavior can occur: in particular, hyperbolic, when this eigenvalue is real, and elliptic, when it is complex. We also study \(\mathcal {P}\)- and \(\mathcal {A}\)-involutes and prove that the side directions of iterated \(\mathcal {A}\)-involutes of polygons with odd number of sides behave ergodically; this generalizes well-known results concerning iterations of the construction of the pedal triangle. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.

Detecting repetitive structures on building footprints for the purposes of 3D modeling and reconstruction

ABSTRACTRepetitive structures of a building share features in terms of geometries and appearance and, therefore, the 3D information for these structures can be transferred from one specification to another for the purpose of 3D modeling and reconstruction once they are identified as repetitive structures. In this paper, a novel approach is proposed for the detection of the repetitive structures specified by the polygons of a building’s footprints. Instead of directly operating on the polygon in 2D space, the polygon is converted into a bend angle function representation in 1D space, whereby an extrusion is represented as a closed polygon intersected by the x-axis and located above it, while an intrusion is represented as a closed polygon below the x-axis. In this way, a polygon of a footprint is decomposed into a number of extrusions and intrusions which can in turn be processed. The task of detecting any repetitive structures specified in a building’s footprints then becomes the task of clustering the intersected polygons in the bend angle function space. The extrusions/intrusions which can be placed in the same clusters can be regarded as repetitive structures. Experiments show that this proposed approach can detect repetitive structures with different sizes, orientations and complexities.

SHAPES OF POLYHEDRA, MIXED VOLUMES AND HYPERBOLIC GEOMETRY

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to .

Pentagrams, inscribed polygons, and Prym varieties

The pentagram map is a discrete integrable system on the moduli space of planar polygons. The corresponding first integrals are so-called monodromy invariants $E_1, O_1, E_2, O_2,\dots$ By analyzing the combinatorics of these invariants, R. Schwartz and S. Tabachnikov have recently proved that for polygons inscribed in a conic section one has $E_k = O_k$ for all $k$. In this paper we give a simple conceptual proof of the Schwartz-Tabachnikov theorem. Our main observation is that for inscribed polygons the corresponding monodromy satisfies a certain self-duality relation. From this we also deduce that the space of inscribed polygons with fixed values of the monodromy invariants is an open dense subset in the Prym variety (i.e., a half-dimensional torus in the Jacobian) of the spectral curve. As a byproduct, we also prove another conjecture of Schwartz and Tabachnikov on positivity of monodromy invariants for convex polygons.

Mobile Robot Path Planning in the Presence of Obstacles and Lack of Information about the Environment

Mobile robot motion is a complex field of research intensively studied during the recent decades. In this article the authors consider the problem of path planning for a mobile robot with two degrees of freedom (ground vehicle) in a priori undetermined environment. The authors formulate the path planning problem and propose a computationally effective algorithm for solving the problem of movement path construction in a priori unknown environment. The proposed algorithm decomposes the path planning problem in two main phases: construction of a graph-based presentation of a free space and graph search. The authors propose a dual graph of a free space polygon triangulation as a means of a free space presentation. This approach allows us to minimize the graph size without an uncontrollable loss of accuracy of environment representation. This approach uses Dijkstra algorithm for the conduct path search on a free space graph, but it is possible to replace this algorithm by another one with account of the particular constraints. The proposed algorithm envisages a continuous collection of the environment information by a robot using its sensors. A brief review of the existing approaches to the problem is presented, as well as a theoretical comparison of these approaches with the proposed one in terms of the computational complexity theory. The proposed algorithm's computational efficiency is demonstrated via a computational experiment, where the proposed and reference algorithms operate on a set of synthetic environment models of different geometric size, but with a similar structure. An experiment proved the algorithm's supremacy in speed.

Affine invariants of generalized polygons and matching under affine transformations

A generalized polygon is an ordered set of vertices. This notion generalizes the concept of the boundary of a polygonal shape because self-intersections are allowed. In this paper we study the problem of matching generalized polygons under affine transformations. Our approach is based on invariants. Firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. These invariant functions are defined as quotients of the so-called Fourier descriptors, also known as discrete Fourier transforms.Each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. Moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.More formally, given a collection {Z1,Z2,…,Zm} of n-sided polygons in the plane and a query polygon W, we give algorithms to find all Zℓ such that f(Zℓ)=W+ΔW, where f is an unknown affine transformation and ΔW=(Δw1,…,Δwn) with |Δwk|≤ρ, where ρ is certain tolerance.

Topological complexity of planar polygon spaces with small genetic code

Abstract We determine lower bounds for the topological complexity of many planar polygon spaces mod isometry. With very few exceptions, the upper and lower bounds given by dimension and cohomology considerations differ by 1. This is true for 130 of the 134 generic 7-gon spaces. Our results apply to spaces of n-gons for all n, but primarily for those whose genetic codes, in the sense of Hausmann and Rodriguez, are moderately small.

A fast direct sampling algorithm for equilateral closed polygons

Sampling equilateral closed polygons is of interest in the statistical study of ring polymers. Over the past 30 years, previous authors have proposed a variety of simple Markov chain algorithms (but have not been able to show that they converge to the correct probability distribution) and complicated direct samplers (which require extended-precision arithmetic to evaluate numerically unstable polynomials). We present a simple direct sampler which is fast and numerically stable, and analyze its runtime using a new formula for the volume of equilateral polygon space as a Dirichlet-type integral.

Fast and Exact Fiber Surfaces for Tetrahedral Meshes.

Isosurfaces are fundamental geometrical objects for the analysis and visualization of volumetric scalar fields. Recent work has generalized them to bivariate volumetric fields with fiber surfaces, the pre-image of polygons in range space. However, the existing algorithm for their computation is approximate, and is limited to closed polygons. Moreover, its runtime performance does not allow instantaneous updates of the fiber surfaces upon user edits of the polygons. Overall, these limitations prevent a reliable and interactive exploration of the space of fiber surfaces. This paper introduces the first algorithm for the exact computation of fiber surfaces in tetrahedral meshes. It assumes no restriction on the topology of the input polygon, handles degenerate cases and better captures sharp features induced by polygon bends. The algorithm also allows visualization of individual fibers on the output surface, better illustrating their relationship with data features in range space. To enable truly interactive exploration sessions, we further improve the runtime performance of this algorithm. In particular, we show that it is trivially parallelizable and that it scales nearly linearly with the number of cores. Further, we study acceleration data-structures both in geometrical domain and range space and we show how to generalize interval trees used in isosurface extraction to fiber surface extraction. Experiments demonstrate the superiority of our algorithm over previous work, both in terms of accuracy and running time, with up to two orders of magnitude speedups. This improvement enables interactive edits of range polygons with instantaneous updates of the fiber surface for exploration purpose. A VTK-based reference implementation is provided as additional material to reproduce our results.

Mutants of compactified representations revisited

We show that the mutants of compactified representations constructed by Franz and Puppe can be written as intersections of real quadrics involving division algebras and as generalizations of polygon spaces. We also show that these manifolds are connected sums of products of spheres.

Topological complexity of some planar polygon spaces

Let \({\overline{M}}_{n,r}\) denote the space of isometry classes of n-gons in the plane with one side of length r and all others of length 1, and assume that \(1\le r<n-3\) and \(n-r\) is not an odd integer. Using known results about the mod-2 cohomology ring, we prove that its topological complexity satisfies \({\text {TC}}({\overline{M}}_{n,r})\ge 2n-6\). Since \({\overline{M}}_{n,r}\) is an \((n-3)\)-manifold, \({\text {TC}}({\overline{M}}_{n,r})\le 2n-5\). So our result is within 1 of being optimal.

The symplectic geometry of closed equilateral random walks in 3-space

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^3$. We study closed random walks using the symplectic geometry of the $(2n-6)$-dimensional quotient of the manifold of polygons by the action of the rotation group $\operatorname {SO}(3)$. The basic objects of study are the moment maps on equilateral random polygon space given by the lengths of any $(n-3)$-tuple of nonintersecting diagonals. The Atiyah-Guillemin-Sternberg theorem shows that the image of such a moment map is a convex polytope in $(n-3)$-dimensional space, while the Duistermaat-Heckman theorem shows that the pushforward measure on this polytope is Lebesgue measure on $\mathbb{R}^{n-3}$. Together, these theorems allow us to define a measure-preserving set of "action-angle" coordinates on the space of closed equilateral polygons. The new coordinate system allows us to make explicit computations of exact expectations for total curvature and for some chord lengths of closed (and confined) equilateral random walks, to give statistical criteria for sampling algorithms on the space of polygons and to prove that the probability that a randomly chosen equilateral hexagon is unknotted is at least $\frac{1}{2}$. We then use our methods to construct a new Markov chain sampling algorithm for equilateral closed polygons, with a simple modification to sample (rooted) confined equilateral closed polygons. We prove rigorously that our algorithm converges geometrically to the standard measure on the space of closed random walks, give a theory of error estimators for Markov chain Monte Carlo integration using our method and analyze the performance of our method. Our methods also apply to open random walks in certain types of confinement, and in general to walks with arbitrary (fixed) edgelengths as well as equilateral walks.