Polygonal Space Research Articles

Moduli Spaces of Polygons and Punctured Riemann Spheres

AbstractThe purpose of this note is to give a simple combinatorial construction of the map from the canonically compactified moduli spaces of punctured complex projective lines to the moduli spaces 𝓟r of polygons with fixed side lengths in the Euclidean space 𝔼3. The advantage of this construction is that one can obtain a complete set of linear relations among the cycles that generate homology of 𝓟r. We also classify moduli spaces of pentagons.

Complexity and Modeling Aspects of Mesh Refinement into Quadrilaterals

We investigate the following mesh refinement problem: Given a mesh of polygons in three-dimensional space, find a decomposition into strictly convex quadrilaterals such that the resulting mesh is conforming and satisfies prescribed local density constraints. The conformal mesh refinement problem is shown to be feasible if and only if a certain system of linear equations over GF(2) has a solution. To improve mesh quality with respect to optimization criteria such as density, angles, and regularity, we introduce a reduction to a minimum cost bidirected flow problem. However, this model is only applicable if the mesh does not contain branching edges, that is, edges incident to more than two polygons. The general case with branchings, however, turns out to be strongly ${\cal NP}$ -hard. To enhance the mesh quality for meshes with branchings, we introduce a two-stage approach which first decomposes the whole mesh into components without branchings, and then uses minimum cost bidirected flows on the components in a second phase.

Predicting geometrical properties of random packed beds from computer simulation

AbstractUsing advances in computational geometry and collision‐detection algorithms, an algorithm was developed to analyze and predict the geometrical properties of a randomly packed structure using packing objects of arbitrary shape. A 3‐D polygonal model was used to describe an arbitrarily complex packing object and a simple container object. The dynamics of the packing process is not simulated, but a search algorithm finds stable equilibrium positions for the packing objects of arbitrary shapes using a collision‐detection algorithm in a 3‐D space. A modified conjugate gradient optimization method determines the packing object's final packing location and orientation. Once the bed is packed, both macroscopic quantities like the overall porosity, the specific surface area, and the number of packing objects per unit volume and microscopic properties like the porosity variation in any direction could be determined. For accurate porosity calculation inside a given 3‐D polygonal sample space, the Sutherland‐Hodgman polygon clipper algorithm was used. Predicted results are validated against available experimental data for spheres, Raschig rings, Pall rings, and cascade minirings.

Frequency relationship between levinson plates and classical thin plates

An exact frequency relationship between Levinson plate theory and Kirchoff plate theory for isotropic plates of general polygonal space and simply supported edges is presented. Levinson plate vibration solutions may be readily obtained from the known Kirchoff solutions.

The fundamental group of the moduli space of polygons in the Euclidean plane

The fundamental group of the moduli space of polygons in the Euclidean plane

The Moduli Space of Boundary Compactifications of SL.2;R/

In an earlier paper, the authors introduced the notion of a boundary compactification of SL(2, R) and SL(2, C), a normal projective embedding of PSL2 arising as the Zariski closure of an orbit in (P1)n under the diagonal action of SL2. Here the moduli space of such boundary compactifications of SL(2, R) is shown to be a contractible hyperbolic orbifold, by using the Schwarz–Christoffel transformation to identify it with a quotient of the moduli space of equi-angular planar polygons.

On Linear Representations of Near Hexagons

We discuss linear representations of near polygons in affine spaces. All linear representations of near hexagons in an affine space of orderq≥ 3 and dimension up to seven are classified. If the dimension of the affine space is at least eight, then the near hexagon necessarily contains a quad of typeT*2(O) and every such quad has a rosette of ovoids. We conjecture that there are no such examples.

Remarks on the topology of spatial polygon spaces

Let Mn be the “polygon space” introduced by Kirwan and Klyachko. In this paper, we give new results on the topology of Mn for odd n. We determine πq(Mn) (q ≤ n − 3). Then we describe Mn in the oriented cobordism ring . We also give new and elementary proofs of the result on the ring structure of H*(Mn/Sn; Q), where Sn denotes the symmetric group acting naturally on Mn.

An algorithm for determining intersection segment-polygon in 3D

In the fields related to geometric modelling, it is fundamental to count on robust. In this work we present a new algorithm for testing the intersection between a segment and a polygon in 3D space. This algorithm is robust and efficient, and it is the basis of most operations developed in geometric modelling. After, we will apply this algorithm to a classic problem in computer graphics: determining the hidden faces of a scene using the ray-casting algorithm. To solve the problem, an object-oriented approach will be used.

Topology of the Configuration Space of Polygons as a Codimension One Submanifold of a Torus

We study the topology of polygons with fixed side length in Euclidean plane by Morse theory. §

The Homology of Singular Polygon Spaces

AbstractLet Mn be the variety of spatial polygons P = (a1, a2, … ,an) whose sides are vectors ai ∈ R3 of length |ai| = 1 (1 ≤ i ≤ n), up to motion in R3. It is known that for odd n, Mn is a smooth manifold, while for even n, Mn has cone-like singular points. For odd n, the rational homology of Mn was determined by Kirwan and Klyachko [6], [9]. The purpose of this paper is to determine the rational homology of Mn for even n. For even n, let be the manifold obtained from Mn by the resolution of the singularities. Then we also determine the integral homology of .

Efficient Algorithms for Computing a Complete Visibility Region in Three-Dimensional Space

Given a set P of polygons in three-dimensional space, two points p and q are said to be visible from each other with respect to P if the line segment joining them does not intersect any polygon in P . A point p is said to be completely visible from an area source S if p is visible from every point in S . The completely visible region CV(S, P) from S with respect to P is defined as the set of all points in three-dimensional space that are completely visible from S .

The cohomology ring of polygon spaces

We compute the integer cohomology rings of the ``polygon spaces'' introduced in [Hausmann,Klyachko,Kapovich-Millson]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Gr\obner bases. Since we do not invert the prime 2, we can tensor with Z/2; halving all degrees we show this produces the Z/2 cohomology rings of planar polygon spaces. In the equilateral case, where there is an action of the symmetric group permuting the edges, we show that the induced action on the integer cohomology is _not_ the standard one, despite it being so on the rational cohomology [Kl]. Finally, our formulae for the Poincar\'e polynomials are more computationally effective than those known [Kl].

96/05142 General shading model for solar building design

This paper describes a general principle for accurate calculation of the effect of shading objects on surfaces exposed to direct solar radiation. The principle used is known as polygon clipping, a technique for determining every subpolygon that arises when two or more polygons overlap one another. In this case, the obstructing objects are approximated by polygons in space, projected as seen from the sun, at any given time of the year, onto the exposed surface of interest, so that every sunlit region and the fully or partly shaded regions of the plane surface can be determined. The technique has been implemented in a PC software application, which can function as a stand-alone design tool or may be integrated with programs for thermal simulation of buildings or solar systems. The paper gives examples of the use of application, shows its functionality when integrated with a thermal simulation tool, and presents comparative validations against the standard ASHRAE algorithms for calculations of overhangs and side fins.

The symplectic geometry of polygons in Euclidean space

The symplectic geometry of polygons in Euclidean space

Topology of equilateral polygon linkages

We study the topology of moduli spaces of polygons with fixed side lengths in the Euclidean plane and space. We study their smoothness, then determine their Euler numbers.

Regular polygons in Euclidean space

Regular polygons in Euclidean space

On the geometry of contact formation cells for systems of polygons

The efficient planning of contact tasks for intelligent robotic systems requires a thorough understanding of the kinematic constraints imposed on the system by the contacts. In this paper, we derive closed-form analytic solutions for the position and orientation of a passive polygon moving in sliding and rolling contact with two or three active polygons whose positions and orientations are independently controlled. This is accomplished by applying a simple elimination procedure to solve the appropriate system of contact constraint equations. We also prove that the set of solutions to the contact constraint equations is a smooth submanifold of the system's configuration space and we study its projection onto the configuration space of the active polygons. By relating these results to the wrench matrices commonly used in grasp analysis, we discover a previously unknown and highly nonintuitive class of nongeneric contact situations. In these situations, for a specific fixed configuration of the active polygons, the passive polygon can maintain three contacts on three mutually nonparallel edges while retaining one degree of freedom of motion. >

On the moduli space of polygons in the Euclidean plane

On the moduli space of polygons in the Euclidean plane

On the moduli space of polygons in the Euclidean plane

On the moduli space of polygons in the Euclidean plane