Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic -metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…
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48 results for “simple polygons”
Study the space of simple polygons and their moduli.
problem Understanding the space of simple polygons and their moduli.
method Local and global descriptions of S(n) and M(n) using Morse theory and asymptotic analysis.
result Completely described S(4) and M(4) using a topological Morse function.
New method finds lattice polygons that can be dissected into triangles with integer areas.
problem Finding lattice polygons that can be dissected into triangles with integer areas.
method A new version of Sperner's Lemma.
result Simple and complete description of lattice polygons that can be dissected into triangles with integer areas.
We give a few simple methods to geometically describe some polygon and chain-spaces in R^d. They are strong enough to give tables of m-gons and m-chains when m <= 6.
Extremal polygons in R^3math.GT
The oriented area function is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function i…
We describe the cohomology ring of the moduli space of a flexible polygon in geometrically meaningful terms. We propose two presentations, both are computation friendly: there are simple rules for cup product.
New property: polygons have a fixed dimension regardless of ambient space dimensions.
problem Understanding the dimension of polygon moduli spaces.
method Generalizing the square bending example to polygons of arbitrary edge lengths.
result There are only finitely many moduli spaces of polygons with given edge lengths, even as ambient dimension increases.
A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. W…
Dancing polygons and rolling balls linked via a special geometric distribution.
problem Understanding the geometric and mechanical relationship between dancing polygons and rolling balls.
method Mapping dancing polygons to trajectories of a rolling ball on a 3D surface, both described by a specific geometric distribution.
result Non-degenerate dancing pairs of polygons exist for all and correspond to rolling ball trajectories.
The abstract proves polygon inscriptions in curves with specific edge ratios.
problem Proving the existence of polygons inscribed in Jordan curves with prescribed edge ratios.
method Using the properties of differentiable curves and proportional side lengths.
result Existence of polygons inscribed in Jordan curves with prescribed edge ratios.
Geometric models for Lie algebras from simple singularities.
problem Classifying simply-laced simple Lie algebras.
method Using polygonal wheels derived from Milnor fibers of simple singularities.
result Geometric root systems are isomorphic to Lie algebras.
The space of polygons up to similarity is studied using the Schwarz-Christoffel formula.
problem Understanding the space of polygons up to similarity.
method Using the Schwarz-Christoffel formula to show homeomorphism and proving embeddings for specific polygons.
result The space of labelled simple polygons up to similarity is homeomorphic to R^(2n-4) for n=3,4,5.
Presented a simple group presentation for degree four cactus group.
problem Presented a simple group presentation for the pure cactus group of degree four.
method Action on hyperbolic plane, Dirichlet polygon construction.
result Isomorphic to the fundamental group of connected sum of five real projective planes.
The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K. Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon. This may be a different knot type…
Proof of Knot Entropy Conjecture for tube lattice polygons.
problem Proving exponential growth rate of knot polygons equals unknot polygons.
method Upper and lower bounds on polygon counts, braid insertions, and pattern theorems.
result Established the Knot Entropy Conjecture for tube lattice polygons.
Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…
Starting by a simple game as a combinatorial data, we build up a cell complex , whose construction resembles combinatorics of the permutohedron. The cell complex proves to be a combinatorial manifold; we call it the \textit{ simple game induced manifold.} By some motivations coming from polygonal linkages, w…
We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately minimize the ropelength (or quotient of length and thickness) for polygons in their kno…
I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve. This result is obtained by approximating the curve with polygons and invoking the r…
Study geometrically characterizes piecewise circular curves with decreasing curvature.
problem Characterizing piecewise circular curves with decreasing curvature.
method Introducing moduli spaces and relating them to Legendrian polygons.
result Proves the moduli space contains a connected component homeomorphic to the Fock-Goncharov space of positive flags.
New methods classify convex lattice polygons for affine dimers.
problem Not all convex lattice polygons are characteristic polygons of affine dimers.
method General constructions and algorithm for finding affine dimers with prescribed polygons.
result All lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.
Exact diameter found for some Riemann surfaces.
problem Calculating the diameter of compact Riemann surfaces exactly.
method Proved for a specific class of surfaces (generalized Bolza surfaces).
result Diameters of generalized Bolza surfaces are equal to their fundamental polygon radii.
The pentagram map's limit point is related to infinitesimal perturbations of polygons.
problem Understanding the limit point of the pentagram map and its relation to polygon perturbations.
method Interpreting Glick's operator as the infinitesimal monodromy of a polygon.
result Glick's operator measures the extent to which a perturbed polygon does not close up.
General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…
Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
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 edges is the -dimensional Riemannian manifold of equilateral closed polygons in …
The paper classifies vertices in planar polygons formed by convex domains.
problem Classifying vertices in planar polygons formed by convex domains.
method Analyzing polygons formed by homothets and translates of a convex domain.
result The number of singular boundary points in a -polygon is between and for a strictly convex domain with singular boundary points.
In this paper, we discuss centroaffine geometry of polygons in -space. For a polygon that is locally convex with respect to an origin together with a transversal vector field , we define the centroaffine dual pair similarly to [6]. We prove that vertices of correspond to flattening points for …
We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…
The pentagram map takes a planar polygon to a polygon whose vertices are the intersection points of consecutive shortest diagonals of . This map is known to interact nicely with Poncelet polygons, i.e. polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of R. Sc…
Optimal Reeb graphs identified for polygon decomposition.
problem Investigating the topological structure of planar polygon decomposition.
method Using oriented Reeb graphs with a marked vertex for height functions.
result Described all possible optimal Reeb graphs for specific polygon configurations.
Study on Poncelet polygons' centers and circumcenters in various geometries.
problem Understanding Poncelet polygons' geometric centers in different geometries.
method Analyzing the Circumcenter of Mass and Center of Mass of Poncelet polygons, proving Dan Reznik's invariants, and exploring spherical geometry.
result Proof of Dan Reznik's invariants for billiard trajectories and insights into Poncelet polygons' centers in spherical geometry.
The study proves analogues of the discrete isoperimetric inequality in hyperbolic geometry.
problem Finding the minimum perimeter for polygons with a fixed area in hyperbolic geometry.
method Proving analogues of the discrete isoperimetric inequality for cyclic and tangential polygons in hyperbolic geometry, considering both single and multiple polygons.
result Established two versions of the isoperimetric inequality for multiple polygons in hyperbolic geometry with certain area or perimeter restrictions.
New formula for spherical polygon area via prequantization.
problem Traditional area formula for spherical polygons requires measuring angles.
method Uses prequantization to create a new formula that doesn't require angle measurement.
result New formula applicable to a wider range of degenerate curves and polygons.
The map S transforms polygon sides, and almost no convex polygons remain convex.
problem Investigating whether convex polygons remain convex under the map S.
method Analyzing the dynamics of the map S and proving properties of the set of polygons that remain convex.
result The set of polygons that remain convex under iterations of S has measure zero and is an algebraic subvariety of codimension two.
In this article we investigate a family of nonlinear evolutions of polygons in the plane called the -polygon flow and obtain some results analogous to results for the smooth curve shortening flow: (1) any planar polygon shrinks to a point and (2) a regular polygon with five or more vertices is asymptotically stable …
The study proves a discrete Blaschke theorem for convex polygons in 2-dimensional space forms.
problem Investigating curvature and circumradius constraints for convex polygons in 2-space forms.
method Defining curvature at each vertex and proving a Blaschke-type theorem.
result The circumradius of a convex polygon satisfies a specific inequality related to its vertex curvatures.
We study polygon spaces arising from planar configurations of necklaces with some of the beads fixed and some of the beads sliding freely. These spaces include configuration spaces of flexible polygons and some other natural polygon spaces. We characterise critical points of the oriented area function in geometric term…
Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of con…
Characterizes polygonal surfaces in pseudo-hyperbolic spaces.
problem Understanding polygonal surfaces in pseudo-hyperbolic spaces.
method Characterizes polygonal surfaces by total curvature finiteness and asymptotic flatness, using comparison of ideal boundaries.
result Polygonal surfaces have parabolic type and polynomial quartic differential.
Maximal distortion between geodesic and Euclidean diameters in polygonal domains is studied.
problem Maximal ratio of geodesic to Euclidean diameters in polygonal domains with holes.
method Analyzes convex polygons with holes, using geometric triangulations as a comparison.
result The supremum of the ratio is between and for convex polygons.
Solitons are special polygon midpoints under affine transformations.
problem Characterizing polygons whose midpoints under affine transformations form a new polygon.
method Analyzing midpoints polygons and their relationship to affine transformations and differential equations.
result A large class of polygons are on an orbit of a one-parameter subgroup of the affine group, and these curves are solutions to a specific differential equation.
Short proof for ideal polygons with near optimal orthogeodesic decomposition.
problem Decomposing ideal polygons into orthogeodesics.
method Short proof with orthogeodesic decomposition of length at most .
result Optimal orthogeodesic decomposition of ideal polygons with length .
We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…
In this paper, we study the symplectic volume of the moduli space of polygons by using Witten's formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the Spherical and Euclidean polygons with fixed perimeter the regular one…
The study connects polygon areas and projective structures in 3D space.
problem Relating polygon areas and projective structures in 3D space.
method Investigates positive tuples of complete flags in R^3 and their associated polygons in RP^2.
result Establishes a relationship between Holmes-Thompson area and projective structures.
Fast algorithm samples confined polygons efficiently.
problem Sampling confined random equilateral closed polygons efficiently.
method Uses symplectic geometry to sample moment polytope, leading to a linear-time algorithm.
result Explicit formulas for expected distances and total curvature of vertices to the origin.
Researchers prove the arc complexes of decorated hyperbolic polygons are balls.
problem Understanding the structure of decorated hyperbolic polygons.
method Combinatorial approach using pseudo-manifolds and shellability.
result Arc complexes of decorated hyperbolic polygons are closed piecewise linear balls.