The article studies polygon flows and their asymptotic behavior.
problem Investigating the behavior of polygon flows over time.
method Analogous to curve shortening flow, the article analyzes the β-polygon flow. result Regular polygons with five or more vertices are asymptotically stable.
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…
Harmonic maps from complex plane to hyperbolic space constructed using heat flow.
problem Constructing harmonic maps from complex plane to hyperbolic space.
method Heat flow method to construct harmonic maps.
result Harmonic maps are unique once the principal part of their Hopf differential is prescribed.
Study shows bi-Hamiltonian structure for polygon evolutions in centro-affine space.
problem Understanding bi-Hamiltonian structure for polygon evolutions.
method Geometric realizations of discrete flows, lifting to pre-symplectic forms.
result Compatibility of two Hamiltonian structures proved straightforward.
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
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…
We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.
We study the moduli spaces of polygons in R^2 and R^3, identifying them with subquotients of 2-Grassmannians using a symplectic version of the Gel'fand-MacPherson correspondence. We show that the bending flows defined by Kapovich-Millson arise as a reduction of the Gel'fand-Cetlin system on the Grassmannian, and with t…
Characterizes how the shape of a polygon affects billiard dynamics.
problem Understanding how the shape of a billiard table influences its dynamics.
method New theorem linking Liouville current support to flat cone metrics.
result Only right-angled tables with affine differences have identical bounce spectra.
A nice trick for studying the billiard flow in a rational polygon is to unfold the polygon along the trajectories. This gives rise to a translation or half-translation surface tiled by the original polygon, or equivalently an Abelian or quadratic differential. Veech surfaces are a special class of translation surfaces …
Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for t < T.
We study the Hamiltonian vector field v=(−∂f/∂w,∂f/∂z) on C2, where f=f(z,w) is a polynomial in two complex variables, which is non-degenerate with respect to its Newton's polygon. We introduce coordinates in four-dimensional neighbourhoods of the "points at infinity", in …
The paper studies circle packings on surfaces with boundary and their total geodesic curvatures.
problem Existence and rigidity of circle packings with conical singularities.
method Variational principle and combinatorial Ricci flow.
result Existence and rigidity of circle packings with prescribed total geodesic curvature.
Any two equivalent discrete curves must have the same invariants at the corresponding points under an affine transformation. In this paper, we construct the moving frame and invariants for the discrete centroaffine curves, which could be used to discriminate the same discrete curves from different graphics, and estimat…
Mirzakhani connects earthquake and Teichmuller flows on surfaces.
problem Understanding the relationship between earthquake and Teichmuller flows.
method Geometric account of connections between flows, avoiding technical prerequisites.
result Mirzakhani's theorem relating earthquake and Teichmuller flows.
Study of straight-line flows on a unique infinite surface.
problem Understanding straight-line flows on a specific infinite surface.
method Geometric description and characterization of periodic and drift orbits; use of rigid symmetries and Veech group.
result Complete characterization of periodic directions and proof of density of periodic and ergodic directions.
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.
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.
The paper explores centroaffine geometry of polygons and their duals.
problem Understanding centroaffine dual pairs of spatial polygons.
method Defining centroaffine dual pairs and proving properties of polygon duals.
result Constant curvature polygons are dual to planar polygons.
Foliation of star-shaped polygons with fixed perimeter and area.
problem Characterizing star-shaped polygons with fixed perimeter and area.
method Analyzing families of star-shaped n-polygons in the Euclidean plane.
result Existence and properties of foliations on the space of star-shaped n-polygons.
The pentagram map preserves Poncelet polygons in convex cases.
problem Characterizing Poncelet polygons using the pentagram map.
method Theory of commuting difference operators, properties of real elliptic curves, and theta functions.
result A convex polygon is Poncelet if and only if it is projectively equivalent to its pentagram image.
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.
Study of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.
For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the…
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.
Paper characterizes critical points of polygon areas, focusing on cocyclic polygons.
problem Characterizing critical points of polygon areas on manifolds of fixed side lengths.
method Alternative demonstration using cocyclic polygons, numerical gradient descent scheme.
result Cocyclic polygons are critical points of polygon areas on manifolds of fixed side lengths.
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 C-polygon is between n and 2(n−1)+m for a strictly convex domain with m singular boundary points. 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 l1-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexe…
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…
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.
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.
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 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.
Study on polygons with fixed edge slopes and their perimeter function.
problem Characterizing and analyzing polygons with prescribed edge slopes.
method Configuration space description and perimeter as a Morse function.
result Characterization and computation of critical points and their Morse indices.
Geometrically describes polygon space cohomology rules.
problem Understanding cohomology of polygon spaces.
method Two geometrically meaningful presentations of cup product rules.
result Simple rules for cup product in polygon spaces.
Oriented area function is a perfect Morse function for polygonal linkages.
problem Understanding the topology of polygonal linkages.
method Generalization of oriented area function as a Morse function.
result Cyclic equilateral polygons are independent generators of configuration space's homology.
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…
Investigates dual foliations of polygon spaces based on area and perimeter.
problem Understanding dual foliations of polygon spaces guided by area and perimeter.
method Investigated topology of leaves, determined homology groups, and extended isoperimetric duality.
result Homology groups and homotopy types of polygon spaces are determined.
Maps complex plane polynomials to light-like polygons in Einstein Universe.
problem Mapping between complex plane polynomials and light-like polygons.
method Constructs geometric homeomorphism between moduli spaces.
result Found minimal Lagrangian maps between ideal polygons.
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 Ω(h1/3) and O(h1/2) for convex polygons. 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.
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.
The study of equal-volume polygons in 3D space and their affine invariants.
problem Estimating projective invariants of planar curves.
method Developing a theory of discrete affine invariants from equal-volume polygons.
result Equal-volume polygons can be used to estimate projective invariants of a planar curve.
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 2log(n). result Optimal orthogeodesic decomposition of ideal polygons with length 2log(n). Let X be a toric surface and u be a normalized symplectic potential on the corresponding polygon P. Suppose that the Riemannian curvature is bounded by a constant C1 and ∫∂Pu dσ<C2, then there exists a constant C3 depending only on C1,C2 and P such that the diameter of X is b…
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…