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.
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.
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.
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…
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.
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.
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.
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…
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.
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
The study of polygon areas with fixed perimeter.
problem Finding the minimum number of critical points for polygon areas.
method Analysis of the configuration space and critical points of the area function.
result Computed indices of critical points (regular stars) on the configuration space.
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.
In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this class of polygons, we define discrete versions of the area evolute, central symmetr…
In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygo…
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.
The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.
problem Finding critical configurations and their properties for polygons on circles or ellipses.
method Computing Morse indices and gradient vector fields for isolated critical points, relating to eigenvalue questions.
result Computed Morse indices and relationships to eigenvalue questions for polygons on circles or ellipses.
We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are co…
It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex polygons in the Lorentzian plane such that their moduli space, if the normals are fixed…
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or…
Affine λ-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.
problem Reconstruction and area estimates for affine λ-equidistants of convex polygons with parallel opposite sides. method Using Wigner caustics and centre symmetry sets.
result Proving a discrete version of the improved isoperimetric inequality.
We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral polygons (which appear as Morse points) are interpreted as independent generato…
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.
A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic, respectively. Convex such n-gons are parametrized by the subspaces of (0,∞)n that contain their side length collections,…
The oriented area function A 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 A i…
Study three discrete envelope types of polygon bisection lines.
problem Understanding different envelope types of polygon bisection lines.
method Examined three distinct notions of discrete envelopes.
result Connected three different notions of discrete envelopes.
Study of hyperbolic polyhedral surfaces with regular faces.
problem Understanding the properties of hyperbolic polyhedral surfaces with regular faces.
method Combinatorial and geometric analysis of hyperbolic polyhedral surfaces with regular faces.
result There is a gap between the areas of non-smooth hyperbolic polyhedral surfaces and smooth hyperbolic surfaces.
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the existence of three different types of circles in the hyperbolic plane. The second …
Lecture notes introduce Abelian differentials and their flat surfaces, focusing on families and Teichmüller dynamics.
problem Study of Abelian differentials and their geometric properties.
method Associate flat surfaces to Abelian differentials and analyze their families under GL2+(R) action. result Properties of orbit of Abelian differentials under Teichmüller dynamics.
We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant b on the space of those isometric deformations which, for conv…
This paper proves that for large n, the regular polygon minimizes the first eigenvalue of the Laplacian.
problem Finding the polygon with the smallest first eigenvalue of the Laplacian for a given area.
method Constructing polygonal manifolds and using spectral theory, tensor calculus, and symmetrization techniques.
result For large n, the regular polygon minimizes the first eigenvalue of the Laplacian.
The paper finds minimal-area metrics on polygons with multiple crossing geodesics.
problem Finding minimal-area metrics on polygons with specific geodesic conditions.
method Applied convex programs to polygons with length conditions on curves.
result The extremal metric coincides with the conformal extremal metric on regular polygons.
The paper studies stability of discrete planar curves using variational methods.
problem Stability of discrete planar curves under area constraints.
method Unified interpretation of discrete curvatures, determination of equilibrium curves, stability analysis.
result Equilibrium curves for the length functional under area-constraint conditions are determined and their stability is studied.
New geometric mechanics approach to elastic curves.
problem Understanding elastic curves in mechanics and geometry.
method Developed a new geometric mechanics perspective on elastic curves.
result Elastic curves are critical points of length under fixed area and volume constraints.
In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to s…
Let α be a polygonal Jordan curve in $\bfR^3$. We show that if α satisfies certain conditions, then the least-area Douglas-Radó disk in $\bfR^3$ with boundary α is unique and is a smooth graph. As our conditions on α are not included amongst previously known conditions for embeddedness, we are enlarging the set…
Transformed geometry into algebra to prove Pick's theorem efficiently.
problem Translating geometric Pick's theorem into formal algebraic proof.
method Formalized geometric Pick's theorem into algebraic proof using Lean.
result Efficient formal proof of Pick's theorem.
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.
Study examines the excluded area between two-dimensional hard particles, identifying key factors affecting its magnitude.
problem Determining the excluded area between two-dimensional hard particles with various orientations and shapes.
method Used principal component analysis and Monte Carlo simulations to analyze randomly generated non-self-intersecting polygons and star lines.
result The minimum excluded area is achieved when particles are antiparallel, and elongation of the particle shape significantly affects the excluded area.
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.
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.
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.
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…
A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of 32π but less than 2π. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result give…
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.