The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.
problem Finding bounds on intersections and inflections for spherical polygons.
method Adapting smooth curve theorems to spherical polygons using discrete tools.
result Proves discrete analogs of four-vertex theorems for spherical polygons.
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…
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.
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…
The paper calculates the Euler characteristic of regular spherical polygon spaces.
problem Determining the Euler characteristic of regular spherical polygon spaces.
method Constructing a manifold Xn and a function μ:XnoR such that μ−1(a)=Mn(a), determining the index of critical points, and using Morse surgeries. result Calculating the Euler characteristic χ(Mn(a)) for all a and odd n. 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…
The abstract proves spherical surface decompositions with conical singularities.
problem Decomposing surfaces with spherical metrics and conical singularities.
method Geometric triangulations and irreducible components of standard shapes.
result Spherical polygons, including half-spherical concave polygons, can be arbitrarily complicated.
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.
Study on least symmetric triangles in spherical geometry.
problem Finding the least symmetric triangle, both in planar and molecular contexts.
method Using Grassmannian correspondence and hyperoctahedral group action, compute the furthest point from the boundary in the Grassmannian.
result Exact computation of least symmetric triangles, including obtuse and acute types.
The study proves a discrete version of Segre's theorem for polygonal curves.
problem Proving a discrete analog of a four-vertex theorem for spherical curves.
method Using the concept of discrete tangent indicatrix of a polygon.
result A polygon with at least four vertices and a non-self-intersecting discrete tangent indicatrix has at least four flattenings.
As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued s…
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…
A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, t…
Defines state sum models with defects in 3-manifolds.
problem Detecting and characterizing defects in 3-manifolds.
method Turaev-Viro-Barrett-Westbury state sum models with defects labeled by bimodule categories and functors.
result State sums are triangulation-independent and can be computed using polygon diagrams.
Paper finds configurations for spherical curves with reductivity four and constructs a reduced curve without certain types of polygons.
problem Unknown configurations for spherical curves with reductivity four and reduced curves without specific polygon types.
method Focused on 5-gons to find unavoidable sets for spherical curves with reductivity four. Constructed a reduced spherical curve without certain types of polygons.
result Found configurations for spherical curves with reductivity four and constructed a reduced curve without specific polygon types.
The study proves rigidity and non-rigidity of spherical caps in mean curvature.
problem Understanding mean curvature rigidity and non-rigidity on spherical caps.
method Used a Tangency Principle to prove rigidity and constructed counterexamples for non-rigidity.
result Contrast between rigidity and non-rigidity phenomena on spherical caps.
Study spectral invariants for polygons and orbisurfaces.
problem Compute spectral invariants for polygons and orbisurfaces.
method Study the heat kernel and asymptotic expansion of the heat trace.
result Explicit formulas for heat invariants of polygons and orbisurfaces.
Local corner-factor conjecture for Neumann jump determinants supported by models.
problem Determining the determinant of Neumann jump operator on piecewise curves.
method Formulated conjecture, supported by three model calculations, and discussed connections.
result Support for the conjectural determinant formula \(\Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^Nα_j^{-1/2}\).
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 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.
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.
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.
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.
A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex Arc(F) consisting of suitable equivalence classes of arcs in F connecting its boundary components. The m…
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.
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.
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.
This paper studies constrained polygonal linkages and their configuration spaces.
problem Understanding the configuration spaces of constrained polygonal linkages.
method The paper uses Bott-Morse functions and critical point analysis to study the configuration spaces.
result The oriented area is a Bott-Morse function with computed indices.
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.
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.
The paper connects polygon spaces with quotient spaces using spin actions and normed division algebras.
problem Understanding correspondences between polygon spaces and quotient spaces.
method Introducing Hopf maps and spin actions on normed division algebras to construct correspondences.
result Extension of polygon space correspondences to higher dimensions and normed division algebras.
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.
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.
We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon s…
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.
In this paper, certain natural and elementary polygonal objects in Euclidean space, {\it the stable polygons}, are introduced, and the novel moduli spaces ${\bfmit M}_{{\bf r}, ε}$ of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and i…
Study of evolutes of polygons and curves in higher dimensions.
problem Understanding evolutes of spatial polygons and curves in higher dimensions.
method Analyzing iterations of evolute transformations and studying properties of evolutes for polygons and curves.
result Eigenvalues of the second evolute map have double multiplicity, and evolutes of certain curves are homothetic to the curves themselves.
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.
Proofs contractibility of geodesic triangulations spaces and non-trivial homotopy groups.
problem Contractibility and homotopy groups of geodesic triangulations.
method Short proofs and existence proofs for specific cases.
result Existence of polygon triangulations with non-trivial nth homotopy groups.
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…
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.
In this paper we show that space of spatial polygons in semi riemann space gives a Kahler manifold. We describe the tangent space and almost complex structure which has many computational advantages.
We prove diffeomorphisms of polygonal linkage moduli spaces to Euclidean spaces.
problem Moduli spaces of self-avoiding polygonal linkages and configurations.
method Construct Lyapunov-Reeb functions to show diffeomorphisms.
result Moduli spaces are diffeomorphic to Euclidean spaces.
We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.
Study of self-dual polygons in higher dimensions, including explicit constructions and dimension calculations.
problem Understanding self-dual polygons in projective spaces of higher dimensions.
method Explicit construction and dimension calculation of moduli spaces of self-dual polygons.
result Provides the dimension of the moduli space for specific cases of n and m.
Algorithm samples polygons of fixed edge lengths in any dimension.
problem Sampling random closed polygons with fixed edge lengths in any dimension.
method Weighted edge vectors on unit sphere, Möbius transformation, reweighting factors.
result Algorithm samples polygons according to standard probability measures efficiently.
Minimal perimeter polygons in punctured discs are found with inscribed horocycles.
problem Finding polygons with minimal perimeter in punctured discs.
method Proving minimal perimeter by inscribed horocycles, generalizing to cone points and geodesic boundaries.
result Minimum perimeter polygons found with inscribed horocycles.