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…
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48 results for “polygonal curves”
For a polygon in Euclidean space we consider a transformation T which is obtained by applying the midpoints polygon construction twice and using an index shift. For a closed polygon this is a curve shortening process. A polygon is called (affine) soliton of the transformation T if its image under T is an affine image o…
Derives conformal parameters of curves using inscribed circular polygons.
problem Characterizing conformal invariants of smooth curves in 3D.
method Limiting process with inscribed circular polygons, based on elementary geometry.
result Derives conformal length, curvature, and torsion via a novel method.
Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine inv…
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.
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.
We find bounds on the difference between the writhing number of a smooth curve, and the writhing number of a polygon inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the computation of writhe.
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.
Smooth curves from polygonal chains with vertex preservation and explicit curvature control.
problem Preserving vertices while smoothing polygonal chains to curves.
method Directional mollification operator for polygonal chains.
result Smooth curves that intersect original vertices and maintain explicit curvature bounds.
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.
We study projectively self-dual polygons and curves in the projective plane. Our results provide a partial answer to problem No 1994-17 in the book of Arnold's problems.
The paper calculates period matrices for specific algebraic curves.
problem Calculating period matrices for a class of algebraic curves.
method Constructing algebraic curves from Euclidean polygons and using symplectic bases.
result Symplectic bases are derived from Euclidean polygons for the constructed curves.
In this work, we discuss graph like image of curves under moment maps and their relation with the Newton polygon of the curve, which has applications to Lagrangian torus fibration of Calabi-Yau manifolds.
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 …
Configuration spaces of plane polygons and a sub-Riemannian approach to the equitangent problemmath.DG
The equitangent locus of a convex plane curve consists of the points from which the two tangent segments to the curve have equal length. The equitangent problem concerns the relation between the curve and its equitangent locus. An equitangent n-gon of a convex curve is a circumscribed n-gon whose vertices belong to the…
Let be a closed polygonal curve in $\RR^3$ consisting of line segments. Assume that is unknotted, so that it is the boundary of an embedded disk in $\RR^3$. This paper considers the question: How many triangles are needed to triangulate a Piecewise-Linear (PL) spanning disk of ? The main result exhibits …
The evolute of a smooth curve in an m-dimensional Euclidean space is the locus of centers of its osculating spheres, and the evolute of a spatial polygon is the polygon whose consecutive vertices are the centers of the spheres through the consecutive (m+1)-tuples of vertices of the original polygon. We study the iterat…
New elastic energy for irregular curves defined through polygonal approximations.
problem Defining elastic energy for irregular curves in any space dimension.
method Relaxation process with -rotation of inscribed polygonals, focusing on geometric curvature distribution.
result Energy finite if and only if curve's arc-length parameterization has second order summability.
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.
New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order , analyzing both Dirichlet and Neumann boundary conditions.
result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
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…
Simplified proof for dimension reduction of polygonal curves.
problem Preserving the continuous Fréchet distance of polygonal curves.
method Sparse oblivious subspace embeddings for generalized dissimilarity measures.
result Generalized dimension reduction technique works for various distance measures.
For Bezier curves, subdivision algorithms create control polygons as piecewise linear (PL) approximations that converge in terms of Hausdorff distance. We prove that the exterior angles of control polygons under subdivision converge to 0 at the rate of , where is the number of subdivisions.…
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…
We study the -median clustering problem for high-dimensional polygonal curves with finite but unbounded number of vertices. We tackle the computational issue that arises from the high number of dimensions by defining a Johnson-Lindenstrauss projection for polygonal curves. We analyze the resulting error in terms of …
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…
We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to …
A new probabilistic polygonal curve representation using Gaussian Mixture Models.
problem Capturing curves with uncertainty in both tangent and normal directions.
method Probabilistic polygonal approximation with Gaussian Mixture Model (GMM).
result The GMM accurately captures the local geometry and uncertainty of curves.
We construct discrete and faithful representations into the isometry group of a hyperbolic space of the fundamental groups of acute negatively curved even-sided polygons of finite groups.
Defines weak normals for irregular curves in high-dimensional spaces.
problem Dealing with irregular curves in high-dimensional Euclidean spaces.
method Using sequences of inscribed polygonals and Gram-Schmidt procedure, introduces a relaxed notion of weak normals.
result Weak normals for irregular curves are the strong limit of approximating polygonals and agree with relaxed energy.
New proof confirms surfaces can be divided into polygons.
problem Estimating the dimension of Thurston spine.
method Existence of an infinite set A, proving codimension is o(g/ log g).
result Proves recent conjecture about Thurston spine dimension.
Algorithm constructs algebraic curves from translation surfaces.
problem Creating algebraic curves from translation surfaces.
method Using discrete Riemann surface theory and Riemann theta functions.
result Algorithm approximates Jacobian varieties of translation surfaces.
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…
Poncelet Curvesmath.DG
We study pairs of curves with Poncelet's porism properties and compute their vertex curves.
problem Understanding pairs of curves with Poncelet's porism properties.
method Developed formulas to compute vertex curves for given envelope curves and vice versa, for all sufficiently regular pairs of Poncelet curves.
result Formulas produce all possible sufficiently regular pairs of Poncelet curves, including sets of curves analogous to pencils of conic sections.
We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the exis…
We study closed smooth convex plane curves enjoying the following property: a pair of points can traverse so that the distances between and along the curve and in the ambient plane do not change; such curves are called {\it bicycle curves}. Motivation for this study comes from the problem how to d…
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…
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…
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 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.
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 paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
problem Understanding self-Bäcklund curves in centroaffine geometry.
method Description of general properties and detailed analysis using elliptic functions.
result Provides a detailed description of self-Bäcklund centroaffine curves in terms of elliptic functions.
Geodesics on polygons in a unit disk are studied with unique metric properties.
problem Characterizing geodesics on polygons within a unit disk.
method Defining a metric on polygons such that geodesics are curves in the family C.
result The constructed metric space is not isometric to any convex domain in R^2.
Paper introduces k-DTW for robust curve comparison.
problem Robust dissimilarity measure for polygonal curves.
method Introduces k-Dynamic Time Warping (k-DTW) as a novel dissimilarity measure.
result k-DTW is more robust to outliers and has stronger metric properties than DTW.
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…
We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserma…
Study symmetries in smoothed polygonal links.
problem Computing Khovanov homology and link invariants.
method Cube of resolutions with combinatorial structure.
result New group-theoretic invariants of links.
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…