arXiv research

We define a computable topological invariant $μ(γ)$ for generic closed planar regular curves $γ$, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves witho…

Proves existence of planar curves with specific curvature.

Study conditions for curvature functions of closed planar curves.

The paper shows how to rearrange arcs to form closed curves.

Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.

The method for approximation of planar curve by circular arcs with length preservation, proposed by I.Kh. Sabitov and A.V. Slovesnov, is analyzed. We extend the applicability of the method, and consider some corollaries, not related to the approximation problem. Inequalities for the length of a convex spiral arc with p…

In this paper we use a gradient flow to deform closed planar curves to curves with least variation of geodesic curvature in the $L^2$ sense. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a mult…

We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$, that is in the case of the classic elastic energy, long-time existence is retrieve…

The article analyzes the stability of a curve shortening flow for planar networks.

The paper proves existence of minimal homotopies for immersed planar curves.

Paper introduces length measures for curves and convex shapes, proving isoperimetric and distance properties.

The secant caustic of a planar curve $M$ is the image of the singular set of the secant map of $M$. We analyse the geometrical properties of the secant caustic of a planar curve, i.e. the number of branches of the secant caustic, the parity of the number of cusps and the number of inflexion points in each branch of thi…

Sharp convergence rate for curvature stability in planar free elastic flow.

Two flows for convex curves converge to circles smoothly.

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with $n$ self-crossings requires $Θ(n^{3/2})$ homotopy moves in the worst case. Our algorithm improves the best previou…

Method for generating new curves from plane curves on cylinders.

Simple closed curves in ε-boundaries separate sets in the plane.

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …

Explains Arnold's J+ invariant for curves, using basic math.

We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a point. Our theorem is the analog of the result of Grayson for curvature flow of clo…

New curve flow preserves area and converges to a circle.

Paper studies planar extensions in o-minimal structures.

The tangential map is a map on the set of smooth planar curves. It satisfies the 3D-consistency property and is closely related to some well-known integrable equations.

Unified approach classifies stable and minimal elastic curves.

We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two boundary curves are either parallel or sufficiently close to parallel, and when th…

The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.

Gradient flow expands curves to round shapes.

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…

The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.

Study on closed curves on negatively curved surfaces, linking number formula, and restrictions.

The paper generalizes Fenchel's theorem for curves with singularities.

Study preserves planar and graphical properties of curves under elastic flow.

Ancient solutions to curve shortening flow are constructed and analyzed.

Deep learning approximates geometric measures of planar curves.

Study on migrating elastic flows of curves across half-planes.

We give a complete description of all order 1 invariants of planar curves.

Study on cr-invariant variational problem for Legendrian curves in 3-sphere.

Analytic non-planar $p$-elasticae are shown to be 3D.

Given a normed plane $\mathcal{P}$, we call $\mathcal{P}$-cycloids the planar curves which are homothetic to their double $\mathcal{P}$-evolutes. It turns out that the radius of curvature and the support function of a $\mathcal{P}$-cycloid satisfy a differential equation of Sturm-Liouville type. By studying this equati…

We present a new implementation of anisotropic mean curvature flow for contour recognition. Our procedure couples the mean curvature flow of planar closed smooth curves, with an external field from a potential of point-wise charges. This coupling constrains the motion when the curve matches a picture placed as backgrou…

A Monge surface is a surface obtained by sweeping a generating plane curve along a trajectory that is orthogonal to the moving plane containing the curve. Locally, they are characterized as being foliated by a family of planar geodesic lines of curvature. We call surfaces with the latter property PGF surfaces, and inve…

Curve shortening flow converges to a point with entropy bound.

Regular shrinkers describe blow-up limits of a finite-time singularity of the motion by curvature of planar network of curves. This follows from Huisken's monotonicity formula. In this paper, we show that there is only one regular shrinker with 2 closed regions. This regular shrinker is the Cisgeminate eye. Moreover, w…

In this paper we prove a universal inequality describing the asymptotic behavior of support points for planar continuous curves. As corollaries we get an analogous result for tangent points of differentiable planar curves and some (partially known) assertions on the asymptotic of the mean value points for various class…

Optimal thresholds ensure curves remain embedded in flows.

The paper studies stability of discrete planar curves using variational methods.

Approximates nonlocal curvature of curves using splines.

Study how $J^+$ invariants change under bifurcations of curves.