Convex Plane Curves Research Articles

Dynamics of area-preserving curvature flow of convex plane curves with small area in an inhomogeneous medium

AbstractIn 1986, Gage studied area-preserving curvature flows in a two-dimensional homogeneous medium and proved that an initially convex closed curve remains convex and converges to a circle as time approaches infinity. However, a medium may not be homogeneous in many applications such as cell motility and motions of active matters. Therefore, this work introduces an area-preserving curvature flow in an inhomogeneous medium, which is a natural extension of an inhomogeneous medium. Through this extension, a closed curve numerically moves towards the higher medium by the flow. To show this fact, the case is considered in which the area enclosed by the closed curve is small. In this situation, it is shown that the dynamics of its center is approximated by a gradient flow determined by the distribution of the inhomogeneous medium and that its center moves toward the critical point of the inhomogeneous medium.

Anisotropic area-preserving nonlocal flow for closed convex plane curves

Abstract We consider an anisotropic area-preserving nonlocal flow for closed convex plane curves, which is a generalization of the model introduced by Pan and Yang (J. Differential Equations 266 (2019), 3764–3786) when τ = 1. Under this flow, the evolving curve maintains its convexity and converges to a homothety of a smooth symmetric strictly convex plane curve in the C ∞ sense. The analysis of the asymptotic behavior of this flow implies the possibility of deforming one curve into another within the framework of Minkowski geometry.

A Schur’s theorem via a monotonicity and the expansion module

Abstract In this paper we present a monotonicity which extends a classical theorem of A. Schur comparing the chord length of a convex plane curve with that of a space curve of smaller curvature. The main result establishes a Schur’s Theorem for spherical curves, which extends the well-known Cauchy’s Arm Lemma. We also extend the result by allowing the curve with the smaller curvature being any curve in 𝕊 n {\mathbb{S}^{n}} .

The evolution of gradient flow minimizing the anisoperimetric ratio of convex plane curves

In this paper, we study the evolution of the gradient flow minimizing the anisoperimetric ratio of convex plane curves. It is shown that for any closed embedded initial curve, which is smooth and convex, the flow exists for all time and the evolving curves converge smoothly to a homothety of boundary of the Wulff shape, which is determined by anisotropic function.

Dissipative hyperbolic mean curvature flow for closed convex plane curves

This paper considers one dimensional dissipative hyperbolic mean curvature flow. To analysis the asymptotic behavior of this dissipative hyperbolic curvature flow, we investigate the evolution of a family of circles. Particularly, we give some propositions and prove the main result. If the minimum of initial velocity is nonnegative, the flow will converge either to a point or a limit curve which has the discontinuous curvature in finite time. If the maximum of initial velocity is positive, the flow will expand firstly and then converge either to a point or a limit curve which has the discontinuous curvature in finite time.

On a nonlocal contracting flow for closed convex plane curves

The aim of this paper is to present a convex curve evolution problem which is determined by both local (curvature [Formula: see text]) and global (area [Formula: see text]) geometric quantities of the evolving curve. This flow will decrease the perimeter and the area of the evolving curve and make the curve more and more circular during the evolution process. And finally, as [Formula: see text] goes to infinity, the limiting curve will be a finite circle in the [Formula: see text] metric.

The evolution of immersed locally convex plane curves driven by anisotropic curvature flow

Abstract In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V = 1 α ψ ( x ) κ α V=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α < 0 \alpha \lt 0 or α > 1 \alpha \gt 1 , where x ∈ [ 0 , 2 m π ] x\in \left[0,2m\pi ] is the tangential angle at the point on evolving curves. For − 1 ≤ α < 0 -1\le \alpha \lt 0 , we show the flow exists globally and the rescaled flow has a full-time convergence. For α < − 1 \alpha \lt -1 or α > 1 \alpha \gt 1 , we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ \psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.

Global existence of solutions of area-preserving curvature flow of a convex plane curve in an inhomogeneous medium

Global existence of solutions of area-preserving curvature flow of a convex plane curve in an inhomogeneous medium

On curves with circles as their isoptics

In the present paper we describe the family of all closed convex plane curves of class C^1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

The reverse isoperimetric inequality for convex plane curves through a length-preserving flow

By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338, 2010), which states that if $$\gamma $$ is a convex curve with length L and enclosed area A, then the best constant $$\varepsilon $$ in the inequality $$\begin{aligned} L^2\le 4\pi A+\varepsilon |{\tilde{A}}| \end{aligned}$$ is $$\pi $$ , where $${\tilde{A}}$$ denotes the oriented area of the locus of its curvature centers.

Symmetrization of convex plane curves

Symmetrization of convex plane curves

Symmetrization of convex plane curves

Symmetrization of convex plane curves

On a family of inverse curvature flows for closed convex plane curves

In this note we introduce a family of flows for closed convex curves in the plane. Along the flows the enclosed area of the curve is increasing, and the curve remains convex and converges to a circle. The flows include various flows studied by various authors as special cases.

Outer Billiards with the Dynamics of a Standard Shift on a Finite Number of Invariant Curves

We give a beautiful explicit example of a convex plane curve such that the outer billiard has a given finite number of invariant curves. Moreover, the dynamics on these curves is a standard shift. This example can be considered as an outer analog of the so-called Gutkin billiard tables. We test total integrability of these billiards, in the region between the two invariant curves. Next, we provide computer simulations on the dynamics in this region. At first glance, the dynamics looks regular but by magnifying the picture we see components of chaotic behavior near the hyperbolic periodic orbits. We believe this is a useful geometric example for coexistence of regular and chaotic behavior of twist maps.

A note on expansion of convex plane curves via inverse curvature flow

Andrews and Bryan (J Reine Angew Math 653:179–187, 2011) discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this comparison function they estimate the length of any chord from below in terms of the arc length between its endpoints and elapsed time. They apply this estimate to short segments and deduce directly that the maximum curvature decays exponentially to the curvature of a circle with the same length. We consider the expansion of convex curves under inverse (mean) curvature flow and show that the above comparison function also works in this case to obtain a new proof of the fact that the flow exists for all times and becomes round in shape, i.e. converges smoothly to the unit circle after an appropriate rescaling.

An anisotropic area-preserving flow for convex plane curves

This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.

Area properties associated with a convex plane curve

Abstract Archimedes knew that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area of the region bounded by the parabola X and chord AB is four thirds of the area of the triangle △ A ⁢ B ⁢ P {\bigtriangleup ABP} . Recently, the first two authors have proved that this fact is the characteristic property of parabolas. In this paper, we study strictly locally convex curves in the plane ℝ 2 {{\mathbb{R}}^{2}} . As a result, generalizing the above mentioned characterization theorem for parabolas, we present two conditions, which are necessary and sufficient, for a strictly locally convex curve in the plane to be an open arc of a parabola.

Integer chords and configurations of lattice points

The area within a closed convex plane curve C may be estimated by enlarging C by a factor R, translating, counting the set J of integer points inside, and scaling back to the original size. This estimate is accurate when C is three times continuously differentiable in a certain sense. The set J is very sensitive to translations of the curve. We show that as R tends to infinity, the domains in which each set J occurs tend to uniform distribution modulo the integer lattice; this was only known for the special case of the circle.

Center of Gravity and a Characterization of Parabolas

Archimedes determined the center of gravity of a parabolic section as follows. For a parabolic section between a parabola and any chord AB on the parabola, let us denote by P the point on the parabola where the tangent is parallel to AB and by V the point where the line through P parallel to the axis of the parabola meets the chord AB. Then the center G of gravity of the section lies on PV called the axis of the parabolic section with PG = 3 PV . In this paper, we study strictly locally convex plane curves satisfying the above center of gravity properties. As a result, we prove that among strictly locally convex plane curves, those properties characterize parabolas.

Configuration Spaces of Plane Polygons and a sub-Riemannian Approach to the Equitangent Problem

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 relationship 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 equitangent locus. We are interested in curves that admit 1-parameter families of equitangent n-gons.We use methods of sub-Riemannian geometry: We define a distribution on the space of polygons and study its bracket generating properties. The 1-parameter families of equitangent polygons correspond to the curves, tangent to this distribution. This distribution is closely related with the Birkhoff distribution on the space of plane polygons with a fixed perimeter length whose study, in the framework of the billiard ball problem, was pioneered by Yu. Baryshnikov, V. Zharnitsky, and J. Landsberg.