Yu. G. Nikonorov
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 classical analytic theorems. Some open questions are formulated.
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Construction of an isometric immersion of a bounded, planar region from a framed curve
本文研究有界平面区域到三维空间的等距浸入构造问题，通过边界标架曲线的相容性与正则性条件保证浸入存在，并将弯曲能精确降维为边界线积分，同时刻画了正则性丧失的几何条件。
Tractrices of planar curves, in particular, a family of tractrices of a circle, are considered. Some new observations (including arc-length parametrization, Chezaro equation) and corrected reference informations are provided. The article is written in Russian.
In this paper, we study planar polygonal curves from the variational methods. We show an unified interpretation of discrete curvatures and the Steiner-type formula by extracting the notion of the discrete curvature vector from the first variation of the length functional. Moreover, we determine the equilibrium curves for the length functional under the area-constraint condition and study their stability.
In this paper we study singular points of the Wigner caustic and affine $λ$--equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.
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 without inflections) whose numbers of crossings are less than or equal to five. Moreover, we discuss the relationship between the number of double tangents and the invariant $μ(γ)$ on a given $γ$.
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 this set. In particular, we investigate in detail some of the geometrical properties of the secant caustic of a rosette, i.e. a smooth regular oriented closed curve with non-vanishing curvature.
Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H--planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaterni…
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 multiply-covered circle. Moreover, we show that curves in any homotopy class with initially small $L^3\lVert k_s\rVert_2^2$ enjoy a uniform length bound under the flow, yielding the convergence result in these cases.
The Geometry of the Centre Symmetry Set of a Planar Curve
该研究聚焦平面曲线的中心对称集这一几何对象，探究其整体几何性质，明确了该集合的奇点数量与渐近线数量相关的几何特征。
Self-affinities of planar curves: towards unified description of aesthetic curves
本文研究平面曲线的自仿射性，重新表述并严格证明了对数美学曲线与抛物线的自仿射特性，提出了一种新的自仿射性，在等仿射几何中统一刻画了常曲率曲线。
Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We consider two conformally equivalent metrics for which the distances between curves are nontrivial. We show that in the case of the simpler of the two metrics, the only minimal geodesics are those corresponding to curve evolution in which the points of the curve move with the same normal speed. An equation for the geodesics and a formula for the sectional curvature are derived; a necessary and sufficient condition for the sectional curvature to be bo…
Planar closed curves with prescribed curvature
本文利用变分方法并结合Struwe的单调性技巧，通过引入参数获得有界Palais-Smale序列，从而证明某些曲率函数类下具有指定曲率的平面闭曲线的存在性。
Approximating the Nonlocal Curvature of Planar Curves
本文研究平面曲线非局部曲率的分解与逼近，通过将定义扩展到曲线外点并利用不完全 beta 函数给出线段公式，证明了 Hölder 正则曲线的非局部曲率可由线性插值样条的非局部曲率逼近。
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 prescribed two-point $G^1$ or $G^2$ Hermite data are derived. We propose a scheme of computer modelling to explore properties of planar curves. As an example, closeness of ovals is tested, leading to some conjectures about closeness conditions.
Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+λf$ to be the curvature in the arc-length of a closed planar curve for all real $λ$. Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied. Finally, the characterization obtained is used to show that a sufficient analogue of the 4-vertex theorem cannot be developed.
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 retrieved.
We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their position and tangent vectors. Among this family, we characterize minimal, constant mean curvature, Hamiltonian stationary, solitons for mean curvature flow and Willmore surfaces in terms of simple properties of the curvatures of the generating curves. As an application, we provide explicitly conformal parametrizations of known and new examples of these classes o…
Hamilton flows on Kähler manifold for which all trajectories are $H$-planar curves (complex analog of geodesics) are considered. These flows are called $H$-planar. The equation which has to obey the Hamiltonian of $H$-planar Hamilton flow is received and the method of finding general solution of this equation is proposed. Trajectories of charged particles in magnetic fields of special form on Kähler manifolds of constant holomorphic sectional curvature are studied. Using the fact that Kähler manifolds of constant holomorphic sectional curvature admit $H$-projective mapping on flat space the eq…
We propose a learning paradigm for the numerical approximation of differential invariants of planar curves. Deep neural-networks' (DNNs) universal approximation properties are utilized to estimate geometric measures. The proposed framework is shown to be a preferable alternative to axiomatic constructions. Specifically, we show that DNNs can learn to overcome instabilities and sampling artifacts and produce consistent signatures for curves subject to a given group of transformations in the plane. We compare the proposed schemes to alternative state-of-the-art axiomatic constructions of differe…
Discrete curve theory in space forms: planar elastic and area-constrained elastic curves
本文提出空间形式中离散弹性与面积约束弹性曲线的概念，推广离散欧几里得曲率方程，利用Bäcklund变换构建离散流，并证明此类曲线可刻画为流不变量的层级元素，同时给出离散Frenet公式与基本定理。
In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from Banach space theory, we study all curvature types of curves in gauge planes, thus generalizing their complete classification for normed planes. We show that (as in the subcase of normed planes) there are four such types, and we call them analogously Minkowski, normal, circular, and arc-length curvature. We study relations between them and extend, based on this,…
In the present paper we study pseudo-Riemannian submanifolds which have 3-planar geodesic normal sections.We consider W-curves (helices) on pseudo-Riemannian submanifolds. Finally, we give neccessary and sufficient condition for a normal section to be a W-curve on pseudo-Riemannian submanifolds.
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 the boundary curves themselves are sufficiently close to each other, we draw specific conclusions about the geometry and topology of the surfaces. We also strength the following result: Let $M$ be any compact minimal annulus with two planar boundary curves of diameters $d_1$ and $d…
Stability analysis for the anisotropic curve shortening flow of planar networks
该文研究三曲线平面网络的各向异性曲线缩短流，通过证明各向异性曲率能量满足Łojasiewicz-Simon梯度不等式，得出接近局部能量极小值的初始数据对应流全局存在且收敛于能量极小值的稳定性结论。
The gradient flow for entropy on closed planar curves
研究闭合平面曲线上熵泛函的 L2 最速下降梯度流，证明在两类初始曲线条件下均光滑收敛为扩张的多重覆盖圆。
On length-preserving and area-preserving inverse curvature flow of planar curves with singularities
本文研究具有奇点的平面曲线的保面积和保长逆曲率流，证明$\u2113$-凸勒让德曲线分别演化为特定半径的圆，并推导出几何不等式。
In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circle $\mathbb{S}^1$ that intuitively represents the length of the portion of curve which tangent vector points in a certain direction. While a planar closed curve is not characterized by its length measure, the fundamental Minkowski-Fenchel-Jessen theorem states that length measures fully characterize convex curves modulo translati…
Geometric Properties of Planar and Spherical Interception Curves
本文研究平面与球面拦截曲线的几何性质，推导平面情形下相关三角形元素极限与双纽线常数等的关联，求解球面类似问题并验证投影保角性与曲线对称性，揭示两类情形的对称/非对称关系及性质的极限推导途径。
The asymptotic of the Mullins-Sekerka and the area-preserving curvature flow in the planar flat torus
本文研究平面平坦环面上 Mullins-Sekerka 流与保面积曲率流的平坦流解的渐近行为，利用尖锐定量 Alexandrov 不等式证明其以指数速度收敛到圆盘、条带或其补集的有限并集。
The free elastic flow for closed planar curves
本文研究闭平面曲线的自由弹性流，证明初始曲线接近圆（可多重覆盖）时，适当重缩放后平滑收敛到唯一圆，确定了其渐近形状。
Geometric reduction of the Newtonian planar three-body problem is investigated in the framework of equivariant Riemannian geometry, which reduces the study of trajectories of three-body motions to the study of their moduli curves, that is, curves which record the change of size and shape, in the moduli space of oriented mass-triangles. The latter space is a Riemannian cone over the shape 2-sphere, and the shape curve is the image curve on this sphere. It is shown that the time parametrized moduli curve is in general determined by the relative geometry of the shape curve and the shape potential…
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