Closed Plane Curves Research Articles

A sharp bound on the number of self-intersections of a trigonometric curve

We obtain a sharp bound on the number of self-intersections of a closed planar curve with trigonometric parameterization. Moreover, we show that a generic curve of this form is normal in the sense of Whitney.

The Gradient Flow for Entropy on Closed Planar Curves

In this paper we consider the steepest descent L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document} or embedded of class W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.

The Cauchy problem of dissipative hyperbolic mean curvature flow

In this paper, the motion of strictly convex closed plane curves under dissipative hyperbolic mean curvature flow is studied. The hyperbolic Monge–Amp re equation is derived by using the support function. The short‐time existence of the flow is proved, and some evolution equations are derived. Furthermore, according to different initial velocities, we discuss the expansion and contraction of the dissipative hyperbolic curvature flow; that is, if the initial velocity , the flow will converge to the limit curve at a finite time; if the initial velocity , the flow will converge to a point ; if the initial velocity , the flow will contract to a limit curve as ; if the initial velocity , the flow will expand to a limit curve as .

Migrating elastic flows

Huisken's problem asks whether there is an elastic flow of closed planar curves that is initially contained in the upper half-plane but ‘migrates’ to the lower half-plane at a positive time. Here we consider variants of Huisken's problem for open curves under the natural boundary condition, and construct various migrating elastic flows both analytically and numerically.

Two nonlocal inverse curvature flows of convex closed plane curves

In this paper, we introduce two [Formula: see text]-type ([Formula: see text]) curvature flows for closed convex planar curves. Along the flows the length of the curve is decreasing while the enclosed area is increasing. Finally, the evolving curves converge smoothly to a finite circle if they do not develop singularity during the evolution process.

A Beautiful Inequality by Saint-Venant and Pólya Revisited

In mathematical physics and beyond, one encounters many beautiful inequalities that relate geometric or physical quantities describing the shape or size of a set. Such isoperimetric inequalities often have a long history and many important applications. For instance, the eponymous and most classical of all isoperimetric inequalities was known already in antiquity. It asserts that among all closed planar curves of a given length, the circles with perimeter equal to that length, and only they, enclose the largest area. Though not nearly as well-known, an isoperimetric inequality conjectured by Saint-Venant in the 1850s and first proved by Pólya almost a century later, is also very beautiful and important. By presenting a short proof as well as two simple physical interpretations, this article illustrates why the result deserves to be cherished by every student of applied analysis.

Aninvestigation Of Winding Number Of A Closed Planar Curve

This paper examines winding number of a closed planar curve through various aspects. It is related to a range on an arc in the complex plane and a point not in the range. Functions of such natures are considered to be continuous real-valued functions. Conclusion was drawn by naming a number of areas of applications.

The evolution of area-preserving and length-preserving inverse curvature flows for immersed locally convex closed plane curves

In this paper, we study an area-preserving inverse curvature flow and a length-preserving inverse curvature flow for immersed locally convex closed plane curves with rotation number m∈N+. The global-in-time flows are shown to converge smoothly to m-fold round circles as time goes to infinity. The sufficient conditions on initial curve are also found to guarantee the occurrence of the flow's singularity at finite time.

The geometry of the Wigner caustic and a\xa0decomposition of a curve into parallel arcs

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data

It is quite common for functional data arising from imaging data to assume values in infinite-dimensional manifolds. Uncovering associations between two or more such nonlinear functional data extracted from the same object across medical imaging modalities can assist development of personalized treatment strategies. We propose a method for canonical correlation analysis between paired probability densities or shapes of closed planar curves, routinely used in biomedical studies, which combines a convenient linearization and dimension reduction of the data using tangent space coordinates. Leveraging the fact that the corresponding manifolds are submanifolds of unit Hilbert spheres, we describe how finite-dimensional representations of the functional data objects can be easily computed, which then facilitates use of standard multivariate canonical correlation analysis methods. We further construct and visualize canonical variate directions directly on the space of densities or shapes. Utility of the method is demonstrated through numerical simulations and performance on a magnetic resonance imaging dataset of glioblastoma multiforme brain tumors.

Affine Subspaces of Curvature Functions from Closed Planar Curves

Given a pair of real functions (k, f), we study the conditions they must satisfy for k+lambda f to be the curvature in the arc-length of a closed planar curve for all real lambda . 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.

Four equivalent properties of integrable billiards

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. Let us say that a strictly geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed plane curves with this property are ellipses. In the present article we show that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net. We also recall Blaschke’s derivation of the Liouville property from the Ivory property and his proof of Weihnacht’s theorem: the only Liouville nets in the plane are nets of confocal conics and their degenerations. This suggests the following generalization of Birkhoff’s conjecture: If an interior neighborhood of a closed strictly geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.

On an area-preserving inverse curvature flow of convex closed plane curves

This paper deals with the 1/κα-type area-preserving nonlocal flow of smooth convex closed plane curves for all constant α>0. Under this flow, the convexity of the evolving curve is preserved. Due to the existence of finite time curvature blow-up examples, it is shown that, if the curvature κ will not blow up in finite time, the evolving curve will converge smoothly to a circle as t→∞.

Fast global SA(2,R) shape registration based on invertible invariant descriptor

Here, we intend to introduce a fast and non-rigid global registration for simple planar closed curves relatively to the planar Special Affine group SA(2,R). In previous work, Ghorbel (1998) has been introduced a complete and stable set of invariant for simple closed planar curves which is invariant jointly under its original parametrization and special planar affine transformations. Such property allows us a robust reconstruction of the considered object up to a special affine transformation. In this paper, several numerical difficulties of the computation of the proposed reconstruction are considered. The robustness of this inverse problem with respect to noise and reasonable deformations of non-rigid shape is demonstrated experimentally. The proposed new registration is based, on the one hand, on the shift theorem relating to the group SA(2,R) and, on the other hand, on the invertibility of the set of invariant. Since this shift theorem allows the extraction of a pose parameters exciting between reference and target objects. The low algorithmic complexity is due to the fact that the computation of the inverse descriptors are based essentially on the Fast Fourier Transformation (FFT) algorithm. Experiments are conducted on different known datasets such as MPEG-7, MCD, Kimia-99, Kimia216, ETH-80 and Swedish leaf datasets. Promising results on the sense of shape retrieval and shape recognition rates will also be demonstrated.

Minimal grid diagrams of 11 crossing prime alternating knots

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentations with 13 arcs.

An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow

<p style='text-indent:20px;'>In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.

A Convex Cover for Closed Unit Curves has Area at Least 0.0975

We combine geometric methods with a numerical box search algorithm to show that the minimal area of a convex set in the plane which can cover every closed plane curve of unit length is at least [Formula: see text]. This improves the best previous lower bound of [Formula: see text]. In fact, we show that the minimal area of the convex hull of circle, equilateral triangle, and rectangle of perimeter [Formula: see text] is between [Formula: see text] and [Formula: see text].

An automated segmentation of NATURA 2000 habitats from Sentinel-2 optical data

<p style='text-indent:20px;'>In this paper, we present a mathematical model and numerical method designed for the segmentation of satellite images, namely to obtain in an automated way borders of Natura 2000 habitats from Sentinel-2 optical data. The segmentation model is based on the evolving closed plane curve approach in the Lagrangian formulation including the efficient treatment of topological changes. The model contains the term expanding the curve in its outer normal direction up to the region of habitat boundary edges, the term attracting the curve accurately to the edges and the smoothing term given by the influence of local curvature. For the numerical solution, we use the flowing finite volume method discretizing the arising advection-diffusion intrinsic partial differential equation including the asymptotically uniform tangential redistribution of curve grid points. We present segmentation results for satellite data from a selected area of Western Slovakia (Záhorie) where the so-called riparian forests represent the important European Natura 2000 habitat. The automatic segmentation results are compared with the semi-automatic segmentation performed by the botany expert and with the GPS tracks obtained in the field. The comparisons show the ability of our numerical model to segment the habitat areas with the accuracy comparable to the pixel resolution of the Sentinel-2 optical data.

On a length-preserving inverse curvature flow of convex closed plane curves

This paper deals with a 1/κα-type length-preserving nonlocal flow of convex closed plane curves for all α>0. Under this flow, the convexity of the evolving curve is preserved. For a global flow, it is shown that the evolving curve converges smoothly to a circle as t→∞. Some numerical blow-up examples and a sufficient condition leading to the global existence of the flow are also constructed.

MORPHOLOGICAL ANALYSIS OF CELLS BY MEANS OF AN ELASTIC METRIC IN THE SHAPE SPACE

Shape analysis is of great importance in many fields, such as computer vision, medical imaging, and computational biology. This analysis can be performed considering shapes as closed planar curves in the shape space. This approach has been used for the first time to obtain the morphological classification of erythrocytes in digital images of sickle cell disease considering the shape space S1, which has the property of being isometric to an infinite-dimensional Grassmann manifold of two-dimensional subspaces (Younes et al., 2008), without taking advantage of all the features offered by the elastic metric related to the possibility of stretching and bending of the curves. In this paper, we study this deformation in the shape space, S2, which is based on the representation of closed planar curves by means of the square-root velocity function (SRVF) (Srivastava et al., 2011), using the elastic metric of this space to obtain more efficient geodesics and geodesic lengths between planar curves. Supervised classification with this approach achieved an accuracy of 94.3%, classification using templates achieved 94.2% and unsupervised clustering in three groups achieved 94.7%, considering three classes of erythrocytes: normal, sickle, and with other deformations. These results are better than those previously achieved in the morphological analysis of erythrocytes and the method can be used in different applications related to the treatment of sickle cell disease, even in cases where it is necessary to study the process of evolution of the deformation, something that can not be done in a natural way in the feature space.