Closed Plane Curves Research Articles

Locally Euclidean metrics with a given geodesic curvature of the boundary

The problem of reconstructing a locally Euclidean metric on a disk from the geodesic curvature of the boundary given in the sought metric is considered. This problem is an analog and a generalization of the classical problem of finding a closed plane curve from its curvature given as a function of the arc length. The solution of this problem in our approach can be interpreted as finding a plane domain with the standard Euclidean metric whose boundary has a given geodesic curvature.

Invariant Signatures of Closed Planar Curves

We prove that any subset of ?2 parametrized by a C 1 periodic function and its derivative is the Euclidean invariant signature of a closed planar curve. This solves a problem posed by Calabi et al. (Int. J. Comput. Vis. 26:107---135, 1998). Based on the proof of this result, we then develop some cautionary examples concerning the application of signature curves for object recognition and symmetry detection as proposed by Calabi et al.

Hyperbolic mean curvature flow: evolution of plane curves

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves S1 × [0,T) → ℝ2 which satisfies the following evolution equation ∂2F∂t2(u,t)=k(u,t)N→(u,t)-∇ρ(u,t),∀(u,t)∈S1×[0,T) with the initial data F(u,0)=F0(u)and∂F∂t(u,0)=f(u)N→0, where k is the mean curvature and ▪ is the unit inner normal vector of the plane curve F(u,t), f(u) and ▪ are the initial velocity and the unit inner normal vector of the initial convex closed curve Fo, respectively, and Δρ is given by ∇ρ≜(∂2F∂s∂t,∂F∂t)T→, in which ▪ stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere equation. Based on this, we show that there exists a class of initial velocities such that the solution of the above initial value problem exists only at a finite time interval [0, Tmax) and when t goes to Tmax, either the solution converges to a point or shocks and other propagating discontinuities are generated. Furthermore, we also consider the hyperbolic mean curvature flow with the dissipative terms and obtain the similar equations about the support functions and the curvature of the curve. In the end, we discuss the close relationship between the hyperbolic mean curvature flow and the equations for the evolving relativistic string in the Minkowski space-time ℝ1,1.

Rotational hypersurfaces of periodic mean curvature

In this paper we discuss rotational hypersurfaces in R n and more specifically rotational hypersurfaces with periodic mean curvature function. We show that, for a given real analytic function H ( s ) on R , every rotational hypersurface M in R n with mean curvature H ( s ) can be extended infinitely in the sense that all coordinate functions of the generating curve of M are defined on all of R as well. For rotational hypersurfaces with periodic mean curvature we present a criterion characterizing the periodicity of such hypersurfaces in terms of their mean curvature function. We also discuss a method to produce families of periodic rotational hypersurfaces where each member of the family has the same mean curvature function. In fact, given any closed planar curve with curvature κ, we prove that there is a family of periodic rotational hypersurfaces such that the mean curvature of each element of the family is explicitly determined by κ. Delaunay's famous result for surfaces of revolution with constant mean curvature is included here as the case where n = 3 and κ is constant.

A Bose type formula for the internal medial axis of an embedded manifold

We establish a three-dimensional variant of the Bose formula for the internal medial axis of a closed plane curve. We generalize the result to dimensions less than or equal to 6, and we apply the result to cut-loci, and spines of 3-manifolds. Also, we describe the difference between the main theorems and recent work of Sedykh.

When Is a Periodic Function the Curvature of a Closed Plane Curve?

(2008). When Is a Periodic Function the Curvature of a Closed Plane Curve? The American Mathematical Monthly: Vol. 115, No. 5, pp. 405-414.

Expanding convex immersed closed plane curves

We study the evolution driven by curvature of a given convex immersed closed plane curve. We show that it will converge to a self-similar solution eventually. This self-similar solution may or may not contain singularities. In case it does, we also have estimate on the curvature blow-up rate.

Comparison study for Level set and Direct Lagrangian methods for computing Willmore flow of closed planar curves

The main goal of this paper is to present results of comparison study for the level set and direct Lagrangian methods for computing evolution of the Willmore flow of embedded planar curves. To perform such a study we construct new numerical approximation schemes for both Lagrangian as well as level set methods based on semi-implicit in time and finite/complementary volume in space discretizations. The Lagrangian scheme is stabilized in tangential direction by the asymptotically uniform grid point redistribution. Both methods are experimentally second order accurate. Moreover, we show precise coincidence of both approaches in case of various elastic curve evolutions provided that solving the linear systems in semi-implicit level set method is done in a precise way, redistancing is performed occasionally and the influence of boundary conditions on the level set function is eliminated.

Hybrid curve fitting

We consider a parameterized family of closed planar curves and introduce an evolution process for identifying a member of the family that approximates a given unorganized point cloud {p i } i =1,..., N . The evolution is driven by the normal velocities at the closest (or foot) points (f i ) to the data points, which are found by approximating the corresponding difference vectors p i -f i in the least-squares sense. In the particular case of parametrically defined curves, this process is shown to be equivalent to normal (or tangent) distance minimization, see [3], [19]. Moreover, it can be generalized to very general representations of curves. These include hybrid curves, which are a collection of parametrically and implicitly defined curve segments, pieced together with certain degrees of geometric continuity.

Geometric dilation of closed planar curves: New lower bounds

Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π / 2 , which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.

A semiclassical approximation for the nonstationary two-dimensional nonlinear Schrödinger equation with an external field in polar coordinates

For the two-dimensional nonlinear Schrodinger equation (NSE) with an external field in polar coordinates, semiclassical solutions asymptotic in small parameter ħ, ħ → 0, have been constructed. The solutions are constructed in the class of functions which are concentrated in a neighborhood of a closed plane curve and have the form of the one-soliton solution of the one-dimensional NSE in the “fast” variable.

Multiregion competition: A level set extension of region competition to multiple region image partitioning

The purpose of this study is to investigate a new representation of a partition of an image domain into a fixed but arbitrary number of regions by explicit correspondence between the regions of segmentation and the regions defined by simple closed planar curves and their intersections, and the use of this representation in the context of region competition to provide a level set multiregion competition algorithm. This formulation leads to a system of coupled curve evolution equations which is easily amenable to a level set implementation and the computed solution is one that minimizes the stated functional. An unambiguous segmentation is garanteed because at all time during curve evolution the evolving regions form a partition of the image domain. We present the multiregion competition algorithm for intensity-based image segmentation and we subsequently extend it to motion/disparity. Finally, we consider an extension of the algorithm to account for images with aberrations such as occlusions. The formulation, the ensuing algorithm, and its implementation have been validated in several experiments on gray level, color, and motion segmentation.

Multiregion level-set partitioning of synthetic aperture radar images

The purpose of this study is to investigate Synthetic Aperture Radar (SAR) image segmentation into a given but arbitrary number of gamma homogeneous regions via active contours and level sets. The segmentation of SAR images is a difficult problem due to the presence of speckle which can be modeled as strong, multiplicative noise. The proposed algorithm consists of evolving simple closed planar curves within an explicit correspondence between the interiors of curves and regions of segmentation to minimize a criterion containing a term of conformity of data to a speckle model of noise and a term of regularization. Results are shown on both synthetic and real images.

Area preserving deformation of multiresolution curves

We describe a method for multiresolution deformation of closed planar curves that keeps the enclosed area constant. We use a wavelet based multiresolution representation of the curves that are represented by a finite number of control points at each level of resolution. A deformation can then be applied to the curve by modifying one or more control points at any level of resolution. This process is generally known as multiresolution editing to which we add the constraint of constant area. The key contribution of this paper is the efficient computation of the area in the wavelet decomposition form: the area is expressed through all levels of resolution as a bilinear form of the coarse and detail coefficients, and recursive formulas are developed to compute the matrix of this bilinear form. A similar result is also given for the bending energy of the curve. The area constraint is maintained through an optimization process. These contributions allow a real time multiresolution deformation with area constraint of complex curves.

A complete and non-overlapping tracing algorithm for closed loops

A procedure for tracing completely closed loops given implicitly by the intersection of two regular surfaces, without resorting to the parametric domain subdivisions or resulting in arc overlapping, is presented. Our primary hypothesis is that the rotation index, a global geometrical property, may be a useful complementary tool to the local differential geometrical properties for improving the efficiency of the well-known marching-based surface–surface intersection algorithms. To validate this hypothesis, we devised a novel approach for incrementally computing the rotation index of a closed plane curve given implicitly while the curve is traced. Moreover, we also proposed its integration in a marching procedure that employs adaptative circular steps.

Geometric constant defining shape transitions of carbon nanotubes under pressure.

We demonstrate that when a single-walled carbon nanotube is under pressure it undergoes a series of shape transitions, first transforming from a circle to an oval and then from an oval to a peanut. Most remarkably, the ratio of the area of the tube cross sections at the second transition over that at the first transition appears as a constant, independent of the tube radius. Its accurate value is computed to be G=0.819 469, by formulating a variational geometry problem to represent single-walled carbon nanotubes with a family of closed plane curves of fixed length and minimum bending energy. The implications of such a geometric constant in designing nanotube electromechanical pressure sensors are discussed.

A Criterion for Quasi-Simple Plane Curves

We give a non-standard criterion for closed plane curves to be quasi-simple, i.e. to be the boundary of a simply connected domain in C. This can be used, for instance, to introduce piecewise close-to-convex functions and to establish the univalence of the extremal polynomials in the maximal range problem for simply connected domains.

On Sampling Closed Planar Curves and Surfaces

On Sampling Closed Planar Curves and Surfaces

Intersecting Curves in the Plane

We prove that for every family of n pairwise intersecting simple closed planar curves in general position, at least (4/5)n2−O(n) points lie on more than one curve. This improves the previous lower bound of (3/4)n2−O(n) due to Richter and Thomassen.

Enhancing CSS-based shape retrieval for objects with shallow concavities

Curvature scale space (CSS) image is a multi-scale organisation of the inflection points of a closed planar curve as it is smoothed. It consists of several arch shape contours, each related to a concavity or a convexity of the curve. The maxima of these contours have already been used as shape descriptors to find similar shapes in large image databases. In this article, we address the problem of shallow concavities. These may give rise to large contours in the CSS image. These contours may then match those corresponding to deep and wide concavities during the matching process. The phenomenon can be explained by recalling the fact that Gaussian smoothing leads to an approximation of geometric heat equation deformation. We have introduced a method to enrich the CSS image and create different contours for different types of concavities. We tested the proposed method on a database of 1100 images of marine creatures. A significant improvement was observed in the performance of the system on shapes with shallow segments.