arXiv research

In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $φ_0$; we refer the piece-wised elastica $Λ_{φ_0}$-elastica or $Λ$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly du…

This paper classifies all planar p-elasticae and their properties.

Classifies stability of flat-core $p$-elasticae pinned at boundaries.

Smooth compactness theorem for elasticae, except straight segments.

Euler derived elastica equation using modern mathematical concepts.

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

New stabilization found in planar elasticae with degenerate diffusion.

Euler's elastica with monotone curvature is uniquely minimal.

Gradient flow of elastic energy converges to elastica.

The elastica is a curve in $\R^3$ that is stationary under variations of the integral of the square of the curvature. Elastica is viewed as a dynamical system that arises from the second order calculus of variations, and its quantization is discussed.

In the previous article (J. Geom. Phys. {\bf 43} (2002) 146), we show the hyperelliptic solutions of a loop soliton as a study of a quantized elastica. This article gives some functional relations in a loop soliton as a quantized elastica.

Classifies pinned $p$-elasticae and finds unique optimality exponents.

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli $\CMeP$ of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere $\PP$. The…

Sub-Riemannian geometry connects bike paths to mathematical curves.

The paper proposes a method to learn 3D object pose manifolds using GANs and elasticae.

Investigates polar tangential angles of curves and their monotonicity.

Closed loop solitons in a plane, whose curvatures obey the modified Korteweg-de Vries equation, were investigated. It was shown that their tangential vectors are expressed by ratio of Weierstrass sigma functions for genus one case and ratio of Baker's sigma functions for the genus two case. This study is closely relate…

The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops …

In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and sho…

Study of closed real plane curves with hyperelliptic genus three solutions.

In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose…

In recent years, total variation (TV) and Euler's elastica (EE) have been successfully applied to image processing tasks such as denoising and inpainting. This paper investigates how to extend TV and EE to the supervised learning settings on high dimensional data. The supervised learning problem can be formulated as an…

The paper finds curves minimizing elastic energy pinned at endpoints.

Unified approach classifies stable and minimal elastic curves.

Isothermic tori with one planar curvature line found and characterized.

Establishes a Li-Yau type inequality for curves in any codimension.

We consider the problem of minimizing the bending or elastic energy among Jordan curves confined in a given open set $Ω$. We prove existence, regularity and some structural properties of minimizers. In particular, when $Ω$ is convex we show that a minimizer is necessarily a convex curve. We also provide an example of a…

The study examines elastic curves with self-intersections and their properties.

For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such a…

Consider the following variational problem: among all curves in $\mathbb{R}^n$ of fixed length with prescribed end points and prescribed tangents at the end points, minimise the $L^\infty$-norm of the curvature. We show that the solutions of this problem, and of a generalised version, are characterised by a system of d…

Optimizes curves on Riemannian manifolds to minimize curvature.

Critical trajectories in a sphere are found for a specific bending functional.

In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop solitons in a plane, \it i.e., loops whose curvatures obey the modified Korteweg-de Vries equations, were investigated for the case related to algebraic curves with genera one and two. This article is a generalization of the previous article …

Approximate 3D elastic curves with exact constraints

For a smooth curve $γ$, we define its elastic energy as $E(γ)= \frac 12 \int_γ k^2 (s) ds$ where $k(s)$ is the curvature. The main purpose of the paper is to prove that among all smooth, simply connected, bounded open sets of prescribed area in $\mathbb{R}^2$, the disc has the boundary with the least elastic energy. In…

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the …

The paper classifies and analyzes the stability of elastic curves with fixed endpoints.

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.

For curves of prescribed length embedded into the unit disc in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds $2π$ and in the large length limit. In the small excess length case, we prove convergence to a fourth order obstacle type problem with integral constraint on…

New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties: not on…

Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.

Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schrödinger hierarchy satisfied by the complex curvature function of the space curve. Rather than working with this infin…

In mathematics, the classical Plateau problem consists of finding the surface of least area that spans a given rigid boundary curve. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in soapy water and then removing it; the shape of the spanning soap film is a solution …

Frustration causes buckling-like behavior in tubular foldable mechanisms.