Euler derived elastica equation using modern mathematical concepts.
problem Euler's original derivation of elastica equation has not been properly interpreted.
method Euler used Noether's theorem and the Goldstein-Petrich scheme.
result Euler's equation is the static modified KdV equation.
Euler's elastica with monotone curvature is uniquely minimal.
problem Global minimality of planar elastica with monotone curvature.
method Proof of global minimality using clamped boundary conditions and length penalization.
result Every planar elastica with non-constant monotone curvature is uniquely minimal.
Sub-Riemannian geometry connects bike paths to mathematical curves.
problem Understanding bike paths and their mathematical properties.
method Relating sub-Riemannian geometry to bicycle motion and curve shapes.
result Geodesics in sub-Riemannian geometry correspond to specific bike paths.
This paper classifies all planar p-elasticae and their properties.
problem Understanding the properties of p-elasticae in the plane.
method Introducing new types of p-elliptic functions to classify all planar p-elasticae.
result Explicit formulae and optimal regularity of all planar p-elasticae are obtained.
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…
New stabilization found in planar elasticae with degenerate diffusion.
problem Existence of local minimizers in degenerate p-elasticae. method Analysis of pinned planar p-elasticae with degenerate diffusion. result Uncountably many local minimizers with diverging energy in degenerate regime.
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…
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 …
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…
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…
Study of closed real plane curves with hyperelliptic genus three solutions.
problem Analyzing real plane curves with specific curvature equations.
method Examined real plane curves associated with the focusing gauged modified KdV equation of genus three.
result Showed closed real plane curves beyond Euler's figure-eight elastica.
The paper proposes a method to learn 3D object pose manifolds using GANs and elasticae.
problem Learning image manifolds of 3D objects with limited data.
method Geom-SGAN and elasticae for geometry-preserving image interpolation.
result The method outperforms state-of-the-art GANs and VAEs in learning rotation paths.
Unified approach classifies stable and minimal elastic curves.
problem Classifying stable and minimal elastic curves under various conditions.
method Unified geometric approach using a `cut-and-paste` trick.
result Complete classification of stable closed p-elasticae and stable pinned p-elasticae. In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle φ0; we refer the piece-wised elastica Λφ0-elastica or Λ-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly du…
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…
Isothermic tori with one planar curvature line found and characterized.
problem Classifying isothermic tori with specific curvature lines.
method Complex analytic methods and explicit theta function formulas.
result Explicit formulas for family of plane curves and their relation to hyperbolic elastica.
Classifies stability of flat-core p-elasticae pinned at boundaries.
problem Stability of flat-core p-elasticae under pinned boundary conditions. method Classification based on previous work for all p∈(1,∞) and d≥2. result Completes the classification of stable pinned p-elasticae in Rd. Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.
problem Minimizing the Euler-Plateau energy with elastic modulus.
method Analyzing the energy functional and its minimizers, considering different boundary conditions and topological constraints.
result Potential minimizers are highly dependent on physical rigidity parameters, and the area of critical surfaces can be computed from boundary data.
Smooth compactness theorem for elasticae, except straight segments.
problem Compactness of elasticae space.
method Smooth compactness theorem proof.
result Smooth stability results for minimizers.
Analytic non-planar p-elasticae are shown to be 3D.
problem Classifying non-planar p-elasticae. method Analyticity and structure results for p-elasticae in Rn. result Every non-planar p-elastica is analytic and three-dimensional. Gradient flow of elastic energy converges to elastica.
problem Optimizing closed curves to minimize elastic energy.
method Proving the existence of a unique global solution and convergence via Łojasiewicz--Simon gradient inequality.
result Convergence to elastica established for the H2(ds)-gradient flow of modified elastic energy. The elastica is a curve in R3 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.
problem Classifying and understanding p-elasticae under pinned boundary conditions. method Classification and analysis of p-elasticae, proving uniqueness and existence. result Discovery of a unique exponent p≃1.5728 for full optimality. 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 …
Investigates polar tangential angles of curves and their monotonicity.
problem Behavior of polar tangential angles of plane curves.
method Proof of monotonicity for certain curves of monotone curvature.
result Nonexistence results for an obstacle problem involving free elasticae.
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…
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
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 …
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…
The paper finds curves minimizing elastic energy pinned at endpoints.
problem Finding curves that minimize elastic energy with fixed endpoints.
method Applying the shooting method to identify and classify critical points.
result Critical points consist of wavelike elasticae, and minimizers have no loops or interior inflection points.
Establishes a Li-Yau type inequality for curves in any codimension.
problem Finding a lower bound for the normalized bending energy of curves in Euclidean space of any codimension.
method Variational approach, Langer-Singer's classification of elasticae, André's algebraic-independence theorem.
result Optimal inequality for any codimension except for planar closed curves with odd multiplicity.
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.
problem Understanding self-intersecting elastic curves in Euclidean space.
method Review of classical elastica theory and discussion of open problems.
result Proof of Li--Yau type inequality for self-intersecting curves.
Consider the following variational problem: among all curves in Rn of fixed length with prescribed end points and prescribed tangents at the end points, minimise the L∞-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.
problem Minimizing curvature on curves with fixed length and endpoints on Riemannian manifolds.
method Solves a second order ODE system derived from the optimization problem.
result Solutions to the optimization problem satisfy a second order ODE system.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
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
problem Designing and approximating 3D elastic curves
method Numerically stable method for recovering 11 parameters
result Fast and stable approximation of arbitrary curves
For a smooth curve γ, we define its elastic energy as E(γ)=21∫γk2(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 R2, the disc has the boundary with the least elastic energy. In…
The paper classifies and analyzes the stability of elastic curves with fixed endpoints.
problem Classification and stability of pinned elasticae.
method Critical points of the length-penalized elastic bending energy among planar curves with fixed endpoints.
result Explicit parametrization and classification of all critical points with a threshold parameter \(\hatλ \simeq 0.70107\).
Study of closed trajectories in hyperbolic plane with specific curvature constraints.
problem Critical trajectories in hyperbolic plane for a specific energy function.
method Classification of critical trajectories based on momentum causal character, proof of existence of closed trajectories.
result Existence of countably many closed trajectories with time-like momentum.
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
Proves Euler characteristic of collapsing Alexandrov spaces.
problem Euler characteristic of collapsing Alexandrov spaces.
method Analyzes strata and fibers of the limit space.
result Euler characteristic equals sum of products of strata and fiber Euler characteristics.
Study on elastic curves pinned at the boundary, focusing on minimizers and their interaction with obstacles.
problem Minimizing elastic bending energy for open planar curves with obstacles.
method Investigation of global minimizers and explicit solutions for different values of the penalization parameter.
result Explicit threshold for λ above which minimizers touch the obstacle, regardless of obstacle shape. Odd-dimensional orbifolds' Euler characteristic equals half of their boundary's.
problem Calculating the Euler characteristic of odd-dimensional orbifolds.
method Proved through mathematical analysis of orbifolds and their boundaries.
result The Euler characteristic of an odd-dimensional orbifold is half of its boundary's.
New evidence supports the Euler class one conjecture for tight contact structures.
problem Euler class one conjecture for taut foliations and tight contact structures.
method Analysis of tight contact structures and counterexamples to the conjecture.
result Counterexamples to the Euler class one conjecture for taut foliations are also Euler classes of tight contact structures.