Plane Curves Research Articles

Lattice points on a curve via ℓ2 decoupling

This paper extends Bombieri and Pila’s estimate of lattice points on curves to arbitrary finite sets by incorporating considerations of minimal separation and the doubling constant. We derive the estimate by establishing the [Formula: see text] decoupling inequality for non-degenerate curves in [Formula: see text]. Additionally, we review the curve-lifting method introduced in Bombieri and Pila’s work and establish the estimate of lattice points in the neighborhood of a planar curve.

Riemann-Hilbert problems, Toeplitz operators and ergosurfaces

The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix M depending on the Weyl coordinates ρ and v, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the (ρ, v) plane on which some elements of M(ρ, v) tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.

Wall crossing for K‐moduli spaces of plane curves

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

Reading the log canonical threshold of a plane curve singularity from its Newton polyhedron

There is a proposition due to Kollár as reported by Kollár (Proceedings of the summer research institute, Santa Cruz, CA, USA, July 9–29, 1995, American Mathematical Society, Providence, 1997) on computing log canonical thresholds of certain hypersurface germs using weighted blowups, which we extend to weighted blowups with non-negative weights. Using this, we show that the log canonical threshold of a convergent complex power series is at most 1/c, where (c,…,c)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(c, \\ldots , c)$$\\end{document} is a point on a facet of its Newton polyhedron. Moreover, in the case n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 2$$\\end{document}, if the power series is weakly normalised with respect to this facet or the point (c, c) belongs to two facets, then we have equality. This generalises a theorem of Varchenko 1982 to non-isolated singularities.

On evolutes of curves in the isotropic plane

On evolutes of curves in the isotropic plane

Geometric Investigation of Three Thin Shells with Ruled Middle Surfaces with the Same Main Frame

It is proved and illustrated that by taking the main frame of the surface, consisting of three plane curves placed in three coordinate planes, three different algebraic surfaces with the same rigid frame can be designed. For the first time, one three of new ruled surfaces in a family of five threes of ruled surfaces, formed on the basis of some shapes of hulls of river and see ships, which, in turn, are projected in the form of algebraic surfaces with a main frame of three superellipses or of three other plane curves, is under consideration in detail with a standpoint of differential geometry. The geometrical properties of the ruled surfaces taken as the middle surfaces of thin shells for industrial and civil engineering are presented. Analytical formulas for determination of force resultants with using the approximate momentless theory of shells of zero Gaussian curvature given by non-orthogonal conjugate curvilinear coordinates are offered for the first time. The results derived using these formulae will help to correct the results obtained by numerical methods.

Exploring helices in Minkowski $3$-space $\mathbb{E}_{1}^{3}$

This study focuses on the geometric exploration of helices in Minkowski $3-$ space. For this purpose, we study the problem of constructing a spacelike general helix, slant helix, or Darboux helix with a timelike principal from a given plane curve, fixed vector, and constant angle. We obtain a parametric representation for the helices whose projection is onto the plane $P$ perpendicular to the fixed vector $U$ share the same fixed vector. In addition, we furnish various illustrative examples showcasing the geometric characteristics of these helices.

Free curves, eigenschemes, and pencils of curves

AbstractLet . A reduced plane curve is free if its associated module of tangent derivations is a free ‐module, or equivalently if the corresponding sheaf of vector fields tangent to splits as a direct sum of line bundles on . In general, free curves are difficult to find, and in this paper, we describe a new method for constructing free curves in . The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves.

Pκ-Curves: Interpolatory curves with curvature approximating a parabola

This paper introduces a novel class of fair and interpolatory planar curves called pκ-curves. These curves are comprised of smoothly stitched Bézier curve segments, where the curvature distribution of each segment is made to closely resemble a parabola, resulting in an aesthetically pleasing shape. Moreover, each segment passes through an interpolated point at a parameter where the parabola has an extremum, encouraging the alignment of interpolated points with curvature extrema. To achieve these properties, we tailor an energy functional that guides the optimization process to obtain the desired curve characteristics. Additionally, we develop an efficient algorithm and an initialization method, enabling interactive modeling of the pκ-curves without the need for global optimization. We provide various examples and comparisons with existing state-of-the-art methods to demonstrate the curve modeling capabilities and visually pleasing appearance of pκ-curves.

An Effective Method for Slicing Triangle Meshes Using a Freeform Curve

Slicing 3D polygonal meshes is a fundamental operation in various applications such as virtual surgery, garment simulation, and game development. Existing methods primarily slice meshes using either a single line or a set of line segments approximating a smooth curve. This paper introduces a novel approach to freely slice a triangle mesh using a freeform curve without discretizing it into line segments. The user draws a stroke on the screen, defining the desired cutting trajectory. Subsequently, a freeform curve approximating this stroke is generated and extended into a ruled surface in the user’s viewing direction. To efficiently compute intersections between the ruled surface and a triangle mesh, the Line–Surface Intersection (LSI) problem is broken down into two subproblems: Plane–Curve Intersection (PCI) followed by Line–Line Intersection (LLI). Intersection points are then connected to form polylines, effectively cutting the mesh into multiple submeshes. To ensure the solidity of the submeshes, cross-sections are generated by trimming the ruled surface along the polylines and merged with the corresponding submeshes. Our method empowers users to slice triangle meshes along arbitrary trajectories encompassing both straight and freely curved paths while preserving efficiency and accuracy. The effectiveness of the proposed approach is demonstrated through experimental results showing various examples of mesh slicing.

Orthogonal trajectories to isoptics of ovals

AbstractFor a given plane curve, consider a one-parameter family of curves consisting of those points at which two support lines to the initial curve intersect at a constant angle. Such curves are well known in differential and convex geometry and called isoptics. In this paper, we describe parametrizations of orthogonal trajectories to isoptics of ovals. We show that such parametrizations can be obtained using solutions to a specific Cauchy problem constructed from the parametrizations of the oval and its isoptics. Moreover, we provide analytical and numerical examples of orthogonal trajectories to isoptics of some ovals.

Analyzing the dynamics of a charged rotating rigid body under constant torques

This study explores the dynamical rotary motion of a charged axisymmetric spinning rigid body (RB) under the effect of a gyrostatic moment (GM). The influence of transverse and invariable body fixed torques (IBFTs), and an electromagnetic force field, is also considered. Euler’s equations of motion (EOM) are utilized to derive the regulating system of motion for the problem in a suitable formulation. Due to the lack of torque exerted along the spin axis and the nearly symmetrical nature of the RB, the spin rate is nearly unchanged. Assuming slight angular deviations of the spin axis relative to a fixed direction in space, it is possible to derive approximate analytical solutions (AS) in closed form for the attitude, translational, and rotational movements. These concise solutions that are expressed in complex form are highly effective in analyzing the maneuvers performed by spinning RBs. The study focuses on deriving the AS for various variables including angular velocities, Euler’s angles, angular momentum, transverse displacements, transverse velocities, axial displacement, and axial velocity. The graphical simulation of the subsequently obtained solutions is presented to show their precision. Furthermore, the positive impacts that alterations in the body’s parameters have on the motion’s behavior are presented graphically. The corresponding phase plane curves, highlighting the influence of different values in relation to the electromagnetic force field, the GM, and the IBFTs are drawn to analyze the stability of the body’s motion. This study has a significant role in various scientific and engineering disciplines. Its importance lies in its ability to optimize mechanical systems, explain celestial motion, and enhance spacecraft performance.

A higher dimensional Marcinkiewicz exponent and the Riemann boundary value problems for polymonogenic functions on fractals domains

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space Rn+1,n≥2 for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.

General rigidity principles for stable and minimal elastic curves

Abstract For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary conditions for global and local minimizers. Our results extend Maddocks’ and Sachkov’s rigidity principles for Euler’s elastica by a new, unified and geometric approach. This in particular leads to complete classification of stable closed p-elasticae for all p ∈ ( 1 , ∞ ) {p\in(1,\infty)} and of stable pinned p-elasticae for p ∈ ( 1 , 2 ] {p\in(1,2]} . Our proof is based on a simple but robust “cut-and-paste” trick without computing the energy nor its second variation, which works well for planar periodic curves but also extends to some non-periodic or non-planar cases. An analytically remarkable point is that our method is directly valid for the highly singular regime p ∈ ( 1 , 3 2 ] {p\in(1,\frac{3}{2}]} in which the second variation may not exist even for smooth variations.

Cost-effectiveness of diagnosis and treatment of early gestational diabetes mellitus: economic evaluation of the TOBOGM study, an international multicenter randomized controlled trial

A recently undertaken multicenter randomized controlled trial (RCT) "Treatment Of BOoking Gestational diabetes Mellitus" (TOBOGM: 2017-2022) found that the diagnosis and treatment of pregnant women with early gestational diabetes mellitus (GDM) improved pregnancy outcomes. Based on data from the trial, this study aimed to assess the cost-effectiveness of diagnosis and treatment of early GDM (from <20 weeks') among women with risk factors for hyperglycemia in pregnancy compared with usual care (no treatment until 24-28 weeks') from a healthcare perspective. Participants' healthcare resource utilization data were collected from their self-reported questionnaires and hospital records, and valued using the unit costs obtained from standard Australian national sources. Costs were reported in US dollars ($) using the purchasing power parity (PPP) estimates to facilitate comparison of costs across countries. Intention-to-treat (ITT) principle was followed. Missing cost data were replaced using multiple imputations. Bootstrapping method was used to estimate the uncertainty around mean cost difference and cost-effectiveness results. Bootstrapped cost-effect pairs were used to plot the cost-effectiveness (CE) plane and cost-effectiveness acceptability curve (CEAC). Diagnosis and treatment of early GDM was more effective and tended to be less costly, i.e., dominant (cost-saving) [-5.6% composite adverse pregnancy outcome (95% CI:-10.1%,-1.2%),-$1373 (95% CI:-$3,749, $642)] compared with usual care. Our findings were confirmed by both the CE plane (88% of the bootstrapped cost-effect pairs fall in the south-west quadrant), and CEAC (the probability of the intervention being cost-effective ranged from 84% at a willingness-to-pay (WTP) threshold value of $10,000-99% at a WTP threshold value of $100,000 per composite adverse pregnancy outcome prevented). Sub-group analyses demonstrated that diagnosis and treatment of early GDM among women in the higher glycemic range (fasting blood glucose 95-109mg/dl [5.3-6.0mmol/L], 1-h blood glucose ≥191mg/dl [10.6mmol/L] and/or 2-h blood glucose 162-199mg/dl [9.0-11.0mmol/L]) was more effective and less costly (dominant) [-7.8% composite adverse pregnancy outcome (95% CI:-14.6%,-0.9%),-$2795 (95% CI:-$6,638,-$533)]; the intervention was more effective and tended to be less costly [-8.9% composite adverse pregnancy outcome (95% CI:-15.1%,-2.6%),-$5548 (95% CI:-$16,740, $1547)] among women diagnosed before 14 weeks' gestation as well. Our findings highlight the potential health and economic benefits from the diagnosis and treatment of early GDM among women with risk factors for hyperglycemia in pregnancy and supports its implementation. Long-term follow-up studies are recommended as a key future area of research to assess the potential long-term health benefits and economic consequences of the intervention. National Health and Medical Research Council (grants 1104231 and 2009326), Region O¨rebro Research Committee (grants Dnr OLL-970566 and OLL-942177), Medical Scientific Fund of the Mayor of Vienna (project 15,205 and project 23,026), South Western Sydney Local Health District Academic Unit (grant 2016), and Western Sydney University Ainsworth Trust Grant (2019).

Primes of the form $X^{3} + NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

Primes of the form $X^{3} + NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

On the ∂—-Equation with L2 Estimates on Singular Complex Spaces

Abstract In this paper, we present the unsolvability of $\overline \partial $-equation with weighted $L^{2}$ estimates involved curvature terms on any singular normal complex space in general. Moreover, in the non-normal case, we also give a complete description on $L^{2}$-solvability of the $\overline \partial $-equation with weighted $L^{2}$ estimates for plane curve singularities and their variants in the higher dimension.

Spectral Curves for Third-Order ODOs

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg–de Vries equation and to seminal works on commuting ODOs by I. Schur and Burchnall and Chaundy. They allow the solvability of the spectral problem Ly=λy, for an algebraic parameter λ and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of an abstract spectral curve Γ. In this work, we use differential resultants to effectively compute the defining ideal of the spectral curve Γ, defined by the centralizer of a third-order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. For this purpose, defining ideals of planar spectral curves associated to commuting pairs are described as radicals of differential elimination ideals. In general, Γ is a non-planar space curve and we provide the first explicit example. As a consequence, the computation of a first-order right factor of L−λ becomes explicit over a new coefficient field containing Γ. Our results establish a new framework appropriate to develop a Picard–Vessiot theory for spectral problems.

More fermionic supersymmetric Wilson loops in four dimensions

We construct supersymmetric fermionic Wilson loops along general curves in four-dimensional N=4 super Yang-Mills theory and along general planar curves in N=2 superconformal SU(N)×SU(N) quiver theory. These loops are generalizations of the Zarembo loops and are cohomologically equivalent to them. In N=4 super Yang-Mills theory, we compute their expectation values and verify the cohomological equivalence relation up to the order g4 in perturbation theory.

On rationally integrable planar dual multibilliards and piecewise smooth projective billiards

The billiard flow in a planar domain Ω acts on the tangent bundle TR2|Ω as geodesic flow with reflections from the boundary. It has the trivial first integral: squared modulus of the velocity. Bolotin’s conjecture, now a joint theorem of Bialy, Mironov and the author, deals with those billiards whose flow admits an additional integral that is polynomial in the velocity and whose restriction to the unit tangent bundle is non-constant. It states that (1) if the boundary of such a billiard is C 2-smooth, nonlinear and connected, then it is a conic; (2) if it is piecewise C 2-smooth and contains a nonlinear arc, then it consists of arcs of conics from a confocal pencil and segments of ‘admissible lines’ for the pencil; (3) the minimal degree of the additional integral is either 2, or 4. In 1997 Sergei Tabachnikov introduced projective billiards: planar curves equipped with a transversal line field, which defines a reflection acting on oriented lines and the projective billiard flow. They are common generalization of billiards on surfaces of constant curvature, but in general may have no canonical conserved quantity. In a previous paper the author classified those C 4-smooth connected nonlinear planar projective billiards whose flow admits a non-constant integral that is a rational 0-homogeneous function of the velocity (with coefficients depending on the position): these billiards are called rationally 0-homogeneously integrable. It was shown that: (1) the underlying curve is a conic; (2) the minimal degree of integral is equal to two, if the billiard is defined by a dual pencil of conics; (3) otherwise it can be arbitrary even number. In the present paper we classify piecewise C4-smooth rationally 0-homogeneously integrable projective billiards. Unexpectedly, we show that such a billiard associated to a dual pencil of conics may have integral of minimal degree 2, 4, or 12. For the proof of main results we prove dual results for the so-called dual multibilliards introduced in the present paper.