Plane Curves Research Articles

On the jerk and snap in motion along non‐lightlike curves in Minkowski 3‐space

In this paper, we study the jerk vector that is the rate of change of the acceleration vector over time. In three‐dimensional space, the decomposition of the jerk vector is a new concept in the field. This decomposition expresses the jerk vector as the sum of three unique components in specific directions: the tangential direction, the radial direction in the osculating plane, and the radial direction in the rectifying plane. The snap vector is the rate of change of the jerk vector over time. In this paper, the authors examine non‐relativistic particles moving along non‐lightlike Frenet curves at low speeds compared to the speed of light in Minkowski 3‐space. They resolve the jerk and snap vectors using Frenet–Serret frames. Additionally, the cases for motion along non‐lightlike Frenet planar curves in the Minkowski 3‐space are given as corollaries. To help understand these results, the paper provides some illustrative examples

Pedal and contrapedal curves in equi-affine plane

In this paper, we define equi-affine pedal and contrapedal curves in equi-affine plane. Then, we also consider the relationships among equi-affine evolutes, involutes, parallels, pedal and contrapedal curves. In addition, we investigate the classifications of singularities of equi-affine pedal and contrapedal curves.

Galois points and Cremona transformations

Abstract In this article, we study Galois points of plane curves and the extension of the corresponding Galois group to $\mathrm{Bir}(\mathbb{P}^2)$ . We prove that if the Galois group has order at most $3$ , it always extends to a subgroup of the Jonquières group associated with the point $P$ . Conversely, with a degree of at least $4$ , we prove that it is false. We provide an example of a Galois extension whose Galois group is extendable to Cremona transformations but not to a group of de Jonquières maps with respect to $P$ . In addition, we also give an example of a Galois extension whose Galois group cannot be extended to Cremona transformations.

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 .

S-shaped rolling gait designed using curve transformations of a snake robot for climbing on a bifurcated pipe

In this work, we focus on overcoming the challenge of a snake robot climbing on the outside of a bifurcated pipe. Inspired by the climbing postures of biological snakes, we propose an S-shaped rolling gait designed using curve transformations. For this gait, the snake robot’s body presenting an S-shaped curve is wrapped mainly around one side of the pipe, which leaves space for the fork of the pipe. To overcome the difficulty in constructing and clarifying the S-shaped curve, we present a method for establishing the transformation between a plane curve and a 3D curve on a cylindrical surface. Therefore, we can intuitively design the curve in 3D space, while analytically calculating the geometric properties of the curve in simple planar coordinate systems. The effectiveness of the proposed gait is verified by actual experiments. In successful configuration scenarios, the snake robot could stably climb on the pipe and efficiently cross or climb to the bifurcation while maintaining its target shape.

Generalized null Cartan helices and principal normal worldsheets in Minkowski 3-space

We give a method for constructing generalized null Cartan helices, which may have singularities, by using regular space-like plane curves and smooth functions in Minkowski 3-space. We also study the principle normal worldsheet, which is deeply related to generalized null Cartan helices from the viewpoint of singularity theory.

A note on complex plane curve singularities up to diffeomorphism and their rigidity

We prove that if two germs of plane curves (C, 0) and (C′,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C',0)$$\\end{document} with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then C is complex isomorphic to C′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C'$$\\end{document} or to C′¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{C'}$$\\end{document}. A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component that is non-factorable. Moreover, we provide a general overview of the different classifications of plane curve singularities.

Counting singular curves with prescribed tangency

We obtain an explicit formula for the characteristic number of degree d curves in P2 with prescribed singularities (of type Ak) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. We are not aware of any explicit formula to enumerate plane curves of degree d with any number of Ak singularities (beyond codimension 8); however, combined with the results of S. Basu and R. Mukherjee ([1], [2], and [3]), this gives us a complete formula for the characteristic number of curves with δ-nodes and one singularity of type Ak, tangent to a given line, provided δ+k≤8. We use a topological method to compute the degenerate contribution to the Euler class. We have made several low degree checks to verify special cases of our result. When the singularities are only nodes, we have verified that our numbers are logically consistent with those computed by L. Caporaso and J. Harris ([7]). We also verify that our answers for the characteristic number of cubics with a cusp tangent to a given line and the characteristic number of quartics with two nodes and a cusp, tangent to a given line is logically consistent with the characteristic number of rational cubics and quartics tangent to a given line that was computed by L. Ernström and G. Kennedy ([9]).

The analytic classification of plane curves

In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allows us to recover some particular cases found in the literature.

Hexagonal pencils of cubic plane curves

This expository paper considers real cubic plane curves as combinations C=a_{1} T_{1} + a_{2} T_{2} of two real triangles meeting in nine distinct points. Our study of hexagonal pencils , defined by C with parameters a_{i} \in \mathbb{R} , picks up on some themes of a recent paper on cubics by Bonifant and Milnor (2017).

The Polya f-curvature of plane curves

We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it Pólya, being the difference between the usual curvature and the inner product of the normal vector field with the Pólya vector field of a given planar function f. We computed it for several examples, since the general problem of vanishing or constant values of this new curvature involves the general expression of f. Mathematics Subject Classification (2010): 53A04, 53A45, 53A55. Received 03 May 2023; Accepted 17 January 2024

Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements

In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals of Levi graphs associated to some curve arrangements in the complex projective plane, like the d-arrangement of curves and the conic-line arrangements. We also discuss the existence of certain induced cycles in the Levi graphs of these arrangements and obtain lower bounds for the regularity of powers of the corresponding binomial edge ideals.

Curves whose curvature depends on their position and null curves

We show that, apart from degeneracies, determining a plane curve whose curvature depends on its position essentially consists of determining a null curve in the Lorentzian 3-space when the null tangent direction depends on its position. We use this point of view to investigate the intrinsic equations for the n-elastic curves. We show how the problem of prescribed null tangent direction in terms of the position can be solved by quadratures when the prescription exhibits sufficient symmetries. This problem is formalized in terms of a convenient contact 3-form.

CurveML: a benchmark for evaluating and training learning-based methods of classification, recognition, and fitting of plane curves

AbstractWe propose CurveML, a benchmark for evaluating and comparing methods for the classification and identification of plane curves represented as point sets. The dataset is composed of 520k curves, of which 280k are generated from specific families characterised by distinctive shapes, and 240k are obtained from Bézier or composite Bézier curves. The dataset was generated starting from the parametric equations of the selected curves making it easily extensible. It is split into training, validation, and test sets to make it usable by learning-based methods, and it contains curves perturbed with different kinds of point set artefacts. To evaluate the detection of curves in point sets, our benchmark includes various metrics with particular care on what concerns the classification and approximation accuracy. Finally, we provide a comprehensive set of accompanying demonstrations, showcasing curve classification, and parameter regression tasks using both ResNet-based and PointNet-based networks. These demonstrations encompass 14 experiments, with each network type comprising 7 runs: 1 for classification and 6 for regression of the 6 defining parameters of plane curves. The corresponding Jupyter notebooks with training procedures, evaluations, and pre-trained models are also included for a thorough understanding of the methodologies employed.

Bounds for Rational Points on Algebraic Curves, Optimal in the Degree, and Dimension Growth

Abstract Bounding the number of rational points of height at most $H$ on irreducible algebraic plane curves of degree $d$ has been an intense topic of investigation since the work by Bombieri and Pila. In this paper we establish optimal dependence on $d$ by showing the upper bound $C d^{2} H^{2/d} (\log H)^{\kappa }$ with some absolute constants $C$ and $\kappa $. This bound is optimal with respect to both $d$ and $H$, except for the constants $C$ and $\kappa $. This answers a question raised by Salberger, leading to a simplified proof of his results on the uniform dimension growth conjectures of Heath-Brown and Serre, and where at the same time we replace the $H^{\varepsilon }$ factor by a power of $\log H$. The main strength of our approach comes from the combination of a new, efficient form of smooth parametrizations of algebraic curves with a century-old criterion of Pólya, which allows us to save one extra power of $d$ compared with the standard approach using Bézout’s theorem.

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.

Pearcey integrals, Stokes lines and exact baryonic layers in the low energy limit of QCD

The first analytic solutions representing baryonic layers living at finite baryon density within a constant magnetic field in the gauged Skyrme model are constructed. A remarkable feature of these configurations is that, if the Skyrme term is neglected, then these baryonic layers in the constant magnetic background cannot be found analytically and their energies grow very fast with the magnetic field. On the other hand, if the Skyrme term is taken into account, the field equations can be solved analytically and the corresponding solutions have a smooth limit for large magnetic fields. Thus, the Skyrme term discloses the universal character of these configurations living at finite Baryon density in a constant magnetic field. The classical gran-canonical partition function of these configurations can be expressed explicitly in terms of the Pearcey integral. This fact allows us to determine analytically the Stokes lines of the partition function and the corresponding dependence on the baryonic chemical potential as well as on the external magnetic field. In this way, we can determine various critical curves in the (μB−Bext) plane which separates different physical behaviors. These families of inhomogeneous baryonic condensates can be also dressed with chiral conformal excitations of the solutions representing modulations of the layers themselves. Some physical consequences are analyzed.

Voronoi cells in metric algebraic geometry of plane curves

Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.

Hardy decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.

Most plane curves over finite fields are not blocking

A plane curve C⊂P2 of degree d is called blocking if every Fq-line in the plane meets C at some Fq-point. We prove that the proportion of blocking curves among those of degree d is o(1) when d≥2q−1 and q→∞. We also show that the same conclusion holds for smooth curves under the somewhat weaker condition d≥3p and d,q→∞. Moreover, the two events in which a random plane curve is smooth and respectively blocking are shown to be asymptotically independent. Extending a classical result on the number of Fq-roots of random polynomials, we find that the limiting distribution of the number of Fq-points in the intersection of a random plane curve and a fixed Fq-line is Poisson with mean 1. We also present an explicit formula for the proportion of blocking curves involving statistics on the number of Fq-points contained in a union of k lines for k=1,2,…,q2+q+1.