Plane Curves Research Articles

Vanishing cycles, plane curve singularities and framed mapping class groups

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type $A_n$ and $D_n$, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least 7.

Hilbert transforms along variable planar curves: Lipschitz regularity

In this paper, we obtain the Lp-boundedness of the Hilbert transform Hγ for 1<p<∞ along a variable plane curve (t,u(x1,x2)γ(t)), where u is a Lipschitz function with small Lipschitz norm, and γ is a general curve satisfying some suitable smoothness and curvature conditions.

Complex planar curves homeomorphic to a line have at most four singular points

We show that a complex planar curve homeomorphic to the projective line has at most four singular points. If it has exactly four then it has degree five and is unique up to a projective equivalence.

Advances in the Theory of Planar Curve Cognates

Abstract Cognate linkages provide the useful property in mechanism design of having the same motion. This paper describes an approach for determining all coupler curve cognates for planar linkages with rotational joints. Although a prior compilation of six-bar cognates due to Dijksman purported to be a complete list, that analysis assumed, without proof, that cognates only arise by permuting link rotations. Our approach eliminates that assumption using arguments concerning the singular foci of the coupler curve to constrain a cognate search and then completing the analysis by solving a precision point problem. This analysis confirms that Dijksman’s list for six-bars is comprehensive. As we further demonstrate on an eight-bar and a ten-bar example, the method greatly constrains the set of permutations of link rotations that can possibly lead to cognates, thereby facilitating the discovery of all cognates that arise in that manner. However, for these higher order linkages, the further step of using a precision point test to eliminate the possibility of any other cognates is still beyond our computational capabilities.

The geometry of the Wigner caustic and a\xa0decomposition of a curve into parallel arcs

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

A Length-Constrained Ideal Curve Flow

Abstract A recent article [1] considered the so-called ‘ideal curve flow’, a sixth-order curvature flow that seeks to deform closed planar curves to curves with least variation of total geodesic curvature in the L2 sense. It was critical in the analysis in that article that there was a length bound on the evolving curves. It is natural to suspect therefore that the length-constrained ideal curve flow should permit a more straightforward analysis, at least in the case of small initial ‘energy’. In this article we show this is indeed the case, with suitable initial data providing a flow that exists for all time and converges smoothly and exponentially to a multiply-covered round circle of the same length and winding number as the initial curve.

Исследование отражения от криволинейных зеркал на плоскости в программе Wolfram Mathematica

In this article, the study of the geometry of the flat shapes reflection from curved lines located in the plane of these shapes continues. The paper is devoted to a more detailed description of reflection from the analytical geometry point of view. In addition, the range of proposed tasks has been significantly expanded.&#x0D; An algorithm for reflecting zero-dimensional and one-dimensional objects from plane curves is compiled, and corresponding illustrations are given.&#x0D; For the first time, the authors have obtained equations that allow us to construct reflections of a point from second-order curves: a circle, an ellipse, a parabola and a hyperbola, as well as from high-order curves: Bernoulli lemniscates and cardioids [17], [24], [13], [25], [23], [22]. In addition, equations for the reflection results of one-dimensional objects: a segment and a circle, from the same plane curves were obtained. Similar studies are being conducted in the works [2], [1], [32], [28], [3], [4]. All equations are accompanied by blueprints of special cases of reflections obtained using the Wolfram Mathematica mathematical package [18], [19]. In addition, the application contains the source codes, which gives the reader to configure the reflection parameters themselves on condition having access this program, as well as visually assess the change in the reflection pattern when changing these parameters for various types of flat mirrors.&#x0D; This article demonstrates the possibilities that the obtained equations provide, and the prospects for further work, which consist in obtaining new equations of objects reflected from other flat mirrors.

Pedal Curves of the Mixed-Type Curves in the Lorentz-Minkowski Plane

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane R12. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve.

Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data

It is quite common for functional data arising from imaging data to assume values in infinite-dimensional manifolds. Uncovering associations between two or more such nonlinear functional data extracted from the same object across medical imaging modalities can assist development of personalized treatment strategies. We propose a method for canonical correlation analysis between paired probability densities or shapes of closed planar curves, routinely used in biomedical studies, which combines a convenient linearization and dimension reduction of the data using tangent space coordinates. Leveraging the fact that the corresponding manifolds are submanifolds of unit Hilbert spheres, we describe how finite-dimensional representations of the functional data objects can be easily computed, which then facilitates use of standard multivariate canonical correlation analysis methods. We further construct and visualize canonical variate directions directly on the space of densities or shapes. Utility of the method is demonstrated through numerical simulations and performance on a magnetic resonance imaging dataset of glioblastoma multiforme brain tumors.

Computing projective equivalences of planar curves birationally equivalent to elliptic and hyperelliptic curves

We present algorithms to find the projective equivalences between two planar curves, birationally equivalent to either elliptic or hyperelliptic curves. The main idea, in both cases, is to find first a corresponding birational mapping between the Weierstrass normal forms of the curves. The computation of this mapping benefits from the abundant theory on elliptic and hyperelliptic curves available in the literature. In particular, we discuss cubic curves in detail, and completely characterize the projective equivalences in terms of the isomorphisms between the Weierstrass forms leaving the infinity point invariant, and the inflection points of one of the curves.

Millimetre-wave range optical properties of BIBO

We present the thorough studies of optical properties of BiB3O6 (BIBO) crystal in the millimeter-wave (subterahertz) range. We observe a large birefringence Δn = nZ -nX = 1.5 and the values of absorption coefficients of all three axes to be less than 0.5 cm−1 at the frequency of 0.3 THz. The difference from visible range in angle ϕ between the dielectric axis z and crystallophysical axis X is found to be more than 6°. The simulated phase-matching curves in the xz plane of the crystal show the optimal value of the angle θ to be around 25.5°±1° for an efficient millimeter-wave generation under the pump of 1064 nm laser radiation.

Symmetry of maximals for fractional ideals of curves

The purpose of this paper is to extend the symmetry of maximals of the ring of a germ of a reducible plane curve proved by Delgado to a relation between the relative maximals of a fractional ideal and the absolute maximals of its dual for any admissible ring. In particular, it includes the case of germs of reduced reducible curve of any codimension. We then apply this symmetry to characterize the elements in the set of values of a fractional ideal from some of its projections and the irreducible absolute maximals of the dual ideal.

Torsion and vertical curvature of motion-trajectory curves

The paper discusses difference between super-elevation of recorded motion-trajectory (MT) curves and constant super-elevation of curve plane and demonstrates that vertical curvature used in railroad literature is not necessarily associated with curve out-of-plane bending or twist. Vertical curvature can be highly nonlinear while the curve remains planar and untwisted. A computational procedure is proposed for identifying MT planar curves using computer-simulation or experimentally recorded data. Data-driven science (DDS) approach allows measuring Frenet vertical-development and super-elevation that define centrifugal inertia force. Difference between motion-independent curve-plane and motion-dependent osculating plane (OP) which contains velocity, acceleration, and inertia centrifugal forces is explained. Three Frenet angles: horizontal-curvature, vertical-development, and bank angles; are used to define planar MT geometry. Zero-torsion and zero-vertical-curvature conditions are derived for identifying planar curves and demonstrating that vertical curvature is not always associated with curve torsion. An orthogonal transformation is used to define OP sweeping angle whose derivative defines curve curvature regardless of curve-plane orientation. Difference between Frenet horizontal-curvature angle and OP sweeping angle is discussed. It is shown that super-elevation of planar-curve surface is equal to Frenet super-elevation if Frenet vertical-development angle is zero. An analytical 3D, yet planar and untwisted, curve is used to demonstrate that planar curves can have non-vanishing and nonlinear curvature, vertical-elevation, vertical curvature, and super-elevation.

Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.

Two-stage spline-approximation in linear structure routing

In the article, computer design of routes of linear structures is considered as a spline approximation problem. A fundamental feature of the corresponding design tasks is that the plan and longitudinal profile of the route consist of elements of a given type. Depending on the type of linear structure, line segments, arcs of circles, parabolas of the second degree, clothoids, etc. are used. In any case, the design result is a curve consisting of the required sequence of elements of a given type. At the points of conjugation, the elements have a common tangent, and in the most difficult case, a common curvature. Such curves are usually called splines. In contrast to other applications of splines in the design of routes of linear structures, it is necessary to take into account numerous restrictions on the parameters of spline elements arising from the need to comply with technical standards in order to ensure the normal operation of the future structure. Technical constraints are formalized as a system of inequalities. The main distinguishing feature of the considered design problems is that the number of elements of the required spline is usually unknown and must be determined in the process of solving the problem. This circumstance fundamentally complicates the problem and does not allow using mathematical models and nonlinear programming algorithms to solve it, since the dimension of the problem is unknown. The article proposes a two-stage scheme for spline approximation of a plane curve. The curve is given by a sequence of points, and the number of spline elements is unknown. At the first stage, the number of spline elements and an approximate solution to the approximation problem are determined. The method of dynamic programming with minimization of the sum of squares of deviations at the initial points is used. At the second stage, the parameters of the spline element are optimized. The algorithms of nonlinear programming are used. They were developed taking into account the peculiarities of the system of constraints. Moreover, at each iteration of the optimization process for the corresponding set of active constraints, a basis is constructed in the null space of the constraint matrix and in the subspace – its complement. This makes it possible to find the direction of descent and solve the problem of excluding constraints from the active set without solving systems of linear equations. As an objective function, along with the traditionally used sum of squares of the deviations of the initial points from the spline, the article proposes other functions taking into account the specificity of a particular project task.

Effectiveness of the Bendixson–Dulac theorem

We illustrate with several new applications the power and elegance of the Bendixson–Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function related with the curvature of the orbits of the vector field as a Dulac function. We get some general results for Liénard type equations and for rigid planar systems. We also present a remarkable phenomenon: for each integer m≥2, we provide a simple 1-parametric differential system for which we prove that it has limit cycles only for the values of the parameter in a subset of an interval of length smaller that 32(3/m)m/2 that decreases exponentially when m grows. One of the strengths of the results presented in this work is that although they are obtained with simple calculations, that can be easily checked by hand, they improve and extend previous studies. Another one is that, for certain systems, it is possible to reduce the question of the number of limit cycles to the study of the shape of a planar curve and the sign of an associated function in one or two variables.

Out-of-Plane Deformation Analysis of the Thin-Walled Closed Curved Box Girder under the Temperature Gradient

For calculating the thin-walled closed curved box girder caused by the temperature gradient of the internal force and displacement, based on the fundamental differential equation of the curve beam and the principle of minimum energy, set a reverse statically indeterminate simply supported curve beam as the basic structure, consider the warping effect of the closed curve box girder, and put forward a kind of plane curve beam temperature deformation simple analytical calculation method. Compared with the finite element calculation results, the relative error of the analytical calculation results is less than 5%. It is concluded that the analytical method has sufficient accuracy in calculating the out-of-plane deformation of the thin-walled closed curved box girder under the temperature gradient.

Recovering affine curves over finite fields from L-functions

Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$.

The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.