Plane Curves Research Articles

Computation of extension spaces for the path algebra of type Ã(n - 1,1) using planar curves

Q is a quiver of type if its graph is of affine type and if its arrows have a certain orientation. We develop a bijection between the set of indecomposable kQ-modules whose dimension vectors are positive real roots of the root system associated to Q and a certain set of planar curves. We prove that the number of self-intersections of the curve which corresponds to the module M is equal to the dimension of . We also prove that, for many pairs of modules (M, N), the number of intersections between the corresponding two curves is equal to the dimension of , where C is the cluster category of kQ-mod.

Resistance-frequency eddy current method for electrical conductivity measurement

In eddy current measurements of electrical conductivity, adverse effects from lift-off effects are inevitable, especially for relatively large lift-offs. In this work, a new method of electrical conductivity measurement, the resistance-frequency eddy current method, is proposed. The method uses an LC resonator as the measurement probe and characterizes the conductivity of the measured material by exploiting the physics that the ratio of the resonant frequency to the resonant resistance of the probe is linearly related to the logarithm of the conductivity. Furthermore, a two-dimensional resistance-frequency plane (RF plane) consisting of resonant frequency and resonant resistance is established. The resistance frequency lift-off curves in the RF plane have good linearity in the med-high lift-off range, which can be used to eliminate lift-off effects. Experimental results show that the proposed method can perform conductivity measurements in the med-high lift-off range of 1 mm to 2.5 mm independent of lift-off, with relative errors within 5 %. Therefore, the proposed method is particularly suitable for conductivity measurements in the med-high lift-off range, especially for metallic materials with thicker surface coatings.

The inverse problem for fractal curves solved with the dynamical approach method

The purpose of the present paper is to present the main applications of a new method for the determination of the fractal structure of plane curves. It is focused on the inverse problem, that is, given a curve in the plane, find its fractal dimension. It is shown that the dynamical approach extends the characterization of a curve as a fractal object introducing the effects of mass density, elastic properties, and transverse geometry. The dynamical dimension characterizes material objects and suggests that biological characterization can be much more complete with the methodology presented here.

Control of transit doses by Monte Carlo simulation inside an ionization casemate housing of a 60Co gamma irradiator

To increase the quality of irradiation at the Moroccan National Institute for Agronomic Research (NIAR) and meet customer needs, A calculation of the transit doses in the NIAR ionization casemate housing of a 60Co gamma irradiator was done in this work using Monte Carlo simulation with GATE/GEANT4 code. These additional doses called in this study transit doses come from the movement of the source cage carrying 8 60Co rods from their position in storage mode to their irradiation position then, from their irradiation position to their position in storage mode at the end of the irradiation process. In this work, the dosimetric mapping of transit doses and the isodose curves in the horizontal plane passing through the irradiation position of the source cage were established. In addition, a calculation of these doses for products placed on two turntables arranged around the NIAR gamma irradiator was also made in this work. The calculations are made taking into account three parameters, namely the number of irradiation fractions, the distance from the irradiator and the activity of the source. The results obtained showed that for a number of fractions of irradiations equal to 2, the transit doses vary between 0.22Gy and 5.68Gy for products placed on one of the two turntables at a distance varying between 47 cm and 147 cm from the gamma irradiator.

A Beautiful Inequality by Saint-Venant and Pólya Revisited

In mathematical physics and beyond, one encounters many beautiful inequalities that relate geometric or physical quantities describing the shape or size of a set. Such isoperimetric inequalities often have a long history and many important applications. For instance, the eponymous and most classical of all isoperimetric inequalities was known already in antiquity. It asserts that among all closed planar curves of a given length, the circles with perimeter equal to that length, and only they, enclose the largest area. Though not nearly as well-known, an isoperimetric inequality conjectured by Saint-Venant in the 1850s and first proved by Pólya almost a century later, is also very beautiful and important. By presenting a short proof as well as two simple physical interpretations, this article illustrates why the result deserves to be cherished by every student of applied analysis.

On Control Polygons of Planar Sextic Pythagorean Hodograph Curves

In this paper, we analyze planar parametric sextic curves to determine conditions for Pythagorean hodograph (PH) curves. By expressing the curves to be analyzed in the complex form, the analysis is conducted in algebraic form. Since sextic PH curves can be classified into two classes according to the degrees of their derivatives’ factors, we introduce auxiliary control points to reconstruct the internal algebraic structure for both classes. We prove that a sextic curve is completely characterized by the lengths of legs and angles formed by the legs of their Bézier control polygons. As such conditions are invariant under rotations and translations, we call them the geometric characteristics of sextic PH curves. We demonstrate that the geometric characteristics form the basis for an easy and intuitive method for identifying sextic PH curves. Benefiting from our results, the computations of the parameters of cusps and/or inflection points can also be simplified.

Algebraic curves admitting non-collinear Galois points

This paper presents a criterion for the existence of a birational embedding into a projective plane with non-collinear Galois points for algebraic curves and describes its application via a novel example of a plane curve with non-collinear Galois points. In addition, this paper presents a new characterisation of the Fermat curve in terms of non-collinear Galois points.

Three-dimensional spline-based computer-generated holography.

Electro-holography is a promising 3D display technology, as it can, in principle, account for all visual cues. Computing the interference patterns to drive them is highly calculation-intensive, requiring the design and development of efficient computer-generated holography (CGH) algorithms to facilitate real-time display. In this work, we propose a new algorithm for computing the CGH for arbitrary 3D curves using splines, as opposed to previous solutions, which could only draw planar curves. The solutions are analytically expressed; we conceived an efficiently computable approximation suitable for GPU implementations. We report over 55-fold speedups over the reference point-wise algorithm, resulting in real-time 4K holographic video generation of complex 3D curved objects. The proposed algorithm is validated numerically and optically on a holographic display setup.

Monodromy and period map of the Winger pencil

AbstractThe sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and three times a conic). This pencil was first considered in a note by R. M. Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain integral representation of the icosahedral group. We find that the elliptic curve comes with a distinguished point of order 3, which proves that the monodromy on this part of the homology is the full congruence subgroup and subsequently identify the base of the pencil with the associated modular curve. We also observe that the Winger pencil “accounts” for the deformation of the Jacobian of Bring's curve as a principal abelian fourfold with an action of the icosahedral group.

Hyperbolic inverse mean curvature flow with forced term: Evolution of plane curves

The motion of plane curves under hyperbolic inverse mean curvature with a constant force field is considered. The preserving convexity of the evolving curve under this flow is proved. Furthermore, we give an example to understand how the forced term affects the evolution of plane curves. Particularly, the asymptotic behavior of the flow is discussed.

Vanishing of higher order Alexander‐type invariants of plane curves

AbstractThe higher order degrees are Alexander‐type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves and in terms of the finiteness and the vanishing properties of the invariants of and , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial is a power of , and we characterize when in terms of the defining equations of and . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.

Elementary Treatise on Melted-like Curves Derived from Center Ellipse and Circle

This study deals with a special case in the geometry of 2D special plane curves. The research treaties the development of a method for locating points and drawing curves derived from moving points on two curved paths of conic sections, namely, circle and ellipse. The paper presents geometrical methods to determine points and nodes and to draw these melted-like curves in two cases. Parametric equations are also used to determine the ordinates of the points of each curve of the two cases of the curves. This research is the initial phase for the new type of curves and awaits future studies that will deal on 3D surfaces, such as a sphere, a right cone, or a right cylinder.

Characteristics of planar sextic indirect-PH curves

<abstract><p>By a fractional quadratic transformation, an indirect-PH curve can have rational offsets. In this paper, I study properties of planar sextic indirect-PH curves, in terms of their Bézier control polygon legs. With our results, sextic Bézier curves can be efficiently tested whether they are indirect-PH curves. The main strategy to achieve our results is using complex representation of planar parametric curve. Sextic indirect-PH curves can be classified into three classes according to different factorizations of their hodographs. Necessary and sufficient conditions for all classes of sextic indirect-PH curves can be described by non-linear complex systems. By analyzing these non-linear systems, algebraic conditions for a sextic Bézier curve to be an indirect-PH curve are first discussed, then geometric characteristics in terms of legs of its control polygon are revealed.</p></abstract>

CONSTRUCTION OF GEODESIC LINES ON SURFACES OF ROTATION OBTAINED BY DISPLACEMENT OF THE MERIDIAN

Geodesic surface lines are analogous to straight lines on a plane. In addition to connecting two points of the surface by the shortest distance, they are the winding trajectories of the reinforcing threads in the strengthening of high-pressure cylinders. Just as a bundle of straight lines can be drawn from a given point on a plane in different directions, so there are geodesic lines on a surface that pass through a given point in different directions. Finding geodesic surface lines in the general case comes down to solving second-order differential equations. The purpose of the study is to investigate geodesic lines on the surface formed by the rotation of a given plane curve around a vertical axis and their transformation when this curve is shifted away from or towards the axis. For surfaces of revolution, the second-order differential equation can be reduced to the first order and even reduced to an integral based on the well-known Clerot formula. However, in this case, geodesic lines in all directions can be constructed only for a limited number of surfaces of rotation, and only limited fragments of geodesic lines can be constructed on the remaining surfaces. The article considers the construction of geodesic lines using the numerical solution of a second-order differential equation. The obtained results were visualized.

Non-intersecting curved paths using Bezier curves in the 2D-Euclidean plane for multiple autonomous robots

Non-intersecting curved paths using Bezier curves in the 2D-Euclidean plane for multiple autonomous robots

Minimal wave speed for a population model with density-dependent migrations and the Allee effect

In this paper, we study the minimal wave speed for a class of population models with density-dependent migration speed and the Allee effect. We first give the exact expression for the minimal wave speed for the case of linear migration speed. This in turn specifies a critical curve in the parameter plane of the migration speed and the Allee effect which separates a region of the linear determinacy of the minimal wave speed from a region of the nonlinear determinacy of the minimal wave speed. Next, for a class of nonlinear migration speed, the corresponding minimal wave speed is shown to enjoy similar qualitative properties as depicted in the case of linear migration speed.

Перспективы применения переходных кривых, совмещенных с кривыми в вертикальной плоскости

This article focuses on the benefits of using spatial geometry in a railway design. The mentioned spatial geometry is a combination of a transitional curve and a curve in a vertical plane. Some advantages and disadvantages of this approach in comparison to the one used on Russian railways today are emphasized. Moreover, research in related fields is being considered and a new conventional scheme of railway geometry has been introduced which uses the above-mentioned spatial geometry, biclothoid curve design and non-linear ramp of a cant in a transitional curve. The conclusion about inappropriateness of some current regulatory requirements for railways in the Russian Federation is made.

$ G^2/C^1 $ Hermite interpolation of offset curves of parametric regular curves

<abstract><p>In this paper we presented a method of $ G^2 $ Hermite interpolation of offset curves of regular plane curves based on approximating the normal vector fields. We showed that our approximant is also $ C^1 $ Hermite interpolation of the offset curve. Our method is capable of achieving circular precision. Another advantage of our method is that if the input curve is a polynomial curve, then our method also yields a polynomial curve. Our approximation method was applied to numerical examples and its numerical results were compared to previous offset approximation methods. It was observed that our method is almost optimal with respect to the number of control points of the approximation curves for the same tolerance.</p></abstract>

The radial curvature of plane curves

The radial curvature of plane curves

Projective plane curves over a finite field with conical components

We present the list of maximal projective plane curves containing conics and those which are arrangements of conics. The number of rational points and the corresponding polynomials are given. The third highest number of points of projective curves of degree d over a finite field Fq (d < [q/3]) is associated only to some linear curves. We show that for q/2 + 2 < d < q, this is no longer the case: the third highest number of points can also be obtained by some curves containing a conic. Throughout this work, we obtain some bounds concerning the number of Fq-points of curves with linear, conic and cubic factors. these bounds apply (not sharply) to irreducible curves