Search

AbstractRational conic fibrations embedded in PN and containing an irreducible curve C transverse to the fibers are classified when C is either a plane curve or a rational normal curve of any degree d. This extends our previous results concerning lines, conics and twisted cubics. To emphasize the role of linear normality, the case in which C is a non-normal rational quartic is also discussed. Moreover, the rank of the d-th jet map of the surface at all points lying on such curves is evaluated. [ABSTRACT FROM AUTHOR]

A method of forming a curved orthogonal coordinate system on a plane and a technique of constructing new surface shapes with curved trapezoidal plans are presented. Multiple examples of curved trapezoidal plans based on different directrix curves and surfaces with the given plans, including combinations of surfaces with different conjugate directrix curves, are illustrated. The proposed technique of surface forming may be used in architecture and construction for development of thin-walled space structures in both urban and industrial buildings. But for the analysis of thin shells, geometric characteristics of the middle surface of the shell are usually used. Vector equation of surfaces with curved trapezoidal plan was used to obtain the formulas for the fundamental form coefficients and surface curvatures. Examples of calculation of the fundamental form coefficients and curvatures of surfaces with particular directrix curves and vertical coordinate functions are presented.

An elementary and self-contained argument for the complete determination of maximal plane curves of degree 3 over F4 will be given, which complements Hirschfeld-Storme-Thas-Voloch’s theorem on a characterization of Hermitian curves in P². This complementary part should be understood as the classification of Sziklai’s example of maximal plane curves of degree q − 1 over Fq. Although two maximal plane curves of degree 3 over F4 up to projective equivalence over F4 appear, they are birationally equivalent over F4 each other. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let X be the normalization of an integral degree d ≤ 9 plane curve Y. We prove that X has a unique gd² if h¹ (P², IZ([d/2]-3)) = 0, where Z is the conductor of Y . Moreover, Y is the unique plane model of X of degree at most d. [ABSTRACT FROM AUTHOR]

There are three well-known Arnold's invariants J + , J − , and St (Arnold strangeness) for plane curves. Two of them, J + and J − , have been successfully quantized. However, the quantization of the Arnold strangeness St remained unrealized. In this paper, we will express a quantization of St by integrating curvatures multiplied by nontrivial densities. The key here is a partition function by Shumakovitch which reformulates the Arnold strangeness. Our quantized Arnold strangeness includes the rotation number, the original Arnold strangeness, and the Tabachnikov invariants as coefficients in its Taylor expansion. [ABSTRACT FROM AUTHOR]

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants A , B and Gauss's constant G, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results. [ABSTRACT FROM AUTHOR]

We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it rotational being the difference between the usual curvature and the inner product of the normal vector field and the angular vector field. But, since the problem of vanishing of this curvature involves complicated expressions we define a second rotational curvature. Both these curvatures are computed for several examples.

Let C : f = 0 be a reduced curve in the complex projective plane. The minimal degree m d r (f) of a Jacobian syzygy for f , which is the same as the minimal degree of a derivation killing f , is an important invariant of the curve C , for instance it can be used to determine whether C is free or nearly free. In this note we study the relations of this invariant m d r (f) with a decomposition of C as a union of two curves C 1 and C 2 , without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer information on this invariant in this setting. Using this, we give some geometrical criteria, the first ones of this type in the existing literature as far as we know , for a line to be a jumping line for the rank 2 vector bundle of logarithmic vector fields along a reduced curve C. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In [8] the first author classified automorphism groups of smooth plane curves of degree not less than four into five types. If the curve has a unique outer Galois point, then the quotient group of its automorphism group by the Galois group at the point, which is called the reduced automorphism group, is a finite subgroup of one-dimensional projective linear group. This article is a sequel of [10] and [9]. In this article, we shall determine the defining equation of the curve when the reduced automorphism group is an icosahedral group and give a description of the full automorphism group. [ABSTRACT FROM AUTHOR]

In [8] the first author classified automorphism groups of smooth plane curves of degree not less than four into five types. If the curve has a unique outer Galois point, then the quotient group of its automorphism group by the Galois group at the point, which is called the reduced automorphism group, is a finite subgroup of one-dimensional projective linear group. This article is a sequel of [10] and [9]. In this article, we shall determine the defining equation of the curve when the reduced automorphism group is an icosahedral group and give a description of the full automorphism group. [ABSTRACT FROM AUTHOR]

A criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented. Surprisingly, the Hermitian function field in positive characteristic becomes the Galois closure of projections of a plane curve from two non-uniform points. [ABSTRACT FROM AUTHOR]

In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants A,B and Gauss’s constant G, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results.

In this paper, we investigate a nonlinear evolution equation which is derived from plane curve by Lie symmetry analysis and theory of planar dynamical system. The one-dimensional optimal system of the equation is given and all bifurcations of the system in different parametric regions are given from which we can guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions and many uncountable infinite traveling wave solutions. Besides, we provide exact expressions of traveling wave solutions under different parametric conditions. [ABSTRACT FROM AUTHOR]

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. [ABSTRACT FROM AUTHOR]

Copyright of Lubrication Engineering (0254-0150) is the property of Editorial Office of LUBRICATION ENGINEERING and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This article presents novel examples of plane curves such that the Galois closures of projections from two points are the same, using a criterion described by Fukasawa, Higashine, and Takahashi. [ABSTRACT FROM AUTHOR]

We study the change of the minimal degree of a logarithmic derivation of a hyperplane arrangement under the addition or the deletion of a hyperplane and give a number of applications. First, we prove the existence of Tjurina maximal line arrangements in a lot of new situations. Then, starting with Ziegler's example of a pair of arrangements of d = 9 lines with n 3 = 6 triple points in addition to some double points, having the same combinatorics, but distinct minimal degree of a logarithmic derivation, we construct new examples of such pairs, for any number d ≥ 9 of lines, and any number n 3 ≥ 6 of triple points. Moreover, we show that such examples are not possible for line arrangements having only double and triple points, with n 3 ≤ 5 . [ABSTRACT FROM AUTHOR]

We consider the parameter space 𝒰 d of smooth plane curves of degree d. The universal smooth plane curve of degree d is a fiber bundle ℰ d → 𝒰 d with fiber diffeomorphic to a surface Σ g . This bundle gives rise to a monodromy homomorphism ρ d : π 1 (𝒰 d) → Mod (Σ g) , where Mod (Σ g) : = π 0 ( Diff + (Σ g)) is the mapping class group of Σ g . The main result of this paper is that the kernel of ρ 4 : π 1 (𝒰 4) → Mod (Σ 3) is isomorphic to F ∞ × ℤ / 3 ℤ , where F ∞ is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement Teich (Σ g) ∖ ℋ g of the hyperelliptic locus ℋ g in Teichmüller space Teich (Σ g) has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry. [ABSTRACT FROM AUTHOR]

We describe the module of integrable derivations in the sense of Hasse-Schmidt of the quotient of the polynomial ring in two variables over an ideal generated by the equation xn - yq. [ABSTRACT FROM AUTHOR]

Copyright of KoG is the property of Croatian Society for Geometry & Graphics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A Riemann surface S having field of moduli ℝ, but not a field of definition, is called pseudo-real. This means that S has anticonformal automorphisms, but none of them is an involution. A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d ≥ 4 in P C 2 . We characterize pseudo-real-plane Riemann surfaces S , whose conformal automorphism group Aut+(S) is PGL3(ℂ)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of P C 2 . In particular, we show that such pseudo-real-plane Riemann surfaces exist only if Aut+(S) is cyclic of even order n dividing the degree d. Explicit families of pseudo-real-plane Riemann surfaces are given for any degree d = 2pm with m > 1 odd, p prime and n = d/p. [ABSTRACT FROM AUTHOR]

Nonsingular plane curves over a finite field F q of degree q + 2 passing through all the F q -points of the plane admit a representation by 3 × 3 matrices over F q. We classify their degenerations by means of the matrix representation. We also discuss the similar problem for the affine-plane filling projective curves of degree q + 1. [ABSTRACT FROM AUTHOR]

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups GA(2) = GL(2, ℝ) ⋉ ℝ2 and GA(3) = GL(3, ℝ) ⋉ ℝ3, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves. [ABSTRACT FROM AUTHOR]

This paper presents a symbolic algorithm to compute the topology of a plane curve. This is a full version of the authors' CASC15 paper. The algorithm mainly involves resultant computations and real root isolation for univariate polynomials. Compared to other symbolic methods based on elimination techniques, the novelty of the proposed method is that the authors use a technique of interval polynomials to solve the system { f (α , y) , ∂ f ∂ y (α , y) } and simultaneously obtain numerous simple roots of f(α, y) = 0 on the α fiber. This significantly improves the efficiency of the lifting step because the authors are no longer required to compute the simple roots of f(α, y) = 0. After the topology is computed, a revised Newton's method is presented to compute an isotopic meshing of the plane algebraic curve. Though the approximation method is numerical, the authors can ensure that the proposed method is a certified one, and the meshing is topologically correct. Several nontrivial examples confirm that the proposed algorithm performs well. [ABSTRACT FROM AUTHOR]

We consider plane curves of the form (ax)α + (by)α = rα defined on the first quadrant of R², where α > 0 and a, b, r > 0. We summarize the outlines of them by using elementary differential calculus. We will in this note understand that they are classified into three types of curves, convex, straight and concave, depending on α. [ABSTRACT FROM AUTHOR]

Let 𝓒 be a plane curve defined over the algebraic closure K of a finite prime field 𝔽p by a separated polynomial, that is 𝓒 : A(Y) = B(X), where A(Y) is an additive polynomial of degree pn and the degree m of B(X) is coprime with p. Plane curves given by separated polynomials are widely studied; however, their automorphism groups are not completely determined. In this paper we compute the full automorphism group of 𝓒 when m ≢ 1 mod pn and B(X) = Xm. Moreover, some sufficient conditions for the automorphism group of 𝓒 to imply that B(X) = Xm are provided. Also, the full automorphism group of the norm-trace curve 𝓒 : X(qr – 1)/(q–1) = Yqr–1 + Yqr–2 + ... + Y is computed. Finally, these results are used to show that certain one-point AG codes have many automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we give a definition of ℤ -valued functions from the ambient isotopy classes of spherical/plane curves derived from chord diagrams, denoted by ∑ i α i x i . Then, we introduce certain elements of the free ℤ -module generated by the chord diagrams with at most l chords, called relators of Type (I) ((SI I), (WI I), (SI I I), or (WI I I), respectively), and introduce another function ∑ i α i x ̃ i derived from ∑ i α i x i . The main result (Theorem 1) shows that if ∑ i α i x ̃ i vanishes for the relators of Type (I) ((SI I), (WI I), (SI I I), or (WI I I), respectively), then ∑ i α i x i is invariant under the Reidemeister move of type RI (strong RI I, weak RI I, strong RI I I, or weak RI I I, respectively) that is defined in [N. Ito and Y. Takimura, (1 , 2) and weak (1 , 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085 14 pp]. [ABSTRACT FROM AUTHOR]

Evolutes and pedals of plane curves have been well investigated since the beginning of the history of differential geometry. However, there might be no direct relationships between the pedal and the evolute of a curve. We introduce families of relatives of pedals and evolutes and investigate some relationships between these families curves. Moreover, we generalize these notions to the category of frontal curves. Then the relation can be completely described in this category. [ABSTRACT FROM AUTHOR]

Abstract We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve C. The relation to the zero set of Bernstein–Sato polynomial of the defining homogeneous polynomial for C is also discussed. When C has some non-weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds. [ABSTRACT FROM AUTHOR]

We prove that a set X, #X = mn, m ≤ n, is the set of intersection points of some two plane algebraic curves of degrees m and n, respectively, if and only if the following conditions are satisfied: a) Any curve of degree m+ n − 3 containing all but one point of X, contains all of X, b) No curve of degree less than m contains all of X. The conditions a) and b) in the"only if" direction of this result follow fromthe Ceyley-Bacharach and Noether theorems, respectively. [ABSTRACT FROM AUTHOR]

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram. [ABSTRACT FROM AUTHOR]

In this paper we use the Apollonius definition of conics to generate algebraic curves in the Minkowski space-time plane M2, which turn out to be different from classical conic sections. We extend and classify this sort of “M-conics”. We discuss the cases of the singularity points of these M-conics, coming from the transition from timelike world to spacelike world through the lightlike one. Finally, we translate the classical concept of Cassini curves with two foci and that of (multifocal) Cassini curves to Minkowski planes M2.