Computing the singularities of rational space curves

In this wor we introduce the definition of fully simple singularities of parameterized curves and explain that this definition is more natural than the definition of simple singularities. The set of fully simple singularities is much smaller than the set of simple ones. We determine and classify all fully simple singularities of plane and space curves, with any number of components. Our classification results imply that any fully simple singularity of a plane or a space curve is quasi-homogeneous (whereas there is a number of non-quasi-homogeneous simple singularities). Another outcome of our classification results is a one-to-one correspondence between the fully simple singularities of plane curves and the classical A-D-E singularities of functions.

Using Smith normal forms and μ-bases to compute all the singularities of rational planar curves

We define the quasi-compact Higgs GC-bundles over singular curves introduced in our previous paper for the Lie group SL(N). The quasi-compact structure means that the automorphism groups of the bundles are reduced to the maximal compact subgroups of GC at marked points of the curves. We demonstrate that in particular cases, this construction leads to the classical integrable systems of the Hitchin type. The examples of the systems are analogs of the classical Calogero-Sutherland systems related to a simple complex Lie group GC with two types of interacting spin variables. These type models were introduced previously by Feher and Pusztai. We construct the Lax operators of the systems as the Higgs fields defined over a singular rational curve. We also construct the hierarchy of independent integrals of motion. Then we pass to a fixed point set of real involution related to one of the complex structures on the moduli space of the Higgs bundles. We prove that the number of independent integrals of motion is equal to the half of dimension of the fixed point set. The latter is a phase space of a real completely integrable system. We construct the classical r-matrix depending on the spectral parameter on a real singular curve, and in this way we prove the complete integrability of the system. We present three equivalent descriptions of the system and establish their equivalence.

It is often necessary to represent a hand-drawn shape accurately. Modeling such shapes manually is both cumbersome and commercially expensive. The user's concern for curves that are easy to manipulate has been a major influence on the development of free-form curves. Rational parametric curves have been receiving considerable attention in the area of geometric modeling because any parametric polynomial curve can be expressed as a rational curve and most polynomial splines have rational extensions. Rational parametric curves can be used to model any object, like ships and airplanes, and even in the medical field for modeling the heart or other parts of the body. In object modeling, the rational cubic spline is most popular because it is the lowest degree that can define space curves and curves with points of zero curvature. The foremost objective of this work is to show how conic sections can adequately be used to represent curves and objects that were previously thought to require rational cubic splines. A single rational cubic curve is represented by two conics by splitting the rational cubic at its mid-point. The degree of smoothness has been taken into account to have a visually pleasing effect. It has been realized that the conic representation is advantageous over the rational cubic one in terms of computational requirements and shape control. All of this work is carried out on both 2D curves and 3D objects.

Given a unicuspidal rational curve C ⊂ P^2 with singular point P, we study the unique pencil Λ_C on P^2 satisfying C ∈ Λ_C and B_s(Λ_C) = {P}. We show that the general member of Λ_C is a rational curve if and only if ν^^~(C) ≥ 0, where ν^^~(C) denotes the self-intersection number of C after the minimal resolution of singularities. We also show that if ν^^~(C) ≥ 0, then Λ_C has a dicritical of degree 1. Note that all currently known unicuspidal rational curves C ⊂ P^2 satisfy ν^^~(C) ≥ 0.

The dependence of the location of singular points of orthorhombic (ORT) media on the stiffness coefficients , and phase velocity at the singularity point is studied under the assumption that are larger than and . In this case, singular points appear only at the intersection of slowness surfaces of S1- and S2-waves. To simplify the presentation of the results, the values , are fixed and changed within the limits at which the stiffness matrix remains positive definite. We define the parameters, , , , which result in 0, 1, or 2 singular points in the symmetry planes of the ORT medium. The types of these singular points and their location on the unit circle are described. It is shown that, by selecting parameters , any singular point in the symmetry plane 13 can be combined with the limiting position of the singularity point in-between the symmetry planes, or this point can be included in the singular curve of the degenerate ORT medium. We derive equations for the semi-axes of an ellipse of conical refraction, which is the image of a singular point from plane 13 in the group domain. Conditions for degeneration of the ellipse of conical refraction into a segment or a point are defined. It is shown that there exists only one ORT medium with a fixed phase velocity of S1- and S2-waves in a given singular direction n. If we present the ORT media as projections of these vectors n onto the plane 12 and mark the value of the Poincaré index of the S1- or S2-wave at the point n, we get 2 regions with indices 1/2 and –1/2 separated by projection of the singular curve in the form of an ellipse or hyperbola. We compute equations for of a degenerate ORT medium in terms of the values, , and velocity of S1-, S2-waves on a singular curve. The singular curve is defined by the intersection of a unit sphere with an elliptic cone. It is proved that a degenerate ORT medium for or is, respectively, a transversally isotropic medium with a vertical or horizontal axis of symmetry. The results are illustrated in several examples.

A quaternion rational surface is a rational surface generated by two rational space curves via quaternion multiplication. In general, the structure of the graded minimal free resolution of a rational surface is unknown. The goal of this paper is to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. We will provide the explicit formulas for the maps of these graded minimal free resolutions. The approach we take is to utilize the information of the μ -bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. In addition, we show that the ideal generated by the first syzygy module expressed in terms of moving planes is exactly the same as the ideal generated by the parametrization in the affine ring.

We study the derived categories of coherent sheaves on some singular projective curves and give a complete description of indecomposable objects using the technique of matrix problems.

We show that on every elliptic K3 surface there are rational curves such that , that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to is dense in the Zariski topology. As an application, we give a simple proof of a theorem of Kobayashi in the elliptic case, that is, there are no globally defined symmetric differential forms.

Rational adaptive blends among obstacles in 3D by contour method

In a projective plane over a finite field, complete (k,n)-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with k close to the order of the plane, is considered a hard problem. A natural candidate to be a small (k,n)-arc with few characters is the set Ω(C) of the points of a plane curve C of degree n (containing no linear components) such that some line meets C transversally in the plane, i.e. in n pairwise distinct points. Let C be either the Hermitian curve of degree q+1 in PG(2,q2r) with r≥1, or the rational BKS curve of degree q+1 in PG(2,qr) with q odd and r≥1. Then Ω(C) has four and seven characters, respectively. Furthermore, Ω(C) is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that Ω(C) is complete for r≥4. For the rational BKS curve, Ω(C) is complete if and only if r is even. If r is odd then the uncovered points by the (q+1)-secants to Ω(C) are exactly the points in PG(2,q) not lying in Ω(C). Adding those points to Ω(C) produces a complete (k,q+1)-arc in PG(2,qr), with k=qr+q. The above results do not hold true for r=2 and there remain open the case r=3 for the Hermitian curve, and the cases r=3,4 for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for PGL(2,q).

Electromagnetic curves and Berry phase construction of a polarized light wave along an optical fiber which is a singular curve on [formula omitted

Mechanical fields over thin elastic surfaces can develop singularities at isolated points and curves in response to constrained deformations (e.g., crumpling and folding of paper), singular body forces and couples, distributions of isolated defects (e.g., dislocations and disclinations), and singular metric anomaly fields (e.g., growth and thermal strains). With such concerns as our motivation, we model thin elastic surfaces as von Kármán plates and generalize the classical von Kármán equations, which are restricted to smooth fields, to fields which are piecewise smooth, and can possibly concentrate at singular curves, in addition to being singular at isolated points. The inhomogeneous sources to the von Kármán equations, given in terms of plastic strains, defect induced incompatibility, and body forces, are likewise allowed to be singular at isolated points and curves in the domain. The generalized framework is used to discuss the singular nature of deformation and stress arising due to conical deformations, folds, and folds terminating at a singular point.

Approximating uniform rational B-spline curves by polynomial B-spline curves

Detecting Cusps and Inflection Points in Curves