Rational Plane Curves Research Articles

Rational adaptive blends among obstacles in 3D by contour method

In this paper we will continue in investigating ‘contour method’ and its using for the computation of rational parameterizations of canal surfaces without a need of sum of squares (SOS) decomposition. Further approaches for constructing flexible smooth transitions between canal surfaces will be presented. Mainly, we focus on one particular application of recently introduced rational envelope curves, newly constructed over an arbitrary planar rational curve in space. Using this type of curves significantly simplifies the previous methods discussed in Bizzarri (2015), and mainly new situations, which could not have been handled with the previous setup, are successfully solved, now. Especially a method for constructing rational adaptive blends which bypass a given obstacle (or more given obstacles when needed) is thoroughly discussed and its functionality is demonstrated on a number of examples. The designed approach works not only for simple obstacles represented by one-dimensional medial axis transforms but also for more general obstacles described by two-dimensional medial surface transforms.

An algebraic framework for computing the topology of offsets to rational curves

A new algebraic framework is introduced for computing the topology of the offset Cδ at distance δ to a rational plane curve C defined by a parameterization (x(t),y(t)). The focus is on computing the topology of Cδ by analyzing the image of the parameterization of Cδ which involves square roots. This framework is mainly intended to deal with curves that bring initially complicated singularities or with curves such that the offset to compute introduces such singularities making approximation techniques difficult to apply in these cases.In this framework the topology of Cδ is determined by computing, among other notable points, its singular, discontinuity and self-intersection points together with analyzing the ordering of these points, according to the values of the parameter t, obtaining in this way the final branching producing the searched topology for Cδ. The computation of the singular and discontinuity points requires determining the real roots of two univariate polynomials. Self-intersection points are characterized as the intersection of two auxiliary algebraic curves and require to compute only one sequence of subresultants. This approach requires only the manipulation of x(t) and y(t) without computing and dealing with the implicit equation of Cδ (known to be typically a huge polynomial difficult to deal with).

Fundamental groups of a class of rational cuspidal plane curves

We compute the presentations of fundamental groups of the complements of a class of rational cuspidal projective plane curves classified by Flenner, Zaidenberg, Fenske and Saito. We use the Zariski–Van Kampen algorithm and exploit the Cremona transformations used in the construction of these curves. We simplify and study the group presentations so obtained and determine if they are abelian, finite or big, i.e. if they contain free non-abelian subgroups. We also study the quotients of these groups to some extent.

Explicit μ-bases for conic sections and planar rational cubic curves

We derive explicit formulas for the μ-bases of conic sections and planar rational cubic curves. Using the μ-bases for planar rational cubic curves, we find explicit formulas for their implicit equations and double points. We also extend the explicit formula for the μ-bases of conic sections to μ-bases for rational curves of degree n in n-dimensions.

On parameterizations of plane rational curves and their syzygies

Let C be a plane rational curve of degree d and be its normalization. We are interested in the splitting type of C, where gives the syzigies of the ideal , and is a parameterization of C. We want to describe in which cases (, via a geometric description; namely we show that if and only if C is the projection of a rational curve on a rational normal surface in .

Symmetry detection of rational space curves

We present a novel, deterministic, and efficient method to detect whether a given rational space curve C is symmetric. The method combines two ideas. On one hand in a similar way to [1], [2], if C is symmetric then the symmetry provides a second parametrization of the curve; furthermore, whenever the first parametrization is proper, i.e. injective except for finitely many parameter values, the latter is also proper and both are related by means of a Mobius transformation [3] that completely determines the symmetry. On the other hand, if C is symmetric then the curvature and torsion of C at corresponding points must coincide. By putting together these two ideas we can give an algorithm to directly find the Mobius transformations defining the symmetries of the curve. From here we can compute these symmetries and its characteristic elements (symmetry axes, symmetry planes, etc.) This completes and improves on an earlier method addressing a similar problem [3]. Keywords Symmetry Detection, Space Curves, Rational Curves References [1] Alcazar J.G. (2014), Efficient detection of symmetries of polynomially parametrized curves, Journal of Computational and Applied Mathematics vol. 255, pp. 715–724. [2] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Similarity of Plane Rational Plane Curves, Journal of Computational and Applied Mathematics, Vol. 269, pp. 1–13. [3] Alcazar J.G., Hermoso C., Muntingh G. (2014), Detecting Symmetries of Rational Plane and Space Curves, to appear in Computer Aided Geometric Design.

Iterative computation of the convex hull of a rational plane curve

Iterative computation of the convex hull of a rational plane curve

On classifications of rational sextic curves

Abstract In this paper, we extend the Yang’s list of reduced sextic plane curves to rational irreducible projective plane curves of type (6, 3, 1).

A new method to compute the singularities of offsets to rational plane curves

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

Equivariantly uniformly rational varieties

We introduce equivariant versions of uniform rationality: given an algebraic group $G$, a $G$-variety is called $G$-uniformly rational (resp. $G$-linearly uniformly rational) if every point has a $G$-invariant open neighborhood equivariantly isomorphic to a $G$-invariant open subset of the affine space endowed with a $G$-action (resp. linear $G$-action). We establish a criterion for $\mathbb{G}_m$-uniform rationality of affine variety equipped with hyperbolic $\mathbb{G}_m$-action with a unique fixed point, formulated in term of their Altmann-Hausen presentation. We prove the $\mathbb{G}_m$-uniform rationality of Koras-Russel threefolds of the first kind and we also give example of non $\mathbb{G}_m$-uniformly rational but smooth rational $\mathbb{G}_m$-threefold associated to pairs of plane rational curves birationally non equivalent to a union of lines.

On Ballico-Hefez curves and associated supersingular surfaces

Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q + 1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingular. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in characteristic 3 and 5.

Chord Diagrams and Generic Real Rational Planar Curves of Degree 4

Chord Diagrams and Generic Real Rational Planar Curves of Degree 4

Syzygies and projective generation of plane rational curves

We investigate the relationship between rational plane curves and the envelopes defined by the syzygies of their parameterizations.

Linear systems associated to unicuspidal rational plane curves

Given a unicuspidal rational curve $C \subset \mathbb{P}^{2}$ with singular point $P$, we study the unique pencil $\Lambda_{C}$ on $\mathbb{P}^{2}$ satisfying $C \in \Lambda_{C}$ and $\mathrm{Bs}(\Lambda_{C})=\{P\}$. We show that the general member of $\Lambda_{C}$ is a rational curve if and only if $\tilde{\nu}(C) \ge 0$, where $\tilde{\nu}(C)$ denotes the self-intersection number of $C$ after the minimal resolution of singularities. We also show that if $\tilde{\nu}(C) \ge0$, then $\Lambda_{C}$ has a dicritical of degree $1$. Note that all currently known unicuspidal rational curves $C \subset \mathbb{P}^{2}$ satisfy $\tilde{\nu}(C) \ge 0$.

Detecting similarity of rational plane curves

A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and the particular case of equal curves yields all symmetries. A complete theoretical description of the method is provided, and the method has been implemented and tested in the Sage system for curves of moderate degrees.

Detecting symmetries of rational plane and space curves

This paper addresses the problem of determining the symmetries of a plane or space curve defined by a rational parametrization. We provide effective methods to compute the involution and rotation symmetries for the planar case. As for space curves, our method finds the involutions in all cases, and all the rotation symmetries in the particular case of Pythagorean-hodograph curves. Our algorithms solve these problems without converting to implicit form. Instead, we make use of a relationship between two proper parametrizations of the same curve, which leads to algorithms that involve only univariate polynomials. These algorithms have been implemented and tested in the Sage system.

On plane rational curves and the splitting of the tangent bundle

Given an immersion ' : P1 ! P2, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as ' : P1 = D ⇢ X ! P2, where X ! P2 is obtained by blowing up r distinct points pi 2 P2. As applications in the case that the points pi are generic, we give a complete determination of the splitting types for such immersions when r  7. The case that D2 = 1 is of particular interest. For r  8 generic points, it is known that there are only finitely many inequivalent ' with D2 = 1, and all of them have balanced splitting. However, for r = 9 generic points we show that there are infinitely many inequivalent ' with D2 = 1 having unbalanced splitting (only two such examples were known previously). We show that these new examples are related to a semi-adjoint formula which we conjecture accounts for all occurrences of unbalanced splitting when D2 = 1 in the case of r = 9 generic points pi . In the last section we apply such results to the study of the resolution of fat point schemes. Mathematics Subject Classification (2010): 14C20 (primary); 13P10, 14J26, 14J60 (secondary).

μ-Bases for complex rational curves

We present a fast algorithm for finding a μ-basis for any rational planar curve that has a complex rational parametrization. We begin by identifying two canonical syzygies that can be extracted directly from any complex rational parametrization without performing any additional calculations. For generic complex rational parametrizations, these two special syzygies form a μ-basis for the corresponding real rational curve. In those anomalous cases where these two canonical syzygies do not form a μ-basis, we show how to quickly calculate a μ-basis by performing Gaussian elimination on these two special syzygies. We also present an algorithm to determine if a real rational planar curve has a complex rational parametrization. Examples are provided to illustrate our methods.

Irreducible Cycles and Points in Special Position in Moduli Spaces for Tropical Curves

In the first part of this paper, we discuss the notion of irreducibility of cycles in the moduli spaces of $n$-marked rational tropical curves. We prove that Psi-classes and vital divisors are irreducible, and that locally irreducible divisors are also globally irreducible for $ n \le 6 $. In the second part of the paper, we show that the locus of point configurations in $({\mathbb R}^2)^n $ in special position for counting rational plane curves (defined in two different ways) can be given the structure a tropical cycle of codimension $1$. In addition, we compute explicitly the weights of this cycle.

Implicitization of curves and (hyper)surfaces using predicted support

We reduce implicitization of rational planar parametric curves and (hyper)surfaces to linear algebra, by interpolating the coefficients of the implicit equation given a superset of its terms. For predicting these terms, we focus on methods that exploit input and output structure in the sense of sparse (or toric) elimination theory, namely by computing the Newton polytope of the implicit polynomial, via sparse resultant theory. Our algorithm works even in the presence of base points but, in this case, the implicit equation shall be obtained as a factor of the produced polynomial. We implement our methods in Maple, and some in Matlab as well, and study their numerical stability and efficiency on several classes of curves and surfaces. We apply our approach to approximate implicitization, and quantify the accuracy of the approximate output, which turns out to be satisfactory on all tested examples. In building a square or rectangular interpolation matrix, an important issue is (over)sampling the given curve or surface: we conclude that unitary complex numbers offer the best tradeoff between speed and accuracy when numerical methods are employed, namely SVD, whereas for exact kernel computation random integers is the method of choice. We compare our prototype to existing software and find that it is rather competitive.