Rational Space Curves Research Articles

Algebraic And Geometric Properties Of A Family of Rational Curves

This paper consists of two components - an application part and a theoretical part, where the former targets the applications of computer aided geometric designs in generating parametric curves, and the latter focuses on the algebraic analysis of rational space curves. At the application level, we construct a family of rational space curves via quaternion products of two generating curves. At the theoretical level, we use algebraic methods to extract a $\mu$-basis for this family of curves, and describe a basis for a special submodule of the syzygy module in terms of a $\mu$-basis for the syzygy module of this family of curves. A commutative diagram is provided to summarize these results.

Restrictions on rational surfaces lying in very general hypersurfaces

AbstractWe study rational surfaces on very general Fano hypersurfaces in$\mathbb {P}^n$, with an eye toward unirationality. We prove that given any fixed family of rational surfaces, a very general hypersurface of degreedsufficiently close tonandnsufficiently large will admit no maps from surfaces in that family. In particular, this shows that for such hypersurfaces, any rational curve in the space of rational curves must meet the boundary. We also prove that for any fixed ratio$\alpha $, a very general hypersurface in$\mathbb {P}^n$of degreedsufficiently close tonwill admit no generically finite maps from a surface satisfying$H^2 \geq \alpha HK$, whereHis the pullback of the hyperplane class from$\mathbb {P}^n$andKis the canonical bundle on the surface.

Counting tropical rational space curves with cross-ratio constraints

This is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

Ruled Surfaces of Revolution with Moving Axes and Angles

A ruled surface of revolution with moving axes and angles is a rational tensor product surface generated from a line and a rational space curve by rotating the line (the directrix) around vectors and angles generated by the rational space curve (the director). Only right circular cylinders and right circular cones are ruled surfaces that are also surfaces of revolution, but we show that a rich collection of other ruled surfaces such as hyperboloids of one sheet, 2-fold Whitney umbrellas, and a wide variety other interesting ruled shapes are ruled surfaces of revolution with moving axes and angles. We present a fast way to compute the implicit equation of a ruled surface of revolution with moving axes and angles from two linearly independent vectors that are perpendicular to the directrix of the surface. We also provide an algorithm for determining whether or not a given rational ruled surface is a ruled surface of revolution with moving axes and angles.

Space Curves and Solitons of the KP Hierarchy. I. The l-th Generalized KdV Hierarchy

It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta functions associated with hyperelliptic curves, and that soliton solutions can be obtained by rational (singular) limits of the corresponding curves. In this paper, we discuss a class of KP solitons in connections with space curves, which are labeled by certain types of numerical semigroups. In particular, we show that some class of the (singular and complex) KP solitons of the $l$-th generalized KdV hierarchy with $l\ge 2$ is related to the rational space curves associated with the numerical semigroup $\langle l,lm+1,\dots, lm+k\rangle$, where $m\ge 1$ and $1\le k\le l-1$. We also calculate the Schur polynomial expansions of the $\tau$-functions for those KP solitons. Moreover, we construct smooth curves by deforming the singular curves associated with the soliton solutions. For these KP solitons, we also construct the space curve from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.

Computing μ-Bases of Univariate Polynomial Matrices Using Polynomial Matrix Factorization

This paper extends the notion of μ-bases to arbitrary univariate polynomial matrices and present an efficient algorithm to compute a μ-basis for a univariate polynomial matrix based on polynomial matrix factorization. Particularly, when applied to polynomial vectors, the algorithm computes a μ-basis of a rational space curve in arbitrary dimension. The authors perform theoretical complexity analysis in this situation and show that the computational complexity of the algorithm is $${\cal O}\left( {d{n^4} + {d^2}{n^3}} \right)$$ , where n is the dimension of the polynomial vector and d is the maximum degree of the polynomials in the vector. In general, the algorithm is n times faster than Song and Goldman’s method, and is more efficient than Hoon Hong’s method when d is relatively large with respect to n. Especially, for computing μ-bases of planar rational curves, the algorithm is among the two fastest algorithms.

Quaternion rational surfaces

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.

Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

Given a polynomial space curve r(ξ) that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves r˜(ξ) with the same rotation–minimizing frame as r(ξ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u(ξ)=r′(ξ)×r″(ξ) and distances from the origin specified in terms of a rational function f(ξ) as f(ξ)/‖u(ξ)‖. An explicit characterization of the rational curves r˜(ξ) generated by a given RRMF curve r(ξ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f(ξ), obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve r(ξ). Criteria for identifying low–degree instances of the curves r˜(ξ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples.

Surfaces of revolution with moving axes and angles

A surface of revolution with moving axes and angles is a rational tensor product surface generated from two rational space curves by rotating one curve (the directrix) around vectors and angles generated by the other curve (the director). Here we introduce these new kinds of rational generalized surfaces of revolution, provide some interesting examples, and investigate their algebraic and geometric properties. In particular, we study the base points and syzygies of these rational surfaces. We construct three special syzygies for a surface of revolution with moving axes and angles from a μ-basis of the directrix, and we show how to compute the implicit equation of these rational surfaces from these three special syzygies. Quaternions and quaternion multiplication are used to represent these surfaces. Examples are provided to illustrate our theorems and flesh out our algorithms.

Rational curves in the logarithmic multiplicative group

The logarithmic multiplicative group is a proper group object in logarithmic schemes, which morally compactifies the usual multiplicative group. We study the structure of the stacks of logarithmic maps from rational curves to this logarithmic torus, and show that in most cases, it is a product of the logarithmic torus with the space of rational curves. This gives a conceptual explanation for earlier results on the moduli spaces of logarithmic stable maps to toric varieties.

Numerical Proper Reparametrization of Space Curves and Surfaces

Simplifying rational parametrizations of surfaces is a basic problem in CAD (computer-aided design). Reducing their tracing index, called proper reparametrization, is an important simplification way. Most existing proper reparametrization work is symbolic. Yet in practical environments the surfaces are usually given with perturbed coefficients hence need a numerical technique of reparametrization considering the intrinsic properness of the perturbed surfaces. We present algorithms for reparametrizing a numerically rational space curve or surface. First, we provide an efficient way to find a parametric support transformation and compute a reparametrization with proper parametric support. Second, we develop a numerical algorithm to further reduce the tracing index, where numerical techniques such as sparse interpolation and approximated GCD computations are involved. We finally provide the error bound between the given rational curve/surface and our reparametrization result.

Translational Surfaces

A translational surface is a rational surface generated from two rational space curves by translating one curve along the other curve. In this paper, we utilize matrices to represent translational surfaces, and give necessary and sufficient conditions for a real rational surface to be a translational surface.

Using Dual Quaternion to Study Translational Surfaces

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces can also be generated from two rational space curves by dual quaternion multiplication. Using the mathematics of dual quaternions, we provide a necessary and sufficient condition for a rational tensor product surface to be a translational surface. Examples are provided to illustrate our theorems and flesh out our algorithms.

Syzygies for translational surfaces

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces are invariant under rigid motions: translating and rotating the two generating curves translates and rotates the translational surface by the same amount. We construct three special syzygies for a translational surface from a μ-basis of one of the generating space curves, and we show how to compute the implicit equation of a translational surface from these three special syzygies. Examples are provided to illustrate our theorems and flesh out our algorithms.

Implicitizing ruled translational surfaces

A ruled translational surface is a rational tensor product surface generated by translating a rational space curve along a straight line or equivalently translating a straight line along a rational space curve. We show how to compute the implicit equation of a ruled translational surface from two linearly independent vectors that are perpendicular to the generating line of the surface.

Similarity detection of rational space curves

We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Gröbner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.

Kempe’s Universality Theorem for Rational Space Curves

We prove that every bounded rational space curve of degree d and circularity c can be drawn by a linkage with 9/2 d - 6c + 1 revolute joints. Our proof is based on two ingredients. The first one is the factorization theory of motion polynomials. The second one is the construction of a motion polynomial of minimum degree with given orbit. Our proof also gives the explicity construction of the linkage.

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.

The compactified jacobian can be nonreduced

We prove by explicit example that the compactified jacobian can be nonreduced. The example is a rational space curve of arithmetic genus 4. This answers a question posed by Cyril D'Souza in 1979.

Symmetry detection of rational space curves from their curvature and torsion

We present a novel, deterministic, and efficient method to detect whether a given rational space curve is symmetric. By using well-known differential invariants of space curves, namely the curvature and torsion, the method is significantly faster, simpler, and more general than an earlier method addressing a similar problem (Alcázar et al., 2014b). To support this claim, we present an analysis of the arithmetic complexity of the algorithm and timings from an implementation in Sage.