Rational Space Curves Research Articles

Strata of rational space curves

The μ-invariant μ=(μ1,μ2,μ3) of a rational space curve gives important information about the curve. In this paper, we describe the structure of all parameterizations that have the same μ-type, what we call a μ-stratum, and as well the closure of strata. Many of our results are based on papers by the second author that appeared in the commutative algebra literature. We also present new results, including an explicit formula for the codimension of the locus of non-proper parametrizations within each μ-stratum and a decomposition of the smallest μ-stratum based on which two-dimensional rational normal scroll the curve lies on.

Using moving planes to implicitize rational surfaces generated from a planar curve and a space curve

A rational surface(1)S(s,t)=(A(s)a(t),B(s)b(t),C(s)c(t),C(s)d(t)) can be generated from a rational planer curve P⁎(s)=(A(s),B(s),C(s)) and a rational space curve P(t)=(a(t),b(t),c(t),d(t)). Let P⁎(s) pass through the point (1,1,1). Then the surface S(s,t) goes through the space curve P(t). Moreover on each z=z⁎-plane, the cross section of the surface S(s,t) is a stretching or shrinking of the planar curve P⁎(s), such that the point (1,1,z⁎,1) travels to the point P(t)∩{z=z⁎}. Using moving planes, we provide a new technique to implicitize this kind of rational surface. We find four moving planes that follow the surface from which we construct a sparse matrix whose size is just the degree of the surface with entries linear in x, y, z, w. We prove that the determinant of this matrix is the exact implicit equation of the surface S(s,t) without any extraneous factors. Examples are presented to illustrate our methods.

Quaternion rational surfaces: Rational surfaces generated from the quaternion product of two rational space curves

A quaternion rational surface is a surface generated from two rational space curves by quaternion multiplication. The goal of this paper is to demonstrate how to apply syzygies to analyze quaternion rational surfaces. We show that we can easily construct three special syzygies for a quaternion rational surface from a μ-basis for one of the generating rational space curves. The implicit equation of any quaternion rational surface can be computed from these three special syzygies and inversion formulas for the non-singular points on quaternion rational surfaces can be constructed. Quaternion rational ruled surfaces are generated from the quaternion product of a straight line and a rational space curve. We investigate special μ-bases for quaternion rational ruled surfaces and use these special μ-bases to provide implicitization and inversion formulas for quaternion rational ruled surfaces. Finally, we show how to determine if a real rational surface is also a quaternion rational surface.

Normal bundle of rational curves and Waring decomposition

The problem of determining the splitting of the normal bundle of rational space curves has been considered in the 80s in a series of papers by Ghione and Sacchiero and by Eisenbud and Van de Ven. With our approach we are able to obtain results for curves embedded in Pm for m⩾3 and we find an interesting interplay between the Waring decomposition of binary forms and the splitting of the normal bundle.

Implicit matrix representations of rational Bézier curves and surfaces

We introduce and study a new implicit representation of rational Bézier curves and surfaces in the 3-dimensional space. Given such a curve or surface, this representation consists of a matrix whose entries depend on the space variables and whose rank drops exactly on this curve or surface. Our approach can be seen as an extension of the moving lines implicitization method introduced by Sederberg, from non-singular matrices to the more general context of singular matrices. In the first part of this paper, we describe the construction of these new implicit matrix representations and their main geometric properties, in particular their ability to solve efficiently the inversion problem. The second part of this paper aims to show that these implicitization matrices adapt geometric problems, such as intersection problems, to the powerful tools of numerical linear algebra, in particular to one of the most important: the singular value decomposition. So, from the singular values of a given implicit matrix representation, we introduce a real evaluation function. We show that the variation of this function is qualitatively comparable to the Euclidean distance function. As an interesting consequence, we obtain a new determinantal formula for implicitizing a rational space curve or surface over the field of real numbers. Then, we show that implicit matrix representations can be used with numerical computations, in particular there is no need for symbolic computations to use them. We give some rigorous results explaining the numerical stability that we have observed in our experiments. We end the paper with a short illustration on ray tracing of parameterized surfaces.

An algorithm to parametrize approximately space curves

We present an algorithm that, given a non-rational irreducible real space curve, satisfying certain conditions, computes a rational parametrization of a space curve near the input one. For a given tolerance ϵ>0, the algorithm checks whether a planar projection of the given space curve is ϵ-rational and, in the affirmative case, generates a planar parametrization that is lifted to a space parametrization. This output rational space curve is of the same degree as the input curve, both have the same structure at infinity, and the Hausdorff distance between their real parts is finite. Moreover, in the examples we check that the distance is small.

Spaces of rational curves on complete intersections

AbstractWe prove that the space of smooth rational curves of degree $e$ on a general complete intersection of multidegree $(d_1, \ldots , d_m)$ in $\mathbb {P}^n$ is irreducible of the expected dimension if $\sum _{i=1}^m d_i \lt (2n+m+1)/3$ and $n$ is sufficiently large. This generalizes a result of Harris, Roth and Starr [Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106], and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum _{i=1}^m d_i$ is small compared to $n$.

Using a bihomogeneous resultant to find the singularities of rational space curves

We provide a new technique to detect the singularities of rational space curves. Given a rational parametrization of a space curve, we first compute a μ-basis for the parametrization. From this μ-basis we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, we construct a new sparse resultant matrix for these three bivariate polynomials. We then compute the parameter values corresponding to the singularities by applying Gaussian elimination to this resultant matrix. Let νQ denote the multiplicity of the singular point Q, and let n be the degree of the curve. We find that when ∑νQ⩽2n−3, the last nonzero row after Gaussian elimination represents a univariate polynomial whose roots are exactly the parameter values of the singularities with the correct multiplicity. Otherwise the last two nonzero rows represent two bivariate polynomials whose common roots provide the parameter values of the singularities. We also show that if R is this resultant matrix, then size(R)−rank(R) gives the total multiplicity ∑νQ(νQ−1) of all the singular points including the infinitely near singular points of a rational space curve and we provide bounds on the expression ∑νQ(νQ−1) for the total multiplicity of all the singular points of a rational space curve. To verify our results, we present several examples to illustrate our methods.

An interpolation scheme for designing rational rotation-minimizing camera motions

When a moving (real or virtual) camera images a stationary object, the use of a rotation-minimizing directed frame (RMDF) to specify the camera orientation along its path yields the least apparent rotation of the image. The construction of such motions, using curves that possess rational RMDFs, is considered herein. In particular, the construction entails interpolation of initial/final camera positions and orientations, together with an initial motion direction. To achieve this, the camera path is described by a rational space curve that has a rational RMDF and interpolates the prescribed data. Numerical experiments are used to illustrate implementation of the method, and sufficient conditions on the two end frame orientations are derived, to ensure the existence of exactly one interpolant. By specifiying a sequence of discrete camera positions/orientations and an initial motion direction, the method can be used to construct general rotation-minimizing camera motions.

Implicitization using univariate resultants

Among several implicitization methods, the method based on resultant computation is a simple and direct one, but it often brings extraneous factors which are difficult to remove. This paper studies a class of rational space curves and rational surfaces by implicitization with univariate resultant computations. This method is more efficient than the other algorithms in finding implicit equations for this class of rational curves and surfaces.

SOME EXAMPLES OF ALGEBRAIC GEODESICS ON QUADRICS. II

In this note we give new examples of algebraic geodesics on some two-dimensional quadrics, namely, on ellipsoids, one-sheet hyperboloids, and hyperbolic paraboloids. It appears that in all considered cases, such geodesics are rational space curves.

Set-theoretic generators of rational space curves

We show how to calculate three low degree set-theoretic generators (i.e., algebraic surfaces) for all rational space curves of low degree (degree ≤ 6 ) as well as for all higher degree rational space curves where at least one element of their μ -basis has degree 1 from a μ -basis of the parametrization. In addition to having low degree, at least two of these surface generators are always ruled surfaces. Whenever possible we also show how to compute two set-theoretic complete intersection generators for these rational space curves from a μ -basis of their parametrization.

SOME EXAMPLES OF ALGEBRAIC GEODESICS ON QUADRICS

In this note we give the conditions for the existence of algebraic geodesics on some two-dimensional quadrics, namely, on hyperbolic paraboloids and elliptic paraboloids. It appears that in some cases, such geodesics are the rational space curves.

Axial moving planes and singularities of rational space curves

Relationships between the singularities of rational space curves and the moving planes that follow these curves are investigated. Given a space curve C with a generic 1–1 rational parametrization F ( s , t ) of homogeneous degree d, we show that if P and Q are two singular points of orders k and k ′ on the space curve C , then there is a moving plane of degree d − k − k ′ with axis P Q ↔ that follows the curve. We also show that a point P is a singular point of order k on the space curve C if and only if there are two axial moving planes L 1 and L 2 of degree d − k such that: (1) the axes of L 1 , L 2 are orthogonal and intersect at P, and (2) the intersection of the moving planes L 1 and L 2 is the cone through the curve C with vertex P together with d − k copies of the plane containing the axes of L 1 and L 2 . In addition, we study relationships between the singularities of rational space curves and generic moving planes that follow these curves. In particular, we show that if p ( s , t ) , q ( s , t ) , r ( s , t ) are a μ-basis for the moving planes that follow a rational space curve F ( s , t ) , then P is a singular point of F ( s , t ) of order k if and only if deg ( gcd ( p ( s , t ) ⋅ P , q ( s , t ) ⋅ P , r ( s , t ) ⋅ P ) ) = k . Moreover, the roots of this gcd are the parameters, counted with proper multiplicity, that correspond to the singularity P. Using these results, we provide straightforward algorithms for finding all the singularities of low degree rational space curves. Our algorithms are easy to implement, requiring only standard techniques from linear algebra. Examples are provided to illustrate these algorithms.

μ-bases for polynomial systems in one variable

The notion of a μ-basis for an arbitrary number of polynomials in one variable is defined. The basic properties of these μ-bases are derived, and an algorithm is presented based on Gaussian Elimination to calculate a μ-basis for any collection of univariate polynomials. These μ-bases are then applied to solve implicitization, inversion and intersection problems for rational space curves. Systems where base points are present are also discussed.

Double structures on rational space curves

AbstractThere are very many non‐reduced schemes. In this paper, we consider two examples to back this statement: we give lists of double scheme structures on a twisted cubic, and we construct rank two bundles on projective 3‐space with prescribed Chern classes, from double structures on smooth rational curves. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Erratum: Cubic fourfolds and spaces of rational curves

Erratum: Cubic fourfolds and spaces of rational curves

Rational space curves are not “unit speed”

Rational space curves are not “unit speed”

Quantum cohomology via [formula omitted]-modules

We propose a new point of view on quantum cohomology, motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is a D -module which “quantizes” a commutative algebra associated to the (uncompactified) space of rational curves. Under appropriate conditions, we show that the associated flat connection may be gauged to the flat connection underlying quantum cohomology. This method clarifies the role of the Birkhoff factorization in the “mirror transformation”, and it gives a new algorithm (requiring construction of a Groebner basis and solution of a system of o.d.e.) for computation of the quantum product.

Cubic fourfolds and spaces of rational curves

For a general nonsingular cubic fourfold $X\subset \PP^5$ and $e\geq 5$ an odd integer, we show that the space $M_e$ parametrizing rational curves of degree $e$ on $X$ is non-uniruled. For $e \geq 6$ an even integer, we prove that the generic fiber dimension of the maximally rationally connected fibration of $M_e$ is at most one, i.e., passing through a very general point of $M_e$ there is at most one rational curve. For $e < 5$ the spaces $M_e$ are fairly well understood and we review what is known.