Pythagorean-hodograph space curves

We investigate the properties of polynomial space curvesr(t)={x(t), y(t), z(t)} whose hodographs (derivatives) satisfy the Pythagorean conditionx′²(t)+y′²(t)+z′²(t)≡σ²(t) for some real polynomial σ(t). The algebraic structure of thecomplete set of regular Pythagorean-hodograph curves in ℝ³ is inherently more complicated than that of the corresponding set in ℝ². We derive a characterization for allcubic Pythagoreanhodograph space curves, in terms of constraints on the Bézier control polygon, and show that such curves correspond geometrically to a family of non-circular helices. Pythagorean-hodograph space curves of higher degree exhibit greater shape flexibility (the quintics, for example, satisfy the general first-order Hermite interpolation problem in ℝ³), but they have no “simple” all-encompassing characterization. We focus on asubset of these higher-order curves that admits a straightforward constructive representation. As distinct from polynomial space curves in general, Pythagorean-hodograph space curves have the following attractive attributes: (i) the arc length of any segment can be determined exactly without numerical quadrature; and (ii) thecanal surfaces based on such curves as spines have precise rational parameterizations.     

First order hermite interpolation with spherical Pythagorean-hodograph curves

The general stereographic projection which maps a point on a sphere with arbitrary radius to a point on a plane stereographically and its inverse projection have the Pythagorean-hodograph (PH) preserving property in the sense that they map a PH curve to another PH curve. Upon this fact, for given spatialC ¹ Hermite data, we construct a spatial PH curve on a sphere that is aC ¹ Hermite interpolant of the given data as follows: First, we solveC ¹ Hermite interpolation problem for the stereographically projected planar data of the given data in ?³ with planar PH curves expressed in the complex representation. Second, we construct spherical PH curves which are interpolants for the given data in ?³ using the inverse general stereographic projection.     

An approach to geometric interpolation by Pythagorean-hodograph curves

The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree n is studied independently of the dimension d????2. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.     

Hypersurfaces with too many rational curves

We study smooth hypersurfaces of degree \(d\ge n+1\) in \(\mathbf{P}^n\) whose spaces of smooth rational curves of low degrees are larger than expected, and show that under certain conditions, the primitive part of the middle cohomology of such hypersurfaces have non-trivial Hodge substructures. As an application, we prove that the space of lines on any smooth Fano hypersurface of degree \(d \le 8\) in \(\mathbf{P}^n\) has the expected dimension \(2n-d-3\) .     

Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C ² Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C ² quintic bases on uniform triple knots are constructed for both open and closed C ² curves, and are used to derive simple explicit formulae for the B–spline control points of C ² PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C ² to C ¹. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.     

Double rational normal curves with linear syzygies

In this note we are looking after nilpotent projective curves without embedded points, which have rational normal curves of degree d as support, are defined (scheme-theoretically) by quadratic equations, have degree 2d and have only linear syzygies. We show that, as expected, no such curve does exist in ℙ ^d , and then consider doublings in a bigger ambient space. The simplest and trivial example is that of a double line in the plane. We show that the only possibility is to take rational normal curves in ℙ ^d embedded further in ℙ^{2 d} and to take a certain doubling in the sense of Ferrand (cf. [5]) in ℙ^{2 d} . These double curves have the Hilbert polynomial H(t)=2dt+1, i.e. they are in the Hilbert scheme of the rational normal curves of degree 2d. Thus, it turns out that they are natural generalizations of the double line in the plane considered as a degenerated conic.The simplest nontrivial example is the curve of degree 4 in ℙ⁴, defined by the ideal (xz−y ², xu−yv, yu−zv, u ², uv, v ²). The double rational curve allow the formulation of a Strong Castelnuovo Lemma in the sense of [7], for sets of points and double points. In the last section we mention some plethysm formulae for symmetric powers. Received: 1 September 2000 / Revised version: 15 January 2001     

On rational cuspidal curves

Let be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that . We show that where is the square of the “golden section”. We also construct examples which show that this estimate is asymptotically sharp. When , we show that and this estimate is sharp. The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types. Received: 11 February 2000 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502     

Generatrices of rational curves

We investigate the one-parametric set of projective subspaces that is generated by a set of rational curves in projective relation. The main theorem connects the algebraic degree of , the number of degenerate subspaces in and the dimension of the variety of all rational curves that can be used to generate . It generalizes classical results and is related to recent investigations on projective motions with trajectories in proper subspaces of the fixed space. Received 9 May 2001.     

On limiting rational curves

This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990.     

Pythagorean-hodograph preserving mappings

We study the scaled Pythagorean-hodograph (PH) preserving mappings. These mappings make offset-rational isothermal surfaces and map PH curves to PH curves. We present a method to produce a great number of the scaled PH preserving mappings. For an application of the PH preserving mappings, we solve the Hermite interpolation problem for PH curves in the space.     

Singular rational curves with points of nearly-maximal weight

In this article we study rational curves with a unique unibranch genus-g singularity, which is of κ-hyperelliptic type in the sense of [27]; we focus on the cases $\kappa =0$ and $\kappa =1$, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.     

Nuclei of normal rational curves

Ak-nucleus of a normal rational curve inPG(n, F) is the intersection over allk-dimensional osculating subspaces of the curve (k {–1,0,...,n– 1}). It is well known that for characteristic zero all nuclei are empty. In case of characteristicp}>0 and #Fn the number of non-zero digits in the representation ofn + 1 in basep equals the number of distinct nuclei. An explicit formula for the dimensions ofk-nuclei is given for #F=Fk + 1.Research supported by the Austrian National Science Fund (FWF), project P-12353-MAT.     

Real rational curves in Grassmannians

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some given general figures. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real. Similarly, given any problem of enumerating -planes incident on some given general subspaces, there are general real subspaces such that each of the (finitely many) incident -planes is real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real.

     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Rational curves and rational singularities

 We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a ℂ^*-action. For varieties with an isolated singularity, covered by a family of rational curves with a general member not passing through the singular point, we show that this singularity is rational. In particular, this provides an explanation of classical results due to H. A. Schwartz and G. H. Halphen on polynomial solutions of the generalized Fermat equation. Received: 7 May 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 14J17, 14L30, 13H10