7 research outputs found
Matrix representations for toric parametrizations
In this paper we show that a surface in P^3 parametrized over a 2-dimensional
toric variety T can be represented by a matrix of linear syzygies if the base
points are finite in number and form locally a complete intersection. This
constitutes a direct generalization of the corresponding result over P^2
established in [BJ03] and [BC05]. Exploiting the sparse structure of the
parametrization, we obtain significantly smaller matrices than in the
homogeneous case and the method becomes applicable to parametrizations for
which it previously failed. We also treat the important case T = P^1 x P^1 in
detail and give numerous examples.Comment: 20 page
Syzygies and the Rees algebra
Let a, b, c be linearly independent homogeneous polynomials in the standard Z-graded ring R {colon equals} k [s, t] with the same degree d and no common divisors. This defines a morphism P1 → P2. The Rees algebra Rees (I) = R ⊕ I ⊕ I2 ⊕ ⋯ of the ideal I = 〈 a, b, c 〉 is the graded R-algebra which can be described as the image of an R-algebra homomorphism h: R [x, y, z] → Rees (I). This paper discusses one result concerning the structure of the kernel of the map h and its relation to the problem of finding the implicit equation of the image of the map given by a, b, c. In particular, we prove a conjecture of Hong, Simis and Vasconcelos. We also relate our results to the theory of adjoint curves and prove a special case of a conjecture of Cox
Syzygies and singularities of tensor product surfaces of bidegree (2,1)
Let U be a basepoint free four-dimensional subspace of the space of sections
of O(2,1) on P^1 x P^1. The sections corresponding to U determine a regular map
p_U: P^1 x P^1 --> P^3. We study the associated bigraded ideal I_U in
k[s,t;u,v] from the standpoint of commutative algebra, proving that there are
exactly six numerical types of possible bigraded minimal free resolution. These
resolutions play a key role in determining the implicit equation of the image
p_U(P^1 x P^1), via work of Buse-Jouanolou, Buse-Chardin, Botbol and
Botbol-Dickenstein-Dohm on the approximation complex. In four of the six cases
I_U has a linear first syzygy; remarkably from this we obtain all differentials
in the minimal free resolution. In particular this allows us to describe the
implicit equation and singular locus of the image.Comment: 35 pages 1 figur
Interrogation of spline surfaces with application to isogeometric design and analysis of lattice-skin structures
A novel surface interrogation technique is proposed to compute the
intersection of curves with spline surfaces in isogeometric analysis. The
intersection points are determined in one-shot without resorting to a
Newton-Raphson iteration or successive refinement. Surface-curve intersection
is required in a wide range of applications, including contact, immersed
boundary methods and lattice-skin structures, and requires usually the solution
of a system of nonlinear equations. It is assumed that the surface is given in
form of a spline, such as a NURBS, T-spline or Catmull-Clark subdivision
surface, and is convertible into a collection of B\'ezier patches. First, a
hierarchical bounding volume tree is used to efficiently identify the B\'ezier
patches with a convex-hull intersecting the convex-hull of a given curve
segment. For ease of implementation convex-hulls are approximated with k-dops
(discrete orientation polytopes). Subsequently, the intersections of the
identified B\'ezier patches with the curve segment are determined with a
matrix-based implicit representation leading to the computation of a sequence
of small singular value decompositions (SVDs). As an application of the
developed interrogation technique the isogeometric design and analysis of
lattice-skin structures is investigated. The skin is a spline surface that is
usually created in a computer-aided design (CAD) system and the periodic
lattice to be fitted consists of unit cells, each containing a small number of
struts. The lattice-skin structure is generated by projecting selected lattice
nodes onto the surface after determining the intersection of unit cell edges
with the surface. For mechanical analysis, the skin is modelled as a
Kirchhoff-Love thin-shell and the lattice as a pin-jointed truss. The two types
of structures are coupled with a standard Lagrange multiplier approach
Implicitizing Rational Curves by the Method of Moving Algebraic Curves
AbstractA functionF(x,y,t)that assigns to each parametertan algebraic curveF(x,y,t)=0is called a moving curve. A moving curveF(x,y,t)is said to follow a rational curvex=x(t)/w(t),y=y(t)/w(t)ifF(x(t)/w(t), y(t)/w(t),t)is identically zero. A new technique for finding the implicit equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2nwith no base points the method of moving conics generates the implicit equation as the determinant of ann×nmatrix, where each entry is a quadratic polynomial inxandy, whereas standard resultant methods generate the implicit equation as the determinant of a 2n× 2nmatrix where each entry is a linear polynomial inxandy. Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines
The Bernstein basis in set-theoretic geometric modelling
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