8,849 research outputs found
A matrix-based approach to properness and inversion problems for rational surfaces
We present a matrix-based algorithm for deciding if the parametrization of a
curve or a surface is invertible or not, and for computing the inverse of the
parametrization if it exists.Comment: 12 pages, latex, revised version accepted for publication in the
Journal AAEC
Lines in Tropical Quadrics
Classical algebraic geometry is the study of curves, surfaces, and other varieties defined as the zero set of polynomial equations. Tropical geometry is a branch of algebraic geometry based on the tropical semiring with operations minimization and addition. We introduce the notions of projective space and tropical projective space, which are well-suited for answering enumerative questions, like ours. We attempt to describe the set of tropical lines contained in a tropical quadric surface in . Analogies with the classical problem and computational techniques based on the idea of a tropical parameterization suggest that the answer is the union of two disjoint conics in
The Relation Between Offset and Conchoid Constructions
The one-sided offset surface Fd of a given surface F is, roughly speaking,
obtained by shifting the tangent planes of F in direction of its oriented
normal vector. The conchoid surface Gd of a given surface G is roughly speaking
obtained by increasing the distance of G to a fixed reference point O by d.
Whereas the offset operation is well known and implemented in most CAD-software
systems, the conchoid operation is less known, although already mentioned by
the ancient Greeks, and recently studied by some authors. These two operations
are algebraic and create new objects from given input objects. There is a
surprisingly simple relation between the offset and the conchoid operation. As
derived there exists a rational bijective quadratic map which transforms a
given surface F and its offset surfaces Fd to a surface G and its conchoidal
surface Gd, and vice versa. Geometric properties of this map are studied and
illustrated at hand of some complete examples. Furthermore rational universal
parameterizations for offsets and conchoid surfaces are provided
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
The Convex Hull of a Variety
We present a characterization, in terms of projective biduality, for the
hypersurfaces appearing in the boundary of the convex hull of a compact real
algebraic variety.Comment: 12 pages, 2 figure
Minimal sections of conic bundles
Let the threefold X be a general smooth conic bundle over the projective
plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this
paper we prove the existence of two natural families C(+) and C(-) of curves on
X, such that the Abel-Jacobi map F sends one of these families onto a copy of
the theta divisor (Theta), and the other -- onto the jacobian J(X). The general
curve C of any of these two families is a section of the conic bundle
projection, and our approach relates such C to a maximal subbundle of a rank 2
vector bundle E(C) on C, or -- to a minimal section of the ruled surface
P(E(C)). The families C(+) and C(-) correspond to the two possible types of
versal deformations of ruled surfaces over curves of fixed genus g(C). As an
application, we find parameterizations of J(X) and (Theta) for certain classes
of Fano threefolds, and study the sets Sing(Theta) of the singularities of
(Theta).Comment: Duke preprint, 29 pages. LaTex 2.0
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