118 research outputs found
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
Conchoid surfaces of spheres
The conchoid of a surface with respect to given fixed point is
roughly speaking the surface obtained by increasing the radius function with
respect to by a constant. This paper studies {\it conchoid surfaces of
spheres} and shows that these surfaces admit rational parameterizations.
Explicit parameterizations of these surfaces are constructed using the
relations to pencils of quadrics in and . Moreover we point to
remarkable geometric properties of these surfaces and their construction
Implicitization of rational surfaces using toric varieties
A parameterized surface can be represented as a projection from a certain
toric surface. This generalizes the classical homogeneous and bihomogeneous
parameterizations. We extend to the toric case two methods for computing the
implicit equation of such a rational parameterized surface. The first approach
uses resultant matrices and gives an exact determinantal formula for the
implicit equation if the parameterization has no base points. In the case the
base points are isolated local complete intersections, we show that the
implicit equation can still be recovered by computing any non-zero maximal
minor of this matrix.
The second method is the toric extension of the method of moving surfaces,
and involves finding linear and quadratic relations (syzygies) among the input
polynomials. When there are no base points, we show that these can be put
together into a square matrix whose determinant is the implicit equation. Its
extension to the case where there are base points is also explored.Comment: 28 pages, 1 figure. Numerous major revisions. New proof of method of
moving surfaces. Paper accepted and to appear in Journal of Algebr
Offsets, Conchoids and Pedal Surfaces
We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces
Common transversals and tangents to two lines and two quadrics in P^3
We solve the following geometric problem, which arises in several
three-dimensional applications in computational geometry: For which
arrangements of two lines and two spheres in R^3 are there infinitely many
lines simultaneously transversal to the two lines and tangent to the two
spheres?
We also treat a generalization of this problem to projective quadrics:
Replacing the spheres in R^3 by quadrics in projective space P^3, and fixing
the lines and one general quadric, we give the following complete geometric
description of the set of (second) quadrics for which the 2 lines and 2
quadrics have infinitely many transversals and tangents: In the
nine-dimensional projective space P^9 of quadrics, this is a curve of degree 24
consisting of 12 plane conics, a remarkably reducible variety.Comment: 26 pages, 9 .eps figures, web page with more pictures and and archive
of computations: http://www.math.umass.edu/~sottile/pages/2l2s
On special quadratic birational transformations of a projective space into a hypersurface
We study transformations as in the title with emphasis on those having smooth
connected base locus, called "special". In particular, we classify all special
quadratic birational maps into a quadric hypersurface whose inverse is given by
quadratic forms by showing that there are only four examples having general
hyperplane sections of Severi varieties as base loci.Comment: Accepted for publication in Rendiconti del Circolo Matematico di
Palerm
Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
Let X -> Y be a fibration whose fibers are complete intersections of two
quadrics. We develop new categorical and algebraic tools---a theory of relative
homological projective duality and the Morita invariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting---to study
semiorthogonal decompositions of the bounded derived category of X. Together
with new results in the theory of quadratic forms, we apply these tools in the
case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers
are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds,
respectively. In the latter two cases, if Y is the projective line over an
algebraically closed field of characteristic zero, we relate rationality
questions to categorical representability of X.Comment: 43 pages, changes made and some material added and corrected in
sections 1, 4, and 5; this is the final version accepted for publication at
Journal de Math\'ematiques Pures et Appliqu\'ee
A sagbi basis for the quantum Grassmannian
The maximal minors of a p by (m + p) matrix of univariate polynomials of
degree n with indeterminate coefficients are themselves polynomials of degree
np. The subalgebra generated by their coefficients is the coordinate ring of
the quantum Grassmannian, a singular compactification of the space of rational
curves of degree np in the Grassmannian of p-planes in (m + p)-space. These
subalgebra generators are shown to form a sagbi basis. The resulting flat
deformation from the quantum Grassmannian to a toric variety gives a new
`Gr\"obner basis style' proof of the Ravi-Rosenthal-Wang formulas in quantum
Schubert calculus. The coordinate ring of the quantum Grassmannian is an
algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and
Koszul, and the ideal of quantum Pl\"ucker relations has a quadratic Gr\"obner
basis. This holds more generally for skew quantum Schubert varieties. These
results are well-known for the classical Schubert varieties (n=0). We also show
that the row-consecutive p by p-minors of a generic matrix form a sagbi basis
and we give a quadratic Gr\"obner basis for their algebraic relations.Comment: 18 pages, 3 eps figure, uses epsf.sty. Dedicated to the memory of
Gian-Carlo Rot
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