10,308 research outputs found
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
Fundamental groups of some special quadric arrangements
In this work we obtain presentations of fundamental groups of the complements
of three families of quadric arrangements in . The first
arrangement is a union of quadrics, which are tangent to each other at two
common points. The second arrangement is composed of quadrics which are
tangent to each other at one common point. The third arrangement is composed of
quadrics, of them are tangent to the 'th one and each one of the
quadrics is transversal to the other ones.Comment: 21 pages, 12 main figures, appears in Revista Mathematica
Groebner bases for spaces of quadrics of codimension 3
Let be an Artinian standard graded -algebra
defined by quadrics. Assume that and that is algebraically
closed of characteristic . We show that is defined by a Gr\"obner
basis of quadrics with, essentially, one exception. The exception is given by
where is a complete intersection of 3 quadrics not containing
the square of a linear form.Comment: Minor changes, to appear in the J. Pure Applied Algebr
Billiard algebra, integrable line congruences, and double reflection nets
The billiard systems within quadrics, playing the role of discrete analogues
of geodesics on ellipsoids, are incorporated into the theory of integrable
quad-graphs. An initial observation is that the Six-pointed star theorem, as
the operational consistency for the billiard algebra, is equivalent to an
integrabilty condition of a line congruence. A new notion of the
double-reflection nets as a subclass of dual Darboux nets associated with
pencils of quadrics is introduced, basic properies and several examples are
presented. Corresponding Yang-Baxter maps, associated with pencils of quadrics
are defined and discussed.Comment: 18 pages, 8 figure
Toric Ideals of Lattice Path Matroids and Polymatroids
We show that the toric ideal of a lattice path polymatroid is generated by
quadrics corresponding to symmetric exchanges, and give a monomial order under
which these quadrics form a Gr\"obner basis. We then obtain an analogous result
for lattice path matroids.Comment: 9 pages, 4 figure
Peterson's Deformations of Higher Dimensional Quadrics
We provide the first explicit examples of deformations of higher dimensional
quadrics: a straightforward generalization of Peterson's explicit 1-dimensional
family of deformations in of 2-dimensional general quadrics with
common conjugate system given by the spherical coordinates on the complex
sphere to an explicit -dimensional
family of deformations in of -dimensional general
quadrics with common conjugate system given by the spherical coordinates on the
complex sphere and non-degenerate joined
second fundamental forms. It is then proven that this family is maximal
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