3,227 research outputs found
Common transversals
AbstractGiven t families, each family consisting of s finite sets, we show that if the families “separate points” in a natural way, and if the union of all the sets in all the families contains more than (s + 1)t − st−1 − 1 elements, then a common transversal of the t families exists. In case each family is a covering family, the bound is st − st−1. Both of these bounds are best possible. This work extends recent work of Longyear [2]
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
Line transversals to disjoint balls
We prove that the set of directions of lines intersecting three disjoint
balls in in a given order is a strictly convex subset of . We then
generalize this result to disjoint balls in . As a consequence, we can
improve upon several old and new results on line transversals to disjoint balls
in arbitrary dimension, such as bounds on the number of connected components
and Helly-type theorems.Comment: 21 pages, includes figure
Geometric Permutations of Non-Overlapping Unit Balls Revisited
Given four congruent balls in that have disjoint
interior and admit a line that intersects them in the order , we show
that the distance between the centers of consecutive balls is smaller than the
distance between the centers of and . This allows us to give a new short
proof that interior-disjoint congruent balls admit at most three geometric
permutations, two if . We also make a conjecture that would imply that
such balls admit at most two geometric permutations, and show that if
the conjecture is false, then there is a counter-example of a highly degenerate
nature
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