1,239 research outputs found
Linear transformation distance for bichromatic matchings
Let be a set of points in general position, where is a
set of blue points and a set of red points. A \emph{-matching}
is a plane geometric perfect matching on such that each edge has one red
endpoint and one blue endpoint. Two -matchings are compatible if their
union is also plane.
The \emph{transformation graph of -matchings} contains one node for each
-matching and an edge joining two such nodes if and only if the
corresponding two -matchings are compatible. In SoCG 2013 it has been shown
by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is
always connected, but its diameter remained an open question. In this paper we
provide an alternative proof for the connectivity of the transformation graph
and prove an upper bound of for its diameter, which is asymptotically
tight
Geometric Spanning Cycles in Bichromatic Point Sets
Given a set of points in the plane each colored either red or blue, we find
non-self-intersecting geometric spanning cycles of the red points and of the
blue points such that each edge of the red spanning cycle is crossed at most
three times by the blue spanning cycle and vice-versa
A Bichromatic Incidence Bound and an Application
We prove a new, tight upper bound on the number of incidences between points
and hyperplanes in Euclidean d-space. Given n points, of which k are colored
red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between
the k red points and m hyperplanes spanned by all n points provided that m =
\Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal
and Aronov.
We use this incidence bound to prove that a set of n points, no more than n-k
of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also
provide an infinite family of counterexamples to a conjecture of Purdy's on the
number of hyperplanes spanned by a set of points in dimensions higher than 3,
and present new conjectures not subject to the counterexample.Comment: 12 page
Atom gratings produced by large angle atom beam splitters
An asymptotic theory of atom scattering by large amplitude periodic
potentials is developed in the Raman-Nath approximation. The atom grating
profile arising after scattering is evaluated in the Fresnel zone for
triangular, sinusoidal, magneto-optical, and bichromatic field potentials. It
is shown that, owing to the scattering in these potentials, two
\QTR{em}{groups} of momentum states are produced rather than two distinct
momentum components. The corresponding spatial density profile is calculated
and found to differ significantly from a pure sinusoid.Comment: 16 pages, 7 figure
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