435 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
Geodesic-Preserving Polygon Simplification
Polygons are a paramount data structure in computational geometry. While the
complexity of many algorithms on simple polygons or polygons with holes depends
on the size of the input polygon, the intrinsic complexity of the problems
these algorithms solve is often related to the reflex vertices of the polygon.
In this paper, we give an easy-to-describe linear-time method to replace an
input polygon by a polygon such that (1)
contains , (2) has its reflex
vertices at the same positions as , and (3) the number of vertices
of is linear in the number of reflex vertices. Since the
solutions of numerous problems on polygons (including shortest paths, geodesic
hulls, separating point sets, and Voronoi diagrams) are equivalent for both
and , our algorithm can be used as a preprocessing
step for several algorithms and makes their running time dependent on the
number of reflex vertices rather than on the size of
Empty Monochromatic Simplices
Let be a -colored (finite) set of points in , , in general position, that is, no {} points of lie in a common
}-dimensional hyperplane. We count the number of empty monochromatic
-simplices determined by , that is, simplices which have only points from
one color class of as vertices and no points of in their interior. For
we provide a lower bound of and
strengthen this to for . On the way we provide various
results on triangulations of point sets in . In particular, for
any constant dimension , we prove that every set of points (
sufficiently large), in general position in , admits a
triangulation with at least simplices
Empty monochromatic simplices
Let S be a k-colored (finite) set of n points in , da parts per thousand yen3, in general position, that is, no (d+1) points of S lie in a common (d-1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3a parts per thousand currency signka parts per thousand currency signd we provide a lower bound of and strengthen this to Omega(n (d-2/3)) for k=2.; On the way we provide various results on triangulations of point sets in . In particular, for any constant dimension da parts per thousand yen3, we prove that every set of n points (n sufficiently large), in general position in , admits a triangulation with at least dn+Omega(logn) simplices.Postprint (author’s final draft
Asymptotic tracking by funnel control with internal models
Funnel control achieves output tracking with guaranteed tracking performance
for unknown systems and arbitrary reference signals. In particular, the
tracking error is guaranteed to satisfy time-varying error bounds for all times
(it evolves in the funnel). However, convergence to zero cannot be guaranteed,
but the error often stays close to the funnel boundary, inducing a
comparatively large feedback gain. This has several disadvantages (e.g. poor
tracking performance and sensitivity to noise due to the underlying high-gain
feedback principle). In this paper, therefore, the usually known reference
signal is taken into account during funnel controller design, i.e. we propose
to combine the well-known internal model principle with funnel control. We
focus on linear systems with linear reference internal models and show that
under mild adjustments of funnel control, we can achieve asymptotic tracking
for a whole class of linear systems (i.e. without relying on the knowledge of
system parameters)
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