1,182 research outputs found
An upper bound on the k-modem illumination problem
A variation on the classical polygon illumination problem was introduced in
[Aichholzer et. al. EuroCG'09]. In this variant light sources are replaced by
wireless devices called k-modems, which can penetrate a fixed number k, of
"walls". A point in the interior of a polygon is "illuminated" by a k-modem if
the line segment joining them intersects at most k edges of the polygon. It is
easy to construct polygons of n vertices where the number of k-modems required
to illuminate all interior points is Omega(n/k). However, no non-trivial upper
bound is known. In this paper we prove that the number of k-modems required to
illuminate any polygon of n vertices is at most O(n/k). For the cases of
illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we
give a tighter bound of 6n/k + 1. Moreover, we present an O(n log n) time
algorithm to achieve this bound.Comment: 9 pages, 4 figure
Epsilon-Unfolding Orthogonal Polyhedra
An unfolding of a polyhedron is produced by cutting the surface and
flattening to a single, connected, planar piece without overlap (except
possibly at boundary points). It is a long unsolved problem to determine
whether every polyhedron may be unfolded. Here we prove, via an algorithm, that
every orthogonal polyhedron (one whose faces meet at right angles) of genus
zero may be unfolded. Our cuts are not necessarily along edges of the
polyhedron, but they are always parallel to polyhedron edges. For a polyhedron
of n vertices, portions of the unfolding will be rectangular strips which, in
the worst case, may need to be as thin as epsilon = 1/2^{Omega(n)}.Comment: 23 pages, 20 figures, 7 references. Revised version improves language
and figures, updates references, and sharpens the conclusio
Ball and Spindle Convexity with respect to a Convex Body
Let be a convex body. We introduce two notions of
convexity associated to C. A set is -ball convex if it is the
intersection of translates of , or it is either , or . The -ball convex hull of two points is called a -spindle. is
-spindle convex if it contains the -spindle of any pair of its points. We
investigate how some fundamental properties of conventional convex sets can be
adapted to -spindle convex and -ball convex sets. We study separation
properties and Carath\'eodory numbers of these two convexity structures. We
investigate the basic properties of arc-distance, a quantity defined by a
centrally symmetric planar disc , which is the length of an arc of a
translate of , measured in the -norm, that connects two points. Then we
characterize those -dimensional convex bodies for which every -ball
convex set is the -ball convex hull of finitely many points. Finally, we
obtain a stability result concerning covering numbers of some -ball convex
sets, and diametrically maximal sets in -dimensional Minkowski spaces.Comment: 27 pages, 5 figure
On the perfect matching of disjoint compact sets by noncrossing line segments in Rn
AbstractLet V be a family of even disjoint line segments in Rn of f-equal width for a direction f∈(Rn)∗(n⩾2), or even disjoint curve segments in Rn of fn-equal width, where fn is the normal direction for bases (n⩾2), or even disjoint twisted triangular prisms in R3 of f3-equal width. We prove that V has a perfect matching by open disjoint line segments in the complementary domain of the union of all the elements of V
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