6 research outputs found
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Given a set of spheres in , with and
odd, having a fixed number of distinct radii , we
show that the worst-case combinatorial complexity of the convex hull
of is
, where
is the number of spheres in with radius .
To prove the lower bound, we construct a set of spheres in
, with odd, where spheres have radius ,
, and , such that their convex hull has combinatorial
complexity
.
Our construction is then generalized to the case where the spheres have
distinct radii.
For the upper bound, we reduce the sphere convex hull problem to the problem
of computing the worst-case combinatorial complexity of the convex hull of a
set of -dimensional convex polytopes lying on parallel hyperplanes
in , where odd, a problem which is of independent
interest. More precisely, we show that the worst-case combinatorial complexity
of the convex hull of a set
of -dimensional convex polytopes lying on parallel hyperplanes of
is
, where
is the number of vertices of .
We end with algorithmic considerations, and we show how our tight bounds for
the parallel polytope convex hull problem, yield tight bounds on the
combinatorial complexity of the Minkowski sum of two convex polytopes in
.Comment: 22 pages, 5 figures, new proof of upper bound for the complexity of
the convex hull of parallel polytopes (the new proof gives upper bounds for
all face numbers of the convex hull of the parallel polytopes
Computing pseudotriangulations via branched coverings
We describe an efficient algorithm to compute a pseudotriangulation of a
finite planar family of pairwise disjoint convex bodies presented by its
chirotope. The design of the algorithm relies on a deepening of the theory of
visibility complexes and on the extension of that theory to the setting of
branched coverings. The problem of computing a pseudotriangulation that
contains a given set of bitangent line segments is also examined.Comment: 66 pages, 39 figure
A convex hull algorithm for discs, and applications
AbstractWe show that the convex hull of a set of discs can be determined in Θ(n log n) time. The algorithm is straightforward and simple to implement. We then show that the convex hull can be used to efficiently solve various problems for a set of discs. This includes O(n log n) algorithms for computing the diameter and the minimum spanning circle, computing the furthest site Voronoi diagram, computing the stabbing region, and establishing the region of intersection of the discs