1,037 research outputs found
Deconstructing Approximate Offsets
We consider the offset-deconstruction problem: Given a polygonal shape Q with
n vertices, can it be expressed, up to a tolerance \eps in Hausdorff distance,
as the Minkowski sum of another polygonal shape P with a disk of fixed radius?
If it does, we also seek a preferably simple-looking solution P; then, P's
offset constitutes an accurate, vertex-reduced, and smoothened approximation of
Q. We give an O(n log n)-time exact decision algorithm that handles any
polygonal shape, assuming the real-RAM model of computation. A variant of the
algorithm, which we have implemented using CGAL, is based on rational
arithmetic and answers the same deconstruction problem up to an uncertainty
parameter \delta; its running time additionally depends on \delta. If the input
shape is found to be approximable, this algorithm also computes an approximate
solution for the problem. It also allows us to solve parameter-optimization
problems induced by the offset-deconstruction problem. For convex shapes, the
complexity of the exact decision algorithm drops to O(n), which is also the
time required to compute a solution P with at most one more vertex than a
vertex-minimal one.Comment: 18 pages, 11 figures, previous version accepted at SoCG 2011,
submitted to DC
Polynomial cubic differentials and convex polygons in the projective plane
We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report.
v2: Corrections in section 5 and related new material in appendix
Computational Aspects of the Hausdorff Distance in Unbounded Dimension
We study the computational complexity of determining the Hausdorff distance
of two polytopes given in halfspace- or vertex-presentation in arbitrary
dimension. Subsequently, a matching problem is investigated where a convex body
is allowed to be homothetically transformed in order to minimize its Hausdorff
distance to another one. For this problem, we characterize optimal solutions,
deduce a Helly-type theorem and give polynomial time (approximation) algorithms
for polytopes
Laminations and groups of homeomorphisms of the circle
If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that
pi_1(M) acts on a circle. Here, we show that some other classes of essential
laminations also give rise to actions on circles. In particular, we show this
for tight essential laminations with solid torus guts. We also show that
pseudo-Anosov flows induce actions on circles. In all cases, these actions can
be made into faithful ones, so pi_1(M) is isomorphic to a subgroup of
Homeo(S^1). In addition, we show that the fundamental group of the Weeks
manifold has no faithful action on S^1. As a corollary, the Weeks manifold does
not admit a tight essential lamination, a pseudo-Anosov flow, or a taut
foliation. Finally, we give a proof of Thurston's universal circle theorem for
taut foliations based on a new, purely topological, proof of the Leaf Pocket
Theorem.Comment: 50 pages, 12 figures. Ver 2: minor improvement
Ergodicity for Infinite Periodic Translation Surfaces
For a Z-cover of a translation surface, which is a lattice surface, and which
admits infinite strips, we prove that almost every direction for the
straightline flow is ergodic
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