56,288 research outputs found
Hyperbolic Dehn filling in dimension four
We introduce and study some deformations of complete finite-volume hyperbolic
four-manifolds that may be interpreted as four-dimensional analogues of
Thurston's hyperbolic Dehn filling.
We construct in particular an analytic path of complete, finite-volume cone
four-manifolds that interpolates between two hyperbolic four-manifolds
and with the same volume . The deformation looks
like the familiar hyperbolic Dehn filling paths that occur in dimension three,
where the cone angle of a core simple closed geodesic varies monotonically from
to . Here, the singularity of is an immersed geodesic surface
whose cone angles also vary monotonically from to . When a cone angle
tends to a small core surface (a torus or Klein bottle) is drilled
producing a new cusp.
We show that various instances of hyperbolic Dehn fillings may arise,
including one case where a degeneration occurs when the cone angles tend to
, like in the famous figure-eight knot complement example.
The construction makes an essential use of a family of four-dimensional
deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.Comment: 60 pages, 23 figures. Final versio
Probing Convex Polygons with a Wedge
Minimizing the number of probes is one of the main challenges in
reconstructing geometric objects with probing devices. In this paper, we
investigate the problem of using an -wedge probing tool to determine
the exact shape and orientation of a convex polygon. An -wedge consists
of two rays emanating from a point called the apex of the wedge and the two
rays forming an angle . To probe with an -wedge, we set the
direction that the apex of the probe has to follow, the line , and the initial orientation of the two rays. A valid -probe of a
convex polygon contains within the -wedge and its outcome
consists of the coordinates of the apex, the orientation of both rays and the
coordinates of the closest (to the apex) points of contact between and each
of the rays.
We present algorithms minimizing the number of probes and prove their
optimality. In particular, we show how to reconstruct a convex -gon (with
all internal angles of size larger than ) using -probes;
if , the reconstruction uses -probes. We show
that both results are optimal. Let be the number of vertices of whose
internal angle is at most , (we show that ). We
determine the shape and orientation of a general convex -gon with
(respectively , ) using (respectively , )
-probes. We prove optimality for the first case. Assuming the algorithm
knows the value of in advance, the reconstruction of with or
can be achieved with probes,- which is optimal.Comment: 31 pages, 27 figure
Algorithms for distance problems in planar complexes of global nonpositive curvature
CAT(0) metric spaces and hyperbolic spaces play an important role in
combinatorial and geometric group theory. In this paper, we present efficient
algorithms for distance problems in CAT(0) planar complexes. First of all, we
present an algorithm for answering single-point distance queries in a CAT(0)
planar complex. Namely, we show that for a CAT(0) planar complex K with n
vertices, one can construct in O(n^2 log n) time a data structure D of size
O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x
and the query point y can be computed in linear time. Our second algorithm
computes the convex hull of a finite set of points in a CAT(0) planar complex.
This algorithm is based on Toussaint's algorithm for computing the convex hull
of a finite set of points in a simple polygon and it constructs the convex hull
of a set of k points in O(n^2 log n + nk log k) time, using a data structure of
size O(n^2 + k)
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