1,966 research outputs found
Relative Convex Hull Determination from Convex Hulls in the Plane
A new algorithm for the determination of the relative convex hull in the
plane of a simple polygon A with respect to another simple polygon B which
contains A, is proposed. The relative convex hull is also known as geodesic
convex hull, and the problem of its determination in the plane is equivalent to
find the shortest curve among all Jordan curves lying in the difference set of
B and A and encircling A. Algorithms solving this problem known from
Computational Geometry are based on the triangulation or similar decomposition
of that difference set. The algorithm presented here does not use such
decomposition, but it supposes that A and B are given as ordered sequences of
vertices. The algorithm is based on convex hull calculations of A and B and of
smaller polygons and polylines, it produces the output list of vertices of the
relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two
typing errors in Definition 2: has to be defined based on , and
has to be defined based on (not just using ). These errors
appeared in the text of the original conference paper, which also contained
the pseudocode of an algorithm where and appeared as correctly
define
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
An Optimal Algorithm for the Separating Common Tangents of two Polygons
We describe an algorithm for computing the separating common tangents of two
simple polygons using linear time and only constant workspace. A tangent of a
polygon is a line touching the polygon such that all of the polygon lies to the
same side of the line. A separating common tangent of two polygons is a tangent
of both polygons where the polygons are lying on different sides of the
tangent. Each polygon is given as a read-only array of its corners. If a
separating common tangent does not exist, the algorithm reports that.
Otherwise, two corners defining a separating common tangent are returned. The
algorithm is simple and implies an optimal algorithm for deciding if the convex
hulls of two polygons are disjoint or not. This was not known to be possible in
linear time and constant workspace prior to this paper.
An outer common tangent is a tangent of both polygons where the polygons are
on the same side of the tangent. In the case where the convex hulls of the
polygons are disjoint, we give an algorithm for computing the outer common
tangents in linear time using constant workspace.Comment: 12 pages, 6 figures. A preliminary version of this paper appeared at
SoCG 201
Design of Combined Coverage Area Reporting and Geo-casting of Queries for Wireless Sensor Networks
In order to efficiently deal with queries or other location dependent information, it is key that the wireless sensor network informs gateways what geographical area is serviced by which gateway. The gateways are then able to e.g. efficiently route queries which are only valid in particular regions of the deployment. The proposed algorithms combine coverage area reporting and geographical routing of queries which are injected by gateways.\u
Approximating the Maximum Overlap of Polygons under Translation
Let and be two simple polygons in the plane of total complexity ,
each of which can be decomposed into at most convex parts. We present an
-approximation algorithm, for finding the translation of ,
which maximizes its area of overlap with . Our algorithm runs in
time, where is a constant that depends only on and .
This suggest that for polygons that are "close" to being convex, the problem
can be solved (approximately), in near linear time
Memory-Constrained Algorithms for Simple Polygons
A constant-workspace algorithm has read-only access to an input array and may
use only O(1) additional words of bits, where is the size of
the input. We assume that a simple -gon is given by the ordered sequence of
its vertices. We show that we can find a triangulation of a plane straight-line
graph in time. We also consider preprocessing a simple polygon for
shortest path queries when the space constraint is relaxed to allow words
of working space. After a preprocessing of time, we are able to solve
shortest path queries between any two points inside the polygon in
time.Comment: Preprint appeared in EuroCG 201
- …