4,039 research outputs found
Zero-Parity Stabbing Information
Everett et al. introduced several varieties of stabbing information for the
lines determined by pairs of vertices of a simple polygon P, and established
their relationships to vertex visibility and other combinatorial data. In the
same spirit, we define the ``zero-parity (ZP) stabbing information'' to be a
natural weakening of their ``weak stabbing information,'' retaining only the
distinction among {zero, odd, even>0} in the number of polygon edges stabbed.
Whereas the weak stabbing information's relation to visibility remains an open
problem, we completely settle the analogous questions for zero-parity
information, with three results: (1) ZP information is insufficient to
distinguish internal from external visibility graph edges; (2) but it does
suffice for all polygons that avoid a certain complex substructure; and (3) the
natural generalization of ZP information to the continuous case of smooth
curves does distinguish internal from external visibility
Meeting in a Polygon by Anonymous Oblivious Robots
The Meeting problem for searchers in a polygon (possibly with
holes) consists in making the searchers move within , according to a
distributed algorithm, in such a way that at least two of them eventually come
to see each other, regardless of their initial positions. The polygon is
initially unknown to the searchers, and its edges obstruct both movement and
vision. Depending on the shape of , we minimize the number of searchers
for which the Meeting problem is solvable. Specifically, if has a
rotational symmetry of order (where corresponds to no
rotational symmetry), we prove that searchers are sufficient, and
the bound is tight. Furthermore, we give an improved algorithm that optimally
solves the Meeting problem with searchers in all polygons whose
barycenter is not in a hole (which includes the polygons with no holes). Our
algorithms can be implemented in a variety of standard models of mobile robots
operating in Look-Compute-Move cycles. For instance, if the searchers have
memory but are anonymous, asynchronous, and have no agreement on a coordinate
system or a notion of clockwise direction, then our algorithms work even if the
initial memory contents of the searchers are arbitrary and possibly misleading.
Moreover, oblivious searchers can execute our algorithms as well, encoding
information by carefully positioning themselves within the polygon. This code
is computable with basic arithmetic operations, and each searcher can
geometrically construct its own destination point at each cycle using only a
compass. We stress that such memoryless searchers may be located anywhere in
the polygon when the execution begins, and hence the information they initially
encode is arbitrary. Our algorithms use a self-stabilizing map construction
subroutine which is of independent interest.Comment: 37 pages, 9 figure
Towards an Iterative Algorithm for the Optimal Boundary Coverage of a 3D Environment
This paper presents a new optimal algorithm for locating a set of sensors in 3D able to see the boundaries of a polyhedral environment. Our approach is iterative and is based on a lower bound on the sensors' number and on a restriction of the original problem requiring each face to be observed in its entirety by at least one sensor. The lower bound allows evaluating the quality of the solution obtained at each step, and halting the algorithm if the solution is satisfactory. The algorithm asymptotically converges to the optimal solution of the unrestricted problem if the faces are subdivided into smaller part
Searching Polyhedra by Rotating Half-Planes
The Searchlight Scheduling Problem was first studied in 2D polygons, where
the goal is for point guards in fixed positions to rotate searchlights to catch
an evasive intruder. Here the problem is extended to 3D polyhedra, with the
guards now boundary segments who rotate half-planes of illumination. After
carefully detailing the 3D model, several results are established. The first is
a nearly direct extension of the planar one-way sweep strategy using what we
call exhaustive guards, a generalization that succeeds despite there being no
well-defined notion in 3D of planar "clockwise rotation". Next follow two
results: every polyhedron with r>0 reflex edges can be searched by at most r^2
suitably placed guards, whereas just r guards suffice if the polyhedron is
orthogonal. (Minimizing the number of guards to search a given polyhedron is
easily seen to be NP-hard.) Finally we show that deciding whether a given set
of guards has a successful search schedule is strongly NP-hard, and that
deciding if a given target area is searchable at all is strongly PSPACE-hard,
even for orthogonal polyhedra. A number of peripheral results are proved en
route to these central theorems, and several open problems remain for future
work.Comment: 45 pages, 26 figure
From isovists to visibility graphs: a methodology for the analysis of architectural space
An isovist, or viewshed, is the area in a spatial environment directly visible from a location within the space. Here we show how a set of isovists can be used to generate a graph of mutual visibility between locations. We demonstrate that this graph can also be constructed without reference to isovists and that we are in fact invoking the more general concept of a visibility graph. Using the visibility graph, we can extend both isovist and current graph-based analyses of architectural space to form a new methodology for the investigation of configurational relationships. The measurement of local and global characteristics of the graph, for each vertex or for the system as a whole, is of interest from an architectural perspective, allowing us to describe a configuration with reference to accessibility and visibility, to compare from location to location within a system, and to compare systems with different geometries. Finally we show that visibility graph properties may be closely related to manifestations of spatial perception, such as way-finding, movement, and space use
A general solution to the hidden-line problem
The requirements for computer-generated perspective projections of three dimensional objects has escalated. A general solution was developed. The theoretical solution to this problem is presented. The method is very efficient as it minimizes the selection of points and comparison of line segments and hence avoids the devastation of square-law growth
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