3,979 research outputs found
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
Lower bounds on the obstacle number of graphs
Given a graph , an {\em obstacle representation} of is a set of points
in the plane representing the vertices of , together with a set of connected
obstacles such that two vertices of are joined by an edge if and only if
the corresponding points can be connected by a segment which avoids all
obstacles. The {\em obstacle number} of is the minimum number of obstacles
in an obstacle representation of . It is shown that there are graphs on
vertices with obstacle number at least
Two-Dimensional Pursuit-Evasion in a Compact Domain with Piecewise Analytic Boundary
In a pursuit-evasion game, a team of pursuers attempt to capture an evader.
The players alternate turns, move with equal speed, and have full information
about the state of the game. We consider the most restictive capture condition:
a pursuer must become colocated with the evader to win the game. We prove two
general results about pursuit-evasion games in topological spaces. First, we
show that one pursuer has a winning strategy in any CAT(0) space under this
restrictive capture criterion. This complements a result of Alexander, Bishop
and Ghrist, who provide a winning strategy for a game with positive capture
radius. Second, we consider the game played in a compact domain in Euclidean
two-space with piecewise analytic boundary and arbitrary Euler characteristic.
We show that three pursuers always have a winning strategy by extending recent
work of Bhadauria, Klein, Isler and Suri from polygonal environments to our
more general setting.Comment: 21 pages, 6 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
Edge Routing with Ordered Bundles
Edge bundling reduces the visual clutter in a drawing of a graph by uniting
the edges into bundles. We propose a method of edge bundling drawing each edge
of a bundle separately as in metro-maps and call our method ordered bundles. To
produce aesthetically looking edge routes it minimizes a cost function on the
edges. The cost function depends on the ink, required to draw the edges, the
edge lengths, widths and separations. The cost also penalizes for too many
edges passing through narrow channels by using the constrained Delaunay
triangulation. The method avoids unnecessary edge-node and edge-edge crossings.
To draw edges with the minimal number of crossings and separately within the
same bundle we develop an efficient algorithm solving a variant of the
metro-line crossing minimization problem. In general, the method creates clear
and smooth edge routes giving an overview of the global graph structure, while
still drawing each edge separately and thus enabling local analysis
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