13 research outputs found
Computational Geometry Column 43
The concept of pointed pseudo-triangulations is defined and a few of its
applications described.Comment: 3 pages, 1 figur
Transforming triangulations on non planar-surfaces
We consider whether any two triangulations of a polygon or a point set on a
non-planar surface with a given metric can be transformed into each other by a
sequence of edge flips. The answer is negative in general with some remarkable
exceptions, such as polygons on the cylinder, and on the flat torus, and
certain configurations of points on the cylinder.Comment: 19 pages, 17 figures. This version has been accepted in the SIAM
Journal on Discrete Mathematics. Keywords: Graph of triangulations,
triangulations on surfaces, triangulations of polygons, edge fli
Transforming triangulations on non planar-surfaces
We consider whether any two triangulations of a polygon or a point set on a non-planar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinde
Transforming Triangulations on Nonplanar Surfaces
We consider whether any two triangulations of a polygon or a point set on a nonplanar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder
Kinetic collision detection between two simple polygons
AbstractWe design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangulation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure when the polygons are moving, and analyze its performance in the kinetic setting
Pseudo-Triangulations, Rigidity and Motion Planning
This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc
Algorithms for Optimizing Search Schedules in a Polygon
In the area of motion planning, considerable work has been done on guarding
problems, where "guards", modelled as points, must guard a polygonal
space from "intruders". Different variants
of this problem involve varying a number of factors. The guards performing
the search may vary in terms of their number, their mobility, and their
range of vision. The model of intruders may or may not allow them to
move. The polygon being searched may have a specified starting point,
a specified ending point, or neither of these. The typical question asked
about one of these problems is whether or not certain polygons can be
searched under a particular guarding paradigm defined by the types
of guards and intruders.
In this thesis, we focus on two cases of a chain of guards searching
a room (polygon with a specific starting point) for mobile intruders.
The intruders must never be allowed to escape through the door undetected.
In the case of the two guard problem, the guards must start at the door
point and move in opposite directions along the boundary of the
polygon, never crossing the door point. At all times, the
guards must be able to see each other. The search is complete once both
guards occupy the same spot elsewhere on the polygon. In the case of
a chain of three guards, consecutive guards in the chain must always
be visible. Again, the search starts at the door point, and the outer
guards of the chain must move from the door in opposite directions.
These outer guards must always remain on the boundary of the polygon.
The search is complete once the chain lies entirely on a portion of
the polygon boundary not containing the door point.
Determining whether a polygon can be searched is a problem in the area
of visibility in polygons; further to that, our work is related
to the area of planning algorithms. We look for ways to find optimal schedules that minimize
the distance or time required to complete the search. This is done
by finding shortest paths in visibility diagrams that indicate valid
positions for the guards. In the case of
the two-guard room search, we are able to find the shortest distance
schedule and the quickest schedule. The shortest distance schedule
is found in O(n^2) time by solving an L_1 shortest path problem
among curved obstacles in two dimensions. The quickest search schedule is
found in O(n^4) time by solving an L_infinity shortest path
problem among curved obstacles in two dimensions.
For the chain of three guards, a search schedule minimizing the total
distance travelled by the outer guards is found in O(n^6) time by
solving an L_1 shortest path problem among curved obstacles in two dimensions