Resolving Loads with Positive Interior Stresses

We consider the pair (pi, fi) as a force with two-dimensional direction vector fi applied at the point pi in the plane. For a given set of forces we ask for a non-crossing geometric graph on the points pi that has the following property: There exists a weight assignment to the edges of the graph, such that for every pi the sum of the weighted edges (seen as vectors) around pi yields −fi. As additional constraint we restrict ourselves to weights that are non-negative on every edge that is not on the convex hull of the point set. We show that (under a generic assumption) for any reasonable set of forces there is exactly one pointed pseudo-triangulation that fulfils the desired properties. Our results will be obtained by linear programming duality over the PPT-polytope. For the case where the forces appear only at convex hull vertices we show that the pseudo-triangulation that resolves the load can be computed as weighted Delaunay triangulation. Our observations lead to a new character...

• Type: Conference
• Year: 2009
• Where: WADS
• Authors: Günter Rote, André Schulz
• Keywords: Algorithms, Convex Hull, Convex Hull Vertices, Pointed Pseudo-triangulations, WADS 2009