Abstract
We consider the pair (p i ,f i ) as a force with two-dimensional direction vector f i applied at the point p i in the plane. For a given set of forces we ask for a non-crossing geometric graph on the points p i that has the following property: There exists a weight assignment to the edges of the graph, such that for every p i the sum of the weighted edges (seen as vectors) around p i yields − f i . 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 characterization of pointed pseudo-triangulations, structures that have been proven to be extremely useful in the design and analysis of efficient geometric algorithms.
As an application, we discuss how to compute the maximal locally convex function for a polygon whose corners lie on its convex hull.
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Rote, G., Schulz, A. (2009). Resolving Loads with Positive Interior Stresses. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_46
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DOI: https://doi.org/10.1007/978-3-642-03367-4_46
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