Parallel-redrawing mechanisms, pseudo-triangulations and kinetic planar graphs

Abstract

We study parallel redrawing graphs: graphs embedded on moving point sets in such a way that edges maintain their slopes all throughout the motion. The configuration space of such a graph is of an oriented-projective nature, and its combinatorial structure relates to rigidity theoretic par rameters of the graph. For an appropriate parametrization the points move with constant speeds on linear trajectories. A special type of kinetic structure emerges, whose events can be analyzed combinatorially. They correspond to collisions of subsets of points, and are in one-toone correspondence with contractions of the underlying graph on rigid components. We show how to process them algorithmically via a parallel redrawing sweep. Of particular interest are those planar graphs which maintain noncrossing edges throughout the motion. Our main result is that they are (essentially) pseudo-triangulation mechanisms: pointed pseudo-triangulations with a convex hull edge removed. These kinetic graph structures have potential applications in morphing of more complex shapes than just simple polygons. © Springer-Verlag Berlin Heidelberg 2005.