Parallel-Redrawing Mechanisms, Pseudo-Triangulations and Kinetic Planar Graphs

摘要

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 parameters 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-to-one 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 non-crossing edges throughout the motion. Our main result is that they are (essentially) pseudo-triangulation mechanisms: pointed pseudo-trian-gulations with a convex hull edge removed. These kinetic graph structures have potential applications in morphing of more complex shapes than just simple polygons.