Kinetic mesh refinement in 2D

We provide a kinetic data structure (KDS) to the planar kinetic mesh refinement problem, which concerns computation of meshes of continuously moving points. Our KDS computes the Delaunay triangulation of a size-optimal well-spaced superset of a set of moving points with algebraic trajectories of constant degree. Our KDS is compact, requiring linear space in the size of the output. It is local, using a point in O(log Delta) certificates. It is responsive, repairing itself in O(log Delta) time per event. It is efficient, processing O(n2 log3 Delta) events in the worst case; this is optimal up to a polylogarithmic factor. Also, our KDS is dynamic, responding to point insertions and deletions in O(log Delta) time. In our bounds Delta stands for the geometric spread, the ratio of the diameter to the closest pair distance. To the best of our knowledge, this is the first KDS for mesh refinement.