Given a finite collection P of points in the plane, three points of P form a Delaunay triangle if their circumscribing circle contains no further points of P. These triangles form the famous Delaunay triangulation which, along with its dual Voronoi diagram, is central to Combinatorial and Computational Geometry.

Despite several decades of intensive study, our understanding of the combinatorial behaviour of Voronoi and Delaunay structures is still far from satisfactory. For example, if the points of P are moving along lines at some uniform speed, it has been long conjectured that the Delaunay triangulation experiences at most near-quadratically many discrete changes during the motion.

Our recent study affirms the above conjecture. This is done in a far more general, purely topological setting, which covers uniform-speed linear motions.
We discuss this work in the view of some other fundamental results on the combinatorial complexity of geometric structures.
No prior knowledge of combinatorial geometry is assumed.