Voronoi Diagrams and Morse Theory of the Distance Function

We consider the (minimal) distance function of a point in the plane to a set P of N points in the plane. The locus of non-di erentiability of this distance function consists (besides of the points of P) exactly of the Voronoi diagram of P. We show that the number of minima (m), maxima (M) and `saddle points' (s) of the distance function satisfy: m s +M = 1 This is similar to the Morse type of statements for di erentiable functions. The saddle points occur exactly where a Delaunay edge cuts the corresponding Voronoi edge in its interior. The set of those edges form a subgraph of the Delaunay graph, which connects all minima and saddle points. This graph devides the plane into regions. In each of the compact regions, there is exactly one maximum, the non compact regions don't contain a local maximum. At the end we classify all those graphs if P contains of 3 or 4 points. Introduction Given a set of N di erent points P = fP1; ; PNg in the plane A 2 we consider the Voronoi diagram of the Euclidean distance function d: VD(P) = fX 2 A 2 j 9i 6= j such that 8k d(X;Pi) = d(X;Pj) d(X;Pk)g: The (closed) Voronoi cells are de ned by: VC(Pi) = fX 2 A 2 j 8k d(X;Pi) d(X;Pk): For the theory of Voronoi diagrams we refer to Aurenhammer [Au], Edelsbrunner [Ed] and the book of Okabe-Books-Sigihara [OBS]. Voronoi diagrams have many applications in mathematics and computer science, but also in geography, biology, cristallography, marketing, cartography, etc. Consider the N distance functions dk(X) = d(X;Pk) 1 A natural function to study is d(X) = minfd1(X); ; dN(X)g. In order to have di erentiability in the poins of P and to have a nicer formula for the gradient, we study D(X) = minfd21(X); ; d 2 N(X)g; which behaves the same, e.g. D and d have the same set of level curves, minima, maxima and `saddle points'. Remark that D(X) = d2i (X) on VC(Pi). The function D is di erentiable on the interior of all the Voronoi cells. The restriction of D to a closed Voronoi cell VC(Pi) is di erentiable and on this set grad D(X) = grad d2i (X) = 2XPi It follows that the set of points where D is not di erentiable is exactly VD(P). The level curves of the distance function d can be considered as wave fronts, which start from the points of P . These wave fronts fd = g bound sets fd g, where the wave front already passed, just as an region passed by a forest re. The change of topology of these regions fd g is studied in this paper. We rst consider an instructive example with 3 points, where two di erent positions of the points of P give rise to di erent topological behaviour. An indicator for toplogical changes is the Euler characteristic . At the beginning the wave fronts surround three di erent regions. So the Euler characteristic = 3. We'll report about the changes in . Next two regions meet in a common point and we get two contractible regions, so = 2. After that the third region meets the other (combined) region, this gives = 1. Figure 1: evolution of a wave front from three points, case A In case A ( gure 1), where one of the angles is obtuse, this region becomes bigger and bigger and does not change anymore. In case B ( gure 2) all the angles are sharp and now the wave fronts meet another time and enclose a region in the middle. The set fd g is `circular' and no longer contractible. Now = 0. 2 Figure 2: evolution of a wave front from three points, case B If one goes on then the enclosed region in the middle disappears and this changes to 1. There is only one region left, which is contractible and there are no changes if increases more. We intend to study this type of process in the paper. The special points, where the wave fronts meet and the points, where they become non-di erentiable, are directly related to the Voronoi diagram. As we see in the above example we need more re ned information than the Voronoi diagram in order to understand the topological behaviour of the wave fronts. Behaviour of D on the plane A . The behaviour of D on the interiors of the Voronoi cells is clear. In the points of P the function D has its minimal value and there are no other critical point in the interior of the Voronoi cells. Next we study the neighborhoods of the points A on the Voronoi diagram. Let eij = VC(Pi) \VC(Pj) be a Voronoi edge between the Voronoi cells of Pi and Pj . Let mij be the perpendicular bisector of PiPj and Qij = mij \ PiPj . The edge eij mij . The position of Qij with respect to eij is important. There are three cases (cf gure 3): 1. Qij lies outside eij , 2. Qij lies in the interior of eij , 3. Qij is a boundary point of eij . In cases (1) and (3) D is monotone on the edge; in case (2) D is not monotone, but increasing from Qij in both directions.