Computational geometry: algorithms and applications (2nd edn.), by M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf. Pp. 367. £20.50. 2000. ISBN 3 540 65620 0 (Springer-Verlag).

Computational geometry: algorithms and applications (2nd edn.), by M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf. Pp. 367. £20.50. 2000. ISBN 3 540 65620 0 (Springer-Verlag). Suppose we are given spot heights for a piece of terrain at a random and not too sparse collection of points—just think of a surface z = f(x, y) where the values of z are given for a finite set of points (x„ v,) in general position. How can we get a reasonably good idea of what the terrain looks like? Over each (JC„ y,) there is a pole (in the telegraph sense) of height / (x„ y,), but what about other points? One natural idea is to triangulate the base plane, using the set P of points (x„ y,) as vertices, and draw the corresponding triangles in 3-space with vertices at the tops of the poles. But how best to triangulate? It is easy to produce situations where a bad choice of triangles results in an unexpected valley or ridge in the landscape. It turns out that a good choice is a triangulation which maximises the minimum angle of any triangle. That is, long thin triangles are a bad thing. What kind of a triangulation is that? It turns out to be something called a Delaunay triangulation, named, I was enlightened to read, after Boris Nikolaevich Delone and spelt in the French manner to make him sound decidedly French instead of Russian. The Delaunay triangulation can also be characterised by requiring that no point of the set P lies inside the circumcircle of any of the triangles. The Delaunay triangulation is dual to the Voronoi diagram of P. This is the subdivision of the plane into regions, each containing one point p of P, where the interior of p's region contains the points of the plane closer to p than to any other point of P. (And Voronoi even sounds Russian.) Voronoi diagrams are extensively used in shape analysis and pattern recognition.