Updating the Number of Crossings in Rectilinear Drawings of the Complete Graph

Let $S$ be a set of $n$ points in general position in the plane. Join every pair of points in $S$ with a straight line segment. Let $\overline{cr}(S)$ be number of pairs of these edges that intersect in their interior. Suppose that this number is known. In this paper we consider the problem of computing $\overline{cr}(S^{\mathrm{\prime }})$, where $S^{\mathrm{\prime }}$ comes from adding, deleting or moving a point from $S$.