Practical methods for set operations on polygons using exact arithmetic

We present here numerical and combinatorial methods that permit the use of exact arithmetic in the construction of unions and intersection of polygonal regions. An argument is given that, even in an exact arithmetic system , rounding of coordinates is necessary. We also argue that it is natural and useful to round to a nonuniform grid, and we give methods for calculating the nearest grid point. The main result is a shortest path rounding algorithm that restores the combinatorial consistency of a polygon after its vertices have been rounded. This algorithm runs in linear time in the number of \near" vertex-edge pairs. It is optimal in the sense that it introduces the minimum combinatorial and geometric changes. We know of no other bounded-error rounding algorithm for nonuniform grids.