Polygon Cutting: Revisited

Abstract

Given a simple polygon P on n vertices v ₀,v ₁,...,v _n –1 with each edge assigned a non-negative weight w _i , we show how to compute in O(n) time a segment v _i b (where b is a point on the boundary) that partitions P into two polygons each having weight at most 2/3 the weight of P. If instead of edge weights, we consider an infinitely additive measure of the interior (such as the area of P), there still exists a segment v _i b that partitions P into two polygons each having measure at most 2/3 the measure of P. In the case where P contains k points in its interior with each point assigned a non-negative weight, then in O(n+k log n) time a segment v _i b can be computed that partitions P into two polygons having weight at most 2/3 the weight of P. In the case of rectilinear polygons, rectilinear cuts having the above properties exist, however, the fraction is 3/4 instead of 2/3. We present examples showing that these bounds are tight in the worst case.

We show that in O(n) time using O(log n) cuts, a simple polygon can be partitioned into two groups G ₁ and G ₂ of pieces, such that the ratio of the area of G ₁ to the area of G ₂ is x : y for any x,y > 0, G ₁ can be made a single piece, and all but possibly one of the cuts are diagonals of the polygon.

Finally, we present an O(n) time algorithm for finding the shortest chord in a convex polygon P that cuts off α times the area of P.

Research supported in part by NSERC (Natural Sciences and Engineering Research Council of Canada.