On fat partitioning, fat covering and the union size of polygons

Abstract

The complexity of the contour of the union of simple polygons can be O(n2) in general. In this paper, a necessary and sufficient condition is given for simple polygons which guarantees smaller union complexity. A δ-corridor in a polygon is a passage between two edges with width/length ratio δ. If a set of polygons with n vertices in total has no δ-corridors, then the union size is O((n log log n)/δ), which is close to optimal in the worst case. The result has many applications to basic problems in computational geometry, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a simple polygon P with n vertices into O(n) convex quadrilaterals, without introducing angles smaller than π/12 radians or narrow corridors. Furthermore, a convex quadrilateral can be covered (but not partitioned) with O(1/δ) triangles without introducing small angles. The maximum overlap of the triangles at any point is two. The algorithms take O(n log2n) and O(n log2n+n/δ time for partitioning and covering, respectively.